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llmeval-env/lib/python3.10/site-packages/sympy/functions/combinatorial/__init__.py

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+# Stub __init__.py for sympy.functions.combinatorial

llmeval-env/lib/python3.10/site-packages/sympy/functions/combinatorial/factorials.py

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+from __future__ import annotations
+from functools import reduce
+
+from sympy.core import S, sympify, Dummy, Mod
+from sympy.core.cache import cacheit
+from sympy.core.function import Function, ArgumentIndexError, PoleError
+from sympy.core.logic import fuzzy_and
+from sympy.core.numbers import Integer, pi, I
+from sympy.core.relational import Eq
+from sympy.external.gmpy import HAS_GMPY, gmpy
+from sympy.ntheory import sieve
+from sympy.polys.polytools import Poly
+
+from math import factorial as _factorial, prod, sqrt as _sqrt
+
+class CombinatorialFunction(Function):
+    """Base class for combinatorial functions. """
+
+    def _eval_simplify(self, **kwargs):
+        from sympy.simplify.combsimp import combsimp
+        # combinatorial function with non-integer arguments is
+        # automatically passed to gammasimp
+        expr = combsimp(self)
+        measure = kwargs['measure']
+        if measure(expr) <= kwargs['ratio']*measure(self):
+            return expr
+        return self
+
+
+###############################################################################
+######################## FACTORIAL and MULTI-FACTORIAL ########################
+###############################################################################
+
+
+class factorial(CombinatorialFunction):
+    r"""Implementation of factorial function over nonnegative integers.
+       By convention (consistent with the gamma function and the binomial
+       coefficients), factorial of a negative integer is complex infinity.
+
+       The factorial is very important in combinatorics where it gives
+       the number of ways in which `n` objects can be permuted. It also
+       arises in calculus, probability, number theory, etc.
+
+       There is strict relation of factorial with gamma function. In
+       fact `n! = gamma(n+1)` for nonnegative integers. Rewrite of this
+       kind is very useful in case of combinatorial simplification.
+
+       Computation of the factorial is done using two algorithms. For
+       small arguments a precomputed look up table is used. However for bigger
+       input algorithm Prime-Swing is used. It is the fastest algorithm
+       known and computes `n!` via prime factorization of special class
+       of numbers, called here the 'Swing Numbers'.
+
+       Examples
+       ========
+
+       >>> from sympy import Symbol, factorial, S
+       >>> n = Symbol('n', integer=True)
+
+       >>> factorial(0)
+       1
+
+       >>> factorial(7)
+       5040
+
+       >>> factorial(-2)
+       zoo
+
+       >>> factorial(n)
+       factorial(n)
+
+       >>> factorial(2*n)
+       factorial(2*n)
+
+       >>> factorial(S(1)/2)
+       factorial(1/2)
+
+       See Also
+       ========
+
+       factorial2, RisingFactorial, FallingFactorial
+    """
+
+    def fdiff(self, argindex=1):
+        from sympy.functions.special.gamma_functions import (gamma, polygamma)
+        if argindex == 1:
+            return gamma(self.args[0] + 1)*polygamma(0, self.args[0] + 1)
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    _small_swing = [
+        1, 1, 1, 3, 3, 15, 5, 35, 35, 315, 63, 693, 231, 3003, 429, 6435, 6435, 109395,
+        12155, 230945, 46189, 969969, 88179, 2028117, 676039, 16900975, 1300075,
+        35102025, 5014575, 145422675, 9694845, 300540195, 300540195
+    ]
+
+    _small_factorials: list[int] = []
+
+    @classmethod
+    def _swing(cls, n):
+        if n < 33:
+            return cls._small_swing[n]
+        else:
+            N, primes = int(_sqrt(n)), []
+
+            for prime in sieve.primerange(3, N + 1):
+                p, q = 1, n
+
+                while True:
+                    q //= prime
+
+                    if q > 0:
+                        if q & 1 == 1:
+                            p *= prime
+                    else:
+                        break
+
+                if p > 1:
+                    primes.append(p)
+
+            for prime in sieve.primerange(N + 1, n//3 + 1):
+                if (n // prime) & 1 == 1:
+                    primes.append(prime)
+
+            L_product = prod(sieve.primerange(n//2 + 1, n + 1))
+            R_product = prod(primes)
+
+            return L_product*R_product
+
+    @classmethod
+    def _recursive(cls, n):
+        if n < 2:
+            return 1
+        else:
+            return (cls._recursive(n//2)**2)*cls._swing(n)
+
+    @classmethod
+    def eval(cls, n):
+        n = sympify(n)
+
+        if n.is_Number:
+            if n.is_zero:
+                return S.One
+            elif n is S.Infinity:
+                return S.Infinity
+            elif n.is_Integer:
+                if n.is_negative:
+                    return S.ComplexInfinity
+                else:
+                    n = n.p
+
+                    if n < 20:
+                        if not cls._small_factorials:
+                            result = 1
+                            for i in range(1, 20):
+                                result *= i
+                                cls._small_factorials.append(result)
+                        result = cls._small_factorials[n-1]
+
+                    # GMPY factorial is faster, use it when available
+                    elif HAS_GMPY:
+                        result = gmpy.fac(n)
+
+                    else:
+                        bits = bin(n).count('1')
+                        result = cls._recursive(n)*2**(n - bits)
+
+                    return Integer(result)
+
+    def _facmod(self, n, q):
+        res, N = 1, int(_sqrt(n))
+
+        # Exponent of prime p in n! is e_p(n) = [n/p] + [n/p**2] + ...
+        # for p > sqrt(n), e_p(n) < sqrt(n), the primes with [n/p] = m,
+        # occur consecutively and are grouped together in pw[m] for
+        # simultaneous exponentiation at a later stage
+        pw = [1]*N
+
+        m = 2 # to initialize the if condition below
+        for prime in sieve.primerange(2, n + 1):
+            if m > 1:
+                m, y = 0, n // prime
+                while y:
+                    m += y
+                    y //= prime
+            if m < N:
+                pw[m] = pw[m]*prime % q
+            else:
+                res = res*pow(prime, m, q) % q
+
+        for ex, bs in enumerate(pw):
+            if ex == 0 or bs == 1:
+                continue
+            if bs == 0:
+                return 0
+            res = res*pow(bs, ex, q) % q
+
+        return res
+
+    def _eval_Mod(self, q):
+        n = self.args[0]
+        if n.is_integer and n.is_nonnegative and q.is_integer:
+            aq = abs(q)
+            d = aq - n
+            if d.is_nonpositive:
+                return S.Zero
+            else:
+                isprime = aq.is_prime
+                if d == 1:
+                    # Apply Wilson's theorem (if a natural number n > 1
+                    # is a prime number, then (n-1)! = -1 mod n) and
+                    # its inverse (if n > 4 is a composite number, then
+                    # (n-1)! = 0 mod n)
+                    if isprime:
+                        return -1 % q
+                    elif isprime is False and (aq - 6).is_nonnegative:
+                        return S.Zero
+                elif n.is_Integer and q.is_Integer:
+                    n, d, aq = map(int, (n, d, aq))
+                    if isprime and (d - 1 < n):
+                        fc = self._facmod(d - 1, aq)
+                        fc = pow(fc, aq - 2, aq)
+                        if d%2:
+                            fc = -fc
+                    else:
+                        fc = self._facmod(n, aq)
+
+                    return fc % q
+
+    def _eval_rewrite_as_gamma(self, n, piecewise=True, **kwargs):
+        from sympy.functions.special.gamma_functions import gamma
+        return gamma(n + 1)
+
+    def _eval_rewrite_as_Product(self, n, **kwargs):
+        from sympy.concrete.products import Product
+        if n.is_nonnegative and n.is_integer:
+            i = Dummy('i', integer=True)
+            return Product(i, (i, 1, n))
+
+    def _eval_is_integer(self):
+        if self.args[0].is_integer and self.args[0].is_nonnegative:
+            return True
+
+    def _eval_is_positive(self):
+        if self.args[0].is_integer and self.args[0].is_nonnegative:
+            return True
+
+    def _eval_is_even(self):
+        x = self.args[0]
+        if x.is_integer and x.is_nonnegative:
+            return (x - 2).is_nonnegative
+
+    def _eval_is_composite(self):
+        x = self.args[0]
+        if x.is_integer and x.is_nonnegative:
+            return (x - 3).is_nonnegative
+
+    def _eval_is_real(self):
+        x = self.args[0]
+        if x.is_nonnegative or x.is_noninteger:
+            return True
+
+    def _eval_as_leading_term(self, x, logx=None, cdir=0):
+        arg = self.args[0].as_leading_term(x)
+        arg0 = arg.subs(x, 0)
+        if arg0.is_zero:
+            return S.One
+        elif not arg0.is_infinite:
+            return self.func(arg)
+        raise PoleError("Cannot expand %s around 0" % (self))
+
+class MultiFactorial(CombinatorialFunction):
+    pass
+
+
+class subfactorial(CombinatorialFunction):
+    r"""The subfactorial counts the derangements of $n$ items and is
+    defined for non-negative integers as:
+
+    .. math:: !n = \begin{cases} 1 & n = 0 \\ 0 & n = 1 \\
+                    (n-1)(!(n-1) + !(n-2)) & n > 1 \end{cases}
+
+    It can also be written as ``int(round(n!/exp(1)))`` but the
+    recursive definition with caching is implemented for this function.
+
+    An interesting analytic expression is the following [2]_
+
+    .. math:: !x = \Gamma(x + 1, -1)/e
+
+    which is valid for non-negative integers `x`. The above formula
+    is not very useful in case of non-integers. `\Gamma(x + 1, -1)` is
+    single-valued only for integral arguments `x`, elsewhere on the positive
+    real axis it has an infinite number of branches none of which are real.
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Subfactorial
+    .. [2] https://mathworld.wolfram.com/Subfactorial.html
+
+    Examples
+    ========
+
+    >>> from sympy import subfactorial
+    >>> from sympy.abc import n
+    >>> subfactorial(n + 1)
+    subfactorial(n + 1)
+    >>> subfactorial(5)
+    44
+
+    See Also
+    ========
+
+    factorial, uppergamma,
+    sympy.utilities.iterables.generate_derangements
+    """
+
+    @classmethod
+    @cacheit
+    def _eval(self, n):
+        if not n:
+            return S.One
+        elif n == 1:
+            return S.Zero
+        else:
+            z1, z2 = 1, 0
+            for i in range(2, n + 1):
+                z1, z2 = z2, (i - 1)*(z2 + z1)
+            return z2
+
+    @classmethod
+    def eval(cls, arg):
+        if arg.is_Number:
+            if arg.is_Integer and arg.is_nonnegative:
+                return cls._eval(arg)
+            elif arg is S.NaN:
+                return S.NaN
+            elif arg is S.Infinity:
+                return S.Infinity
+
+    def _eval_is_even(self):
+        if self.args[0].is_odd and self.args[0].is_nonnegative:
+            return True
+
+    def _eval_is_integer(self):
+        if self.args[0].is_integer and self.args[0].is_nonnegative:
+            return True
+
+    def _eval_rewrite_as_factorial(self, arg, **kwargs):
+        from sympy.concrete.summations import summation
+        i = Dummy('i')
+        f = S.NegativeOne**i / factorial(i)
+        return factorial(arg) * summation(f, (i, 0, arg))
+
+    def _eval_rewrite_as_gamma(self, arg, piecewise=True, **kwargs):
+        from sympy.functions.elementary.exponential import exp
+        from sympy.functions.special.gamma_functions import (gamma, lowergamma)
+        return (S.NegativeOne**(arg + 1)*exp(-I*pi*arg)*lowergamma(arg + 1, -1)
+                + gamma(arg + 1))*exp(-1)
+
+    def _eval_rewrite_as_uppergamma(self, arg, **kwargs):
+        from sympy.functions.special.gamma_functions import uppergamma
+        return uppergamma(arg + 1, -1)/S.Exp1
+
+    def _eval_is_nonnegative(self):
+        if self.args[0].is_integer and self.args[0].is_nonnegative:
+            return True
+
+    def _eval_is_odd(self):
+        if self.args[0].is_even and self.args[0].is_nonnegative:
+            return True
+
+
+class factorial2(CombinatorialFunction):
+    r"""The double factorial `n!!`, not to be confused with `(n!)!`
+
+    The double factorial is defined for nonnegative integers and for odd
+    negative integers as:
+
+    .. math:: n!! = \begin{cases} 1 & n = 0 \\
+                    n(n-2)(n-4) \cdots 1 & n\ \text{positive odd} \\
+                    n(n-2)(n-4) \cdots 2 & n\ \text{positive even} \\
+                    (n+2)!!/(n+2) & n\ \text{negative odd} \end{cases}
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Double_factorial
+
+    Examples
+    ========
+
+    >>> from sympy import factorial2, var
+    >>> n = var('n')
+    >>> n
+    n
+    >>> factorial2(n + 1)
+    factorial2(n + 1)
+    >>> factorial2(5)
+    15
+    >>> factorial2(-1)
+    1
+    >>> factorial2(-5)
+    1/3
+
+    See Also
+    ========
+
+    factorial, RisingFactorial, FallingFactorial
+    """
+
+    @classmethod
+    def eval(cls, arg):
+        # TODO: extend this to complex numbers?
+
+        if arg.is_Number:
+            if not arg.is_Integer:
+                raise ValueError("argument must be nonnegative integer "
+                                    "or negative odd integer")
+
+            # This implementation is faster than the recursive one
+            # It also avoids "maximum recursion depth exceeded" runtime error
+            if arg.is_nonnegative:
+                if arg.is_even:
+                    k = arg / 2
+                    return 2**k * factorial(k)
+                return factorial(arg) / factorial2(arg - 1)
+
+
+            if arg.is_odd:
+                return arg*(S.NegativeOne)**((1 - arg)/2) / factorial2(-arg)
+            raise ValueError("argument must be nonnegative integer "
+                                "or negative odd integer")
+
+
+    def _eval_is_even(self):
+        # Double factorial is even for every positive even input
+        n = self.args[0]
+        if n.is_integer:
+            if n.is_odd:
+                return False
+            if n.is_even:
+                if n.is_positive:
+                    return True
+                if n.is_zero:
+                    return False
+
+    def _eval_is_integer(self):
+        # Double factorial is an integer for every nonnegative input, and for
+        # -1 and -3
+        n = self.args[0]
+        if n.is_integer:
+            if (n + 1).is_nonnegative:
+                return True
+            if n.is_odd:
+                return (n + 3).is_nonnegative
+
+    def _eval_is_odd(self):
+        # Double factorial is odd for every odd input not smaller than -3, and
+        # for 0
+        n = self.args[0]
+        if n.is_odd:
+            return (n + 3).is_nonnegative
+        if n.is_even:
+            if n.is_positive:
+                return False
+            if n.is_zero:
+                return True
+
+    def _eval_is_positive(self):
+        # Double factorial is positive for every nonnegative input, and for
+        # every odd negative input which is of the form -1-4k for an
+        # nonnegative integer k
+        n = self.args[0]
+        if n.is_integer:
+            if (n + 1).is_nonnegative:
+                return True
+            if n.is_odd:
+                return ((n + 1) / 2).is_even
+
+    def _eval_rewrite_as_gamma(self, n, piecewise=True, **kwargs):
+        from sympy.functions.elementary.miscellaneous import sqrt
+        from sympy.functions.elementary.piecewise import Piecewise
+        from sympy.functions.special.gamma_functions import gamma
+        return 2**(n/2)*gamma(n/2 + 1) * Piecewise((1, Eq(Mod(n, 2), 0)),
+                (sqrt(2/pi), Eq(Mod(n, 2), 1)))
+
+
+###############################################################################
+######################## RISING and FALLING FACTORIALS ########################
+###############################################################################
+
+
+class RisingFactorial(CombinatorialFunction):
+    r"""
+    Rising factorial (also called Pochhammer symbol [1]_) is a double valued
+    function arising in concrete mathematics, hypergeometric functions
+    and series expansions. It is defined by:
+
+    .. math:: \texttt{rf(y, k)} = (x)^k = x \cdot (x+1) \cdots (x+k-1)
+
+    where `x` can be arbitrary expression and `k` is an integer. For
+    more information check "Concrete mathematics" by Graham, pp. 66
+    or visit https://mathworld.wolfram.com/RisingFactorial.html page.
+
+    When `x` is a `~.Poly` instance of degree $\ge 1$ with a single variable,
+    `(x)^k = x(y) \cdot x(y+1) \cdots x(y+k-1)`, where `y` is the
+    variable of `x`. This is as described in [2]_.
+
+    Examples
+    ========
+
+    >>> from sympy import rf, Poly
+    >>> from sympy.abc import x
+    >>> rf(x, 0)
+    1
+    >>> rf(1, 5)
+    120
+    >>> rf(x, 5) == x*(1 + x)*(2 + x)*(3 + x)*(4 + x)
+    True
+    >>> rf(Poly(x**3, x), 2)
+    Poly(x**6 + 3*x**5 + 3*x**4 + x**3, x, domain='ZZ')
+
+    Rewriting is complicated unless the relationship between
+    the arguments is known, but rising factorial can
+    be rewritten in terms of gamma, factorial, binomial,
+    and falling factorial.
+
+    >>> from sympy import Symbol, factorial, ff, binomial, gamma
+    >>> n = Symbol('n', integer=True, positive=True)
+    >>> R = rf(n, n + 2)
+    >>> for i in (rf, ff, factorial, binomial, gamma):
+    ...  R.rewrite(i)
+    ...
+    RisingFactorial(n, n + 2)
+    FallingFactorial(2*n + 1, n + 2)
+    factorial(2*n + 1)/factorial(n - 1)
+    binomial(2*n + 1, n + 2)*factorial(n + 2)
+    gamma(2*n + 2)/gamma(n)
+
+    See Also
+    ========
+
+    factorial, factorial2, FallingFactorial
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Pochhammer_symbol
+    .. [2] Peter Paule, "Greatest Factorial Factorization and Symbolic
+           Summation", Journal of Symbolic Computation, vol. 20, pp. 235-268,
+           1995.
+
+    """
+
+    @classmethod
+    def eval(cls, x, k):
+        x = sympify(x)
+        k = sympify(k)
+
+        if x is S.NaN or k is S.NaN:
+            return S.NaN
+        elif x is S.One:
+            return factorial(k)
+        elif k.is_Integer:
+            if k.is_zero:
+                return S.One
+            else:
+                if k.is_positive:
+                    if x is S.Infinity:
+                        return S.Infinity
+                    elif x is S.NegativeInfinity:
+                        if k.is_odd:
+                            return S.NegativeInfinity
+                        else:
+                            return S.Infinity
+                    else:
+                        if isinstance(x, Poly):
+                            gens = x.gens
+                            if len(gens)!= 1:
+                                raise ValueError("rf only defined for "
+                                            "polynomials on one generator")
+                            else:
+                                return reduce(lambda r, i:
+                                              r*(x.shift(i)),
+                                              range(int(k)), 1)
+                        else:
+                            return reduce(lambda r, i: r*(x + i),
+                                          range(int(k)), 1)
+
+                else:
+                    if x is S.Infinity:
+                        return S.Infinity
+                    elif x is S.NegativeInfinity:
+                        return S.Infinity
+                    else:
+                        if isinstance(x, Poly):
+                            gens = x.gens
+                            if len(gens)!= 1:
+                                raise ValueError("rf only defined for "
+                                            "polynomials on one generator")
+                            else:
+                                return 1/reduce(lambda r, i:
+                                                r*(x.shift(-i)),
+                                                range(1, abs(int(k)) + 1), 1)
+                        else:
+                            return 1/reduce(lambda r, i:
+                                            r*(x - i),
+                                            range(1, abs(int(k)) + 1), 1)
+
+        if k.is_integer == False:
+            if x.is_integer and x.is_negative:
+                return S.Zero
+
+    def _eval_rewrite_as_gamma(self, x, k, piecewise=True, **kwargs):
+        from sympy.functions.elementary.piecewise import Piecewise
+        from sympy.functions.special.gamma_functions import gamma
+        if not piecewise:
+            if (x <= 0) == True:
+                return S.NegativeOne**k*gamma(1 - x) / gamma(-k - x + 1)
+            return gamma(x + k) / gamma(x)
+        return Piecewise(
+            (gamma(x + k) / gamma(x), x > 0),
+            (S.NegativeOne**k*gamma(1 - x) / gamma(-k - x + 1), True))
+
+    def _eval_rewrite_as_FallingFactorial(self, x, k, **kwargs):
+        return FallingFactorial(x + k - 1, k)
+
+    def _eval_rewrite_as_factorial(self, x, k, **kwargs):
+        from sympy.functions.elementary.piecewise import Piecewise
+        if x.is_integer and k.is_integer:
+            return Piecewise(
+                (factorial(k + x - 1)/factorial(x - 1), x > 0),
+                (S.NegativeOne**k*factorial(-x)/factorial(-k - x), True))
+
+    def _eval_rewrite_as_binomial(self, x, k, **kwargs):
+        if k.is_integer:
+            return factorial(k) * binomial(x + k - 1, k)
+
+    def _eval_rewrite_as_tractable(self, x, k, limitvar=None, **kwargs):
+        from sympy.functions.special.gamma_functions import gamma
+        if limitvar:
+            k_lim = k.subs(limitvar, S.Infinity)
+            if k_lim is S.Infinity:
+                return (gamma(x + k).rewrite('tractable', deep=True) / gamma(x))
+            elif k_lim is S.NegativeInfinity:
+                return (S.NegativeOne**k*gamma(1 - x) / gamma(-k - x + 1).rewrite('tractable', deep=True))
+        return self.rewrite(gamma).rewrite('tractable', deep=True)
+
+    def _eval_is_integer(self):
+        return fuzzy_and((self.args[0].is_integer, self.args[1].is_integer,
+                          self.args[1].is_nonnegative))
+
+
+class FallingFactorial(CombinatorialFunction):
+    r"""
+    Falling factorial (related to rising factorial) is a double valued
+    function arising in concrete mathematics, hypergeometric functions
+    and series expansions. It is defined by
+
+    .. math:: \texttt{ff(x, k)} = (x)_k = x \cdot (x-1) \cdots (x-k+1)
+
+    where `x` can be arbitrary expression and `k` is an integer. For
+    more information check "Concrete mathematics" by Graham, pp. 66
+    or [1]_.
+
+    When `x` is a `~.Poly` instance of degree $\ge 1$ with single variable,
+    `(x)_k = x(y) \cdot x(y-1) \cdots x(y-k+1)`, where `y` is the
+    variable of `x`. This is as described in
+
+    >>> from sympy import ff, Poly, Symbol
+    >>> from sympy.abc import x
+    >>> n = Symbol('n', integer=True)
+
+    >>> ff(x, 0)
+    1
+    >>> ff(5, 5)
+    120
+    >>> ff(x, 5) == x*(x - 1)*(x - 2)*(x - 3)*(x - 4)
+    True
+    >>> ff(Poly(x**2, x), 2)
+    Poly(x**4 - 2*x**3 + x**2, x, domain='ZZ')
+    >>> ff(n, n)
+    factorial(n)
+
+    Rewriting is complicated unless the relationship between
+    the arguments is known, but falling factorial can
+    be rewritten in terms of gamma, factorial and binomial
+    and rising factorial.
+
+    >>> from sympy import factorial, rf, gamma, binomial, Symbol
+    >>> n = Symbol('n', integer=True, positive=True)
+    >>> F = ff(n, n - 2)
+    >>> for i in (rf, ff, factorial, binomial, gamma):
+    ...  F.rewrite(i)
+    ...
+    RisingFactorial(3, n - 2)
+    FallingFactorial(n, n - 2)
+    factorial(n)/2
+    binomial(n, n - 2)*factorial(n - 2)
+    gamma(n + 1)/2
+
+    See Also
+    ========
+
+    factorial, factorial2, RisingFactorial
+
+    References
+    ==========
+
+    .. [1] https://mathworld.wolfram.com/FallingFactorial.html
+    .. [2] Peter Paule, "Greatest Factorial Factorization and Symbolic
+           Summation", Journal of Symbolic Computation, vol. 20, pp. 235-268,
+           1995.
+
+    """
+
+    @classmethod
+    def eval(cls, x, k):
+        x = sympify(x)
+        k = sympify(k)
+
+        if x is S.NaN or k is S.NaN:
+            return S.NaN
+        elif k.is_integer and x == k:
+            return factorial(x)
+        elif k.is_Integer:
+            if k.is_zero:
+                return S.One
+            else:
+                if k.is_positive:
+                    if x is S.Infinity:
+                        return S.Infinity
+                    elif x is S.NegativeInfinity:
+                        if k.is_odd:
+                            return S.NegativeInfinity
+                        else:
+                            return S.Infinity
+                    else:
+                        if isinstance(x, Poly):
+                            gens = x.gens
+                            if len(gens)!= 1:
+                                raise ValueError("ff only defined for "
+                                            "polynomials on one generator")
+                            else:
+                                return reduce(lambda r, i:
+                                              r*(x.shift(-i)),
+                                              range(int(k)), 1)
+                        else:
+                            return reduce(lambda r, i: r*(x - i),
+                                          range(int(k)), 1)
+                else:
+                    if x is S.Infinity:
+                        return S.Infinity
+                    elif x is S.NegativeInfinity:
+                        return S.Infinity
+                    else:
+                        if isinstance(x, Poly):
+                            gens = x.gens
+                            if len(gens)!= 1:
+                                raise ValueError("rf only defined for "
+                                            "polynomials on one generator")
+                            else:
+                                return 1/reduce(lambda r, i:
+                                                r*(x.shift(i)),
+                                                range(1, abs(int(k)) + 1), 1)
+                        else:
+                            return 1/reduce(lambda r, i: r*(x + i),
+                                            range(1, abs(int(k)) + 1), 1)
+
+    def _eval_rewrite_as_gamma(self, x, k, piecewise=True, **kwargs):
+        from sympy.functions.elementary.piecewise import Piecewise
+        from sympy.functions.special.gamma_functions import gamma
+        if not piecewise:
+            if (x < 0) == True:
+                return S.NegativeOne**k*gamma(k - x) / gamma(-x)
+            return gamma(x + 1) / gamma(x - k + 1)
+        return Piecewise(
+            (gamma(x + 1) / gamma(x - k + 1), x >= 0),
+            (S.NegativeOne**k*gamma(k - x) / gamma(-x), True))
+
+    def _eval_rewrite_as_RisingFactorial(self, x, k, **kwargs):
+        return rf(x - k + 1, k)
+
+    def _eval_rewrite_as_binomial(self, x, k, **kwargs):
+        if k.is_integer:
+            return factorial(k) * binomial(x, k)
+
+    def _eval_rewrite_as_factorial(self, x, k, **kwargs):
+        from sympy.functions.elementary.piecewise import Piecewise
+        if x.is_integer and k.is_integer:
+            return Piecewise(
+                (factorial(x)/factorial(-k + x), x >= 0),
+                (S.NegativeOne**k*factorial(k - x - 1)/factorial(-x - 1), True))
+
+    def _eval_rewrite_as_tractable(self, x, k, limitvar=None, **kwargs):
+        from sympy.functions.special.gamma_functions import gamma
+        if limitvar:
+            k_lim = k.subs(limitvar, S.Infinity)
+            if k_lim is S.Infinity:
+                return (S.NegativeOne**k*gamma(k - x).rewrite('tractable', deep=True) / gamma(-x))
+            elif k_lim is S.NegativeInfinity:
+                return (gamma(x + 1) / gamma(x - k + 1).rewrite('tractable', deep=True))
+        return self.rewrite(gamma).rewrite('tractable', deep=True)
+
+    def _eval_is_integer(self):
+        return fuzzy_and((self.args[0].is_integer, self.args[1].is_integer,
+                          self.args[1].is_nonnegative))
+
+
+rf = RisingFactorial
+ff = FallingFactorial
+
+###############################################################################
+########################### BINOMIAL COEFFICIENTS #############################
+###############################################################################
+
+
+class binomial(CombinatorialFunction):
+    r"""Implementation of the binomial coefficient. It can be defined
+    in two ways depending on its desired interpretation:
+
+    .. math:: \binom{n}{k} = \frac{n!}{k!(n-k)!}\ \text{or}\
+                \binom{n}{k} = \frac{(n)_k}{k!}
+
+    First, in a strict combinatorial sense it defines the
+    number of ways we can choose `k` elements from a set of
+    `n` elements. In this case both arguments are nonnegative
+    integers and binomial is computed using an efficient
+    algorithm based on prime factorization.
+
+    The other definition is generalization for arbitrary `n`,
+    however `k` must also be nonnegative. This case is very
+    useful when evaluating summations.
+
+    For the sake of convenience, for negative integer `k` this function
+    will return zero no matter the other argument.
+
+    To expand the binomial when `n` is a symbol, use either
+    ``expand_func()`` or ``expand(func=True)``. The former will keep
+    the polynomial in factored form while the latter will expand the
+    polynomial itself. See examples for details.
+
+    Examples
+    ========
+
+    >>> from sympy import Symbol, Rational, binomial, expand_func
+    >>> n = Symbol('n', integer=True, positive=True)
+
+    >>> binomial(15, 8)
+    6435
+
+    >>> binomial(n, -1)
+    0
+
+    Rows of Pascal's triangle can be generated with the binomial function:
+
+    >>> for N in range(8):
+    ...     print([binomial(N, i) for i in range(N + 1)])
+    ...
+    [1]
+    [1, 1]
+    [1, 2, 1]
+    [1, 3, 3, 1]
+    [1, 4, 6, 4, 1]
+    [1, 5, 10, 10, 5, 1]
+    [1, 6, 15, 20, 15, 6, 1]
+    [1, 7, 21, 35, 35, 21, 7, 1]
+
+    As can a given diagonal, e.g. the 4th diagonal:
+
+    >>> N = -4
+    >>> [binomial(N, i) for i in range(1 - N)]
+    [1, -4, 10, -20, 35]
+
+    >>> binomial(Rational(5, 4), 3)
+    -5/128
+    >>> binomial(Rational(-5, 4), 3)
+    -195/128
+
+    >>> binomial(n, 3)
+    binomial(n, 3)
+
+    >>> binomial(n, 3).expand(func=True)
+    n**3/6 - n**2/2 + n/3
+
+    >>> expand_func(binomial(n, 3))
+    n*(n - 2)*(n - 1)/6
+
+    References
+    ==========
+
+    .. [1] https://www.johndcook.com/blog/binomial_coefficients/
+
+    """
+
+    def fdiff(self, argindex=1):
+        from sympy.functions.special.gamma_functions import polygamma
+        if argindex == 1:
+            # https://functions.wolfram.com/GammaBetaErf/Binomial/20/01/01/
+            n, k = self.args
+            return binomial(n, k)*(polygamma(0, n + 1) - \
+                polygamma(0, n - k + 1))
+        elif argindex == 2:
+            # https://functions.wolfram.com/GammaBetaErf/Binomial/20/01/02/
+            n, k = self.args
+            return binomial(n, k)*(polygamma(0, n - k + 1) - \
+                polygamma(0, k + 1))
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    @classmethod
+    def _eval(self, n, k):
+        # n.is_Number and k.is_Integer and k != 1 and n != k
+
+        if k.is_Integer:
+            if n.is_Integer and n >= 0:
+                n, k = int(n), int(k)
+
+                if k > n:
+                    return S.Zero
+                elif k > n // 2:
+                    k = n - k
+
+                if HAS_GMPY:
+                    return Integer(gmpy.bincoef(n, k))
+
+                d, result = n - k, 1
+                for i in range(1, k + 1):
+                    d += 1
+                    result = result * d // i
+                return Integer(result)
+            else:
+                d, result = n - k, 1
+                for i in range(1, k + 1):
+                    d += 1
+                    result *= d
+                return result / _factorial(k)
+
+    @classmethod
+    def eval(cls, n, k):
+        n, k = map(sympify, (n, k))
+        d = n - k
+        n_nonneg, n_isint = n.is_nonnegative, n.is_integer
+        if k.is_zero or ((n_nonneg or n_isint is False)
+                and d.is_zero):
+            return S.One
+        if (k - 1).is_zero or ((n_nonneg or n_isint is False)
+                and (d - 1).is_zero):
+            return n
+        if k.is_integer:
+            if k.is_negative or (n_nonneg and n_isint and d.is_negative):
+                return S.Zero
+            elif n.is_number:
+                res = cls._eval(n, k)
+                return res.expand(basic=True) if res else res
+        elif n_nonneg is False and n_isint:
+            # a special case when binomial evaluates to complex infinity
+            return S.ComplexInfinity
+        elif k.is_number:
+            from sympy.functions.special.gamma_functions import gamma
+            return gamma(n + 1)/(gamma(k + 1)*gamma(n - k + 1))
+
+    def _eval_Mod(self, q):
+        n, k = self.args
+
+        if any(x.is_integer is False for x in (n, k, q)):
+            raise ValueError("Integers expected for binomial Mod")
+
+        if all(x.is_Integer for x in (n, k, q)):
+            n, k = map(int, (n, k))
+            aq, res = abs(q), 1
+
+            # handle negative integers k or n
+            if k < 0:
+                return S.Zero
+            if n < 0:
+                n = -n + k - 1
+                res = -1 if k%2 else 1
+
+            # non negative integers k and n
+            if k > n:
+                return S.Zero
+
+            isprime = aq.is_prime
+            aq = int(aq)
+            if isprime:
+                if aq < n:
+                    # use Lucas Theorem
+                    N, K = n, k
+                    while N or K:
+                        res = res*binomial(N % aq, K % aq) % aq
+                        N, K = N // aq, K // aq
+
+                else:
+                    # use Factorial Modulo
+                    d = n - k
+                    if k > d:
+                        k, d = d, k
+                    kf = 1
+                    for i in range(2, k + 1):
+                        kf = kf*i % aq
+                    df = kf
+                    for i in range(k + 1, d + 1):
+                        df = df*i % aq
+                    res *= df
+                    for i in range(d + 1, n + 1):
+                        res = res*i % aq
+
+                    res *= pow(kf*df % aq, aq - 2, aq)
+                    res %= aq
+
+            else:
+                # Binomial Factorization is performed by calculating the
+                # exponents of primes <= n in `n! /(k! (n - k)!)`,
+                # for non-negative integers n and k. As the exponent of
+                # prime in n! is e_p(n) = [n/p] + [n/p**2] + ...
+                # the exponent of prime in binomial(n, k) would be
+                # e_p(n) - e_p(k) - e_p(n - k)
+                M = int(_sqrt(n))
+                for prime in sieve.primerange(2, n + 1):
+                    if prime > n - k:
+                        res = res*prime % aq
+                    elif prime > n // 2:
+                        continue
+                    elif prime > M:
+                        if n % prime < k % prime:
+                            res = res*prime % aq
+                    else:
+                        N, K = n, k
+                        exp = a = 0
+
+                        while N > 0:
+                            a = int((N % prime) < (K % prime + a))
+                            N, K = N // prime, K // prime
+                            exp += a
+
+                        if exp > 0:
+                            res *= pow(prime, exp, aq)
+                            res %= aq
+
+            return S(res % q)
+
+    def _eval_expand_func(self, **hints):
+        """
+        Function to expand binomial(n, k) when m is positive integer
+        Also,
+        n is self.args[0] and k is self.args[1] while using binomial(n, k)
+        """
+        n = self.args[0]
+        if n.is_Number:
+            return binomial(*self.args)
+
+        k = self.args[1]
+        if (n-k).is_Integer:
+            k = n - k
+
+        if k.is_Integer:
+            if k.is_zero:
+                return S.One
+            elif k.is_negative:
+                return S.Zero
+            else:
+                n, result = self.args[0], 1
+                for i in range(1, k + 1):
+                    result *= n - k + i
+                return result / _factorial(k)
+        else:
+            return binomial(*self.args)
+
+    def _eval_rewrite_as_factorial(self, n, k, **kwargs):
+        return factorial(n)/(factorial(k)*factorial(n - k))
+
+    def _eval_rewrite_as_gamma(self, n, k, piecewise=True, **kwargs):
+        from sympy.functions.special.gamma_functions import gamma
+        return gamma(n + 1)/(gamma(k + 1)*gamma(n - k + 1))
+
+    def _eval_rewrite_as_tractable(self, n, k, limitvar=None, **kwargs):
+        return self._eval_rewrite_as_gamma(n, k).rewrite('tractable')
+
+    def _eval_rewrite_as_FallingFactorial(self, n, k, **kwargs):
+        if k.is_integer:
+            return ff(n, k) / factorial(k)
+
+    def _eval_is_integer(self):
+        n, k = self.args
+        if n.is_integer and k.is_integer:
+            return True
+        elif k.is_integer is False:
+            return False
+
+    def _eval_is_nonnegative(self):
+        n, k = self.args
+        if n.is_integer and k.is_integer:
+            if n.is_nonnegative or k.is_negative or k.is_even:
+                return True
+            elif k.is_even is False:
+                return  False
+
+    def _eval_as_leading_term(self, x, logx=None, cdir=0):
+        from sympy.functions.special.gamma_functions import gamma
+        return self.rewrite(gamma)._eval_as_leading_term(x, logx=logx, cdir=cdir)

llmeval-env/lib/python3.10/site-packages/sympy/functions/combinatorial/numbers.py

@@ -0,0 +1,2563 @@
+"""
+This module implements some special functions that commonly appear in
+combinatorial contexts (e.g. in power series); in particular,
+sequences of rational numbers such as Bernoulli and Fibonacci numbers.
+
+Factorials, binomial coefficients and related functions are located in
+the separate 'factorials' module.
+"""
+from math import prod
+from collections import defaultdict
+from typing import Tuple as tTuple
+
+from sympy.core import S, Symbol, Add, Dummy
+from sympy.core.cache import cacheit
+from sympy.core.expr import Expr
+from sympy.core.function import ArgumentIndexError, Function, expand_mul
+from sympy.core.logic import fuzzy_not
+from sympy.core.mul import Mul
+from sympy.core.numbers import E, I, pi, oo, Rational, Integer
+from sympy.core.relational import Eq, is_le, is_gt
+from sympy.external.gmpy import SYMPY_INTS
+from sympy.functions.combinatorial.factorials import (binomial,
+    factorial, subfactorial)
+from sympy.functions.elementary.exponential import log
+from sympy.functions.elementary.piecewise import Piecewise
+from sympy.ntheory.primetest import isprime, is_square
+from sympy.polys.appellseqs import bernoulli_poly, euler_poly, genocchi_poly
+from sympy.utilities.enumerative import MultisetPartitionTraverser
+from sympy.utilities.exceptions import sympy_deprecation_warning
+from sympy.utilities.iterables import multiset, multiset_derangements, iterable
+from sympy.utilities.memoization import recurrence_memo
+from sympy.utilities.misc import as_int
+
+from mpmath import mp, workprec
+from mpmath.libmp import ifib as _ifib
+
+
+def _product(a, b):
+    return prod(range(a, b + 1))
+
+
+# Dummy symbol used for computing polynomial sequences
+_sym = Symbol('x')
+
+
+#----------------------------------------------------------------------------#
+#                                                                            #
+#                           Carmichael numbers                               #
+#                                                                            #
+#----------------------------------------------------------------------------#
+
+def _divides(p, n):
+    return n % p == 0
+
+class carmichael(Function):
+    r"""
+    Carmichael Numbers:
+
+    Certain cryptographic algorithms make use of big prime numbers.
+    However, checking whether a big number is prime is not so easy.
+    Randomized prime number checking tests exist that offer a high degree of
+    confidence of accurate determination at low cost, such as the Fermat test.
+
+    Let 'a' be a random number between $2$ and $n - 1$, where $n$ is the
+    number whose primality we are testing. Then, $n$ is probably prime if it
+    satisfies the modular arithmetic congruence relation:
+
+    .. math :: a^{n-1} = 1 \pmod{n}
+
+    (where mod refers to the modulo operation)
+
+    If a number passes the Fermat test several times, then it is prime with a
+    high probability.
+
+    Unfortunately, certain composite numbers (non-primes) still pass the Fermat
+    test with every number smaller than themselves.
+    These numbers are called Carmichael numbers.
+
+    A Carmichael number will pass a Fermat primality test to every base $b$
+    relatively prime to the number, even though it is not actually prime.
+    This makes tests based on Fermat's Little Theorem less effective than
+    strong probable prime tests such as the Baillie-PSW primality test and
+    the Miller-Rabin primality test.
+
+    Examples
+    ========
+
+    >>> from sympy import carmichael
+    >>> carmichael.find_first_n_carmichaels(5)
+    [561, 1105, 1729, 2465, 2821]
+    >>> carmichael.find_carmichael_numbers_in_range(0, 562)
+    [561]
+    >>> carmichael.find_carmichael_numbers_in_range(0,1000)
+    [561]
+    >>> carmichael.find_carmichael_numbers_in_range(0,2000)
+    [561, 1105, 1729]
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Carmichael_number
+    .. [2] https://en.wikipedia.org/wiki/Fermat_primality_test
+    .. [3] https://www.jstor.org/stable/23248683?seq=1#metadata_info_tab_contents
+    """
+
+    @staticmethod
+    def is_perfect_square(n):
+        sympy_deprecation_warning(
+        """
+is_perfect_square is just a wrapper around sympy.ntheory.primetest.is_square
+so use that directly instead.
+        """,
+        deprecated_since_version="1.11",
+        active_deprecations_target='deprecated-carmichael-static-methods',
+        )
+        return is_square(n)
+
+    @staticmethod
+    def divides(p, n):
+        sympy_deprecation_warning(
+        """
+        divides can be replaced by directly testing n % p == 0.
+        """,
+        deprecated_since_version="1.11",
+        active_deprecations_target='deprecated-carmichael-static-methods',
+        )
+        return n % p == 0
+
+    @staticmethod
+    def is_prime(n):
+        sympy_deprecation_warning(
+        """
+is_prime is just a wrapper around sympy.ntheory.primetest.isprime so use that
+directly instead.
+        """,
+        deprecated_since_version="1.11",
+        active_deprecations_target='deprecated-carmichael-static-methods',
+        )
+        return isprime(n)
+
+    @staticmethod
+    def is_carmichael(n):
+        if n >= 0:
+            if (n == 1) or isprime(n) or (n % 2 == 0):
+                return False
+
+            divisors = [1, n]
+
+            # get divisors
+            divisors.extend([i for i in range(3, n // 2 + 1, 2) if n % i == 0])
+
+            for i in divisors:
+                if is_square(i) and i != 1:
+                    return False
+                if isprime(i):
+                    if not _divides(i - 1, n - 1):
+                        return False
+
+            return True
+
+        else:
+            raise ValueError('The provided number must be greater than or equal to 0')
+
+    @staticmethod
+    def find_carmichael_numbers_in_range(x, y):
+        if 0 <= x <= y:
+            if x % 2 == 0:
+                return [i for i in range(x + 1, y, 2) if carmichael.is_carmichael(i)]
+            else:
+                return [i for i in range(x, y, 2) if carmichael.is_carmichael(i)]
+
+        else:
+            raise ValueError('The provided range is not valid. x and y must be non-negative integers and x <= y')
+
+    @staticmethod
+    def find_first_n_carmichaels(n):
+        i = 1
+        carmichaels = []
+
+        while len(carmichaels) < n:
+            if carmichael.is_carmichael(i):
+                carmichaels.append(i)
+            i += 2
+
+        return carmichaels
+
+
+#----------------------------------------------------------------------------#
+#                                                                            #
+#                           Fibonacci numbers                                #
+#                                                                            #
+#----------------------------------------------------------------------------#
+
+
+class fibonacci(Function):
+    r"""
+    Fibonacci numbers / Fibonacci polynomials
+
+    The Fibonacci numbers are the integer sequence defined by the
+    initial terms `F_0 = 0`, `F_1 = 1` and the two-term recurrence
+    relation `F_n = F_{n-1} + F_{n-2}`.  This definition
+    extended to arbitrary real and complex arguments using
+    the formula
+
+    .. math :: F_z = \frac{\phi^z - \cos(\pi z) \phi^{-z}}{\sqrt 5}
+
+    The Fibonacci polynomials are defined by `F_1(x) = 1`,
+    `F_2(x) = x`, and `F_n(x) = x*F_{n-1}(x) + F_{n-2}(x)` for `n > 2`.
+    For all positive integers `n`, `F_n(1) = F_n`.
+
+    * ``fibonacci(n)`` gives the `n^{th}` Fibonacci number, `F_n`
+    * ``fibonacci(n, x)`` gives the `n^{th}` Fibonacci polynomial in `x`, `F_n(x)`
+
+    Examples
+    ========
+
+    >>> from sympy import fibonacci, Symbol
+
+    >>> [fibonacci(x) for x in range(11)]
+    [0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
+    >>> fibonacci(5, Symbol('t'))
+    t**4 + 3*t**2 + 1
+
+    See Also
+    ========
+
+    bell, bernoulli, catalan, euler, harmonic, lucas, genocchi, partition, tribonacci
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Fibonacci_number
+    .. [2] https://mathworld.wolfram.com/FibonacciNumber.html
+
+    """
+
+    @staticmethod
+    def _fib(n):
+        return _ifib(n)
+
+    @staticmethod
+    @recurrence_memo([None, S.One, _sym])
+    def _fibpoly(n, prev):
+        return (prev[-2] + _sym*prev[-1]).expand()
+
+    @classmethod
+    def eval(cls, n, sym=None):
+        if n is S.Infinity:
+            return S.Infinity
+
+        if n.is_Integer:
+            if sym is None:
+                n = int(n)
+                if n < 0:
+                    return S.NegativeOne**(n + 1) * fibonacci(-n)
+                else:
+                    return Integer(cls._fib(n))
+            else:
+                if n < 1:
+                    raise ValueError("Fibonacci polynomials are defined "
+                       "only for positive integer indices.")
+                return cls._fibpoly(n).subs(_sym, sym)
+
+    def _eval_rewrite_as_sqrt(self, n, **kwargs):
+        from sympy.functions.elementary.miscellaneous import sqrt
+        return 2**(-n)*sqrt(5)*((1 + sqrt(5))**n - (-sqrt(5) + 1)**n) / 5
+
+    def _eval_rewrite_as_GoldenRatio(self,n, **kwargs):
+        return (S.GoldenRatio**n - 1/(-S.GoldenRatio)**n)/(2*S.GoldenRatio-1)
+
+
+#----------------------------------------------------------------------------#
+#                                                                            #
+#                               Lucas numbers                                #
+#                                                                            #
+#----------------------------------------------------------------------------#
+
+
+class lucas(Function):
+    """
+    Lucas numbers
+
+    Lucas numbers satisfy a recurrence relation similar to that of
+    the Fibonacci sequence, in which each term is the sum of the
+    preceding two. They are generated by choosing the initial
+    values `L_0 = 2` and `L_1 = 1`.
+
+    * ``lucas(n)`` gives the `n^{th}` Lucas number
+
+    Examples
+    ========
+
+    >>> from sympy import lucas
+
+    >>> [lucas(x) for x in range(11)]
+    [2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123]
+
+    See Also
+    ========
+
+    bell, bernoulli, catalan, euler, fibonacci, harmonic, genocchi, partition, tribonacci
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Lucas_number
+    .. [2] https://mathworld.wolfram.com/LucasNumber.html
+
+    """
+
+    @classmethod
+    def eval(cls, n):
+        if n is S.Infinity:
+            return S.Infinity
+
+        if n.is_Integer:
+            return fibonacci(n + 1) + fibonacci(n - 1)
+
+    def _eval_rewrite_as_sqrt(self, n, **kwargs):
+       from sympy.functions.elementary.miscellaneous import sqrt
+       return 2**(-n)*((1 + sqrt(5))**n + (-sqrt(5) + 1)**n)
+
+
+#----------------------------------------------------------------------------#
+#                                                                            #
+#                             Tribonacci numbers                             #
+#                                                                            #
+#----------------------------------------------------------------------------#
+
+
+class tribonacci(Function):
+    r"""
+    Tribonacci numbers / Tribonacci polynomials
+
+    The Tribonacci numbers are the integer sequence defined by the
+    initial terms `T_0 = 0`, `T_1 = 1`, `T_2 = 1` and the three-term
+    recurrence relation `T_n = T_{n-1} + T_{n-2} + T_{n-3}`.
+
+    The Tribonacci polynomials are defined by `T_0(x) = 0`, `T_1(x) = 1`,
+    `T_2(x) = x^2`, and `T_n(x) = x^2 T_{n-1}(x) + x T_{n-2}(x) + T_{n-3}(x)`
+    for `n > 2`.  For all positive integers `n`, `T_n(1) = T_n`.
+
+    * ``tribonacci(n)`` gives the `n^{th}` Tribonacci number, `T_n`
+    * ``tribonacci(n, x)`` gives the `n^{th}` Tribonacci polynomial in `x`, `T_n(x)`
+
+    Examples
+    ========
+
+    >>> from sympy import tribonacci, Symbol
+
+    >>> [tribonacci(x) for x in range(11)]
+    [0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149]
+    >>> tribonacci(5, Symbol('t'))
+    t**8 + 3*t**5 + 3*t**2
+
+    See Also
+    ========
+
+    bell, bernoulli, catalan, euler, fibonacci, harmonic, lucas, genocchi, partition
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Generalizations_of_Fibonacci_numbers#Tribonacci_numbers
+    .. [2] https://mathworld.wolfram.com/TribonacciNumber.html
+    .. [3] https://oeis.org/A000073
+
+    """
+
+    @staticmethod
+    @recurrence_memo([S.Zero, S.One, S.One])
+    def _trib(n, prev):
+        return (prev[-3] + prev[-2] + prev[-1])
+
+    @staticmethod
+    @recurrence_memo([S.Zero, S.One, _sym**2])
+    def _tribpoly(n, prev):
+        return (prev[-3] + _sym*prev[-2] + _sym**2*prev[-1]).expand()
+
+    @classmethod
+    def eval(cls, n, sym=None):
+        if n is S.Infinity:
+            return S.Infinity
+
+        if n.is_Integer:
+            n = int(n)
+            if n < 0:
+                raise ValueError("Tribonacci polynomials are defined "
+                       "only for non-negative integer indices.")
+            if sym is None:
+                return Integer(cls._trib(n))
+            else:
+                return cls._tribpoly(n).subs(_sym, sym)
+
+    def _eval_rewrite_as_sqrt(self, n, **kwargs):
+        from sympy.functions.elementary.miscellaneous import cbrt, sqrt
+        w = (-1 + S.ImaginaryUnit * sqrt(3)) / 2
+        a = (1 + cbrt(19 + 3*sqrt(33)) + cbrt(19 - 3*sqrt(33))) / 3
+        b = (1 + w*cbrt(19 + 3*sqrt(33)) + w**2*cbrt(19 - 3*sqrt(33))) / 3
+        c = (1 + w**2*cbrt(19 + 3*sqrt(33)) + w*cbrt(19 - 3*sqrt(33))) / 3
+        Tn = (a**(n + 1)/((a - b)*(a - c))
+            + b**(n + 1)/((b - a)*(b - c))
+            + c**(n + 1)/((c - a)*(c - b)))
+        return Tn
+
+    def _eval_rewrite_as_TribonacciConstant(self, n, **kwargs):
+        from sympy.functions.elementary.integers import floor
+        from sympy.functions.elementary.miscellaneous import cbrt, sqrt
+        b = cbrt(586 + 102*sqrt(33))
+        Tn = 3 * b * S.TribonacciConstant**n / (b**2 - 2*b + 4)
+        return floor(Tn + S.Half)
+
+
+#----------------------------------------------------------------------------#
+#                                                                            #
+#                           Bernoulli numbers                                #
+#                                                                            #
+#----------------------------------------------------------------------------#
+
+
+class bernoulli(Function):
+    r"""
+    Bernoulli numbers / Bernoulli polynomials / Bernoulli function
+
+    The Bernoulli numbers are a sequence of rational numbers
+    defined by `B_0 = 1` and the recursive relation (`n > 0`):
+
+    .. math :: n+1 = \sum_{k=0}^n \binom{n+1}{k} B_k
+
+    They are also commonly defined by their exponential generating
+    function, which is `\frac{x}{1 - e^{-x}}`. For odd indices > 1,
+    the Bernoulli numbers are zero.
+
+    The Bernoulli polynomials satisfy the analogous formula:
+
+    .. math :: B_n(x) = \sum_{k=0}^n (-1)^k \binom{n}{k} B_k x^{n-k}
+
+    Bernoulli numbers and Bernoulli polynomials are related as
+    `B_n(1) = B_n`.
+
+    The generalized Bernoulli function `\operatorname{B}(s, a)`
+    is defined for any complex `s` and `a`, except where `a` is a
+    nonpositive integer and `s` is not a nonnegative integer. It is
+    an entire function of `s` for fixed `a`, related to the Hurwitz
+    zeta function by
+
+    .. math:: \operatorname{B}(s, a) = \begin{cases}
+              -s \zeta(1-s, a) & s \ne 0 \\ 1 & s = 0 \end{cases}
+
+    When `s` is a nonnegative integer this function reduces to the
+    Bernoulli polynomials: `\operatorname{B}(n, x) = B_n(x)`. When
+    `a` is omitted it is assumed to be 1, yielding the (ordinary)
+    Bernoulli function which interpolates the Bernoulli numbers and is
+    related to the Riemann zeta function.
+
+    We compute Bernoulli numbers using Ramanujan's formula:
+
+    .. math :: B_n = \frac{A(n) - S(n)}{\binom{n+3}{n}}
+
+    where:
+
+    .. math :: A(n) = \begin{cases} \frac{n+3}{3} &
+        n \equiv 0\ \text{or}\ 2 \pmod{6} \\
+        -\frac{n+3}{6} & n \equiv 4 \pmod{6} \end{cases}
+
+    and:
+
+    .. math :: S(n) = \sum_{k=1}^{[n/6]} \binom{n+3}{n-6k} B_{n-6k}
+
+    This formula is similar to the sum given in the definition, but
+    cuts `\frac{2}{3}` of the terms. For Bernoulli polynomials, we use
+    Appell sequences.
+
+    For `n` a nonnegative integer and `s`, `a`, `x` arbitrary complex numbers,
+
+    * ``bernoulli(n)`` gives the nth Bernoulli number, `B_n`
+    * ``bernoulli(s)`` gives the Bernoulli function `\operatorname{B}(s)`
+    * ``bernoulli(n, x)`` gives the nth Bernoulli polynomial in `x`, `B_n(x)`
+    * ``bernoulli(s, a)`` gives the generalized Bernoulli function
+      `\operatorname{B}(s, a)`
+
+    .. versionchanged:: 1.12
+        ``bernoulli(1)`` gives `+\frac{1}{2}` instead of `-\frac{1}{2}`.
+        This choice of value confers several theoretical advantages [5]_,
+        including the extension to complex parameters described above
+        which this function now implements. The previous behavior, defined
+        only for nonnegative integers `n`, can be obtained with
+        ``(-1)**n*bernoulli(n)``.
+
+    Examples
+    ========
+
+    >>> from sympy import bernoulli
+    >>> from sympy.abc import x
+    >>> [bernoulli(n) for n in range(11)]
+    [1, 1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66]
+    >>> bernoulli(1000001)
+    0
+    >>> bernoulli(3, x)
+    x**3 - 3*x**2/2 + x/2
+
+    See Also
+    ========
+
+    andre, bell, catalan, euler, fibonacci, harmonic, lucas, genocchi,
+    partition, tribonacci, sympy.polys.appellseqs.bernoulli_poly
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Bernoulli_number
+    .. [2] https://en.wikipedia.org/wiki/Bernoulli_polynomial
+    .. [3] https://mathworld.wolfram.com/BernoulliNumber.html
+    .. [4] https://mathworld.wolfram.com/BernoulliPolynomial.html
+    .. [5] Peter Luschny, "The Bernoulli Manifesto",
+           https://luschny.de/math/zeta/The-Bernoulli-Manifesto.html
+    .. [6] Peter Luschny, "An introduction to the Bernoulli function",
+           https://arxiv.org/abs/2009.06743
+
+    """
+
+    args: tTuple[Integer]
+
+    # Calculates B_n for positive even n
+    @staticmethod
+    def _calc_bernoulli(n):
+        s = 0
+        a = int(binomial(n + 3, n - 6))
+        for j in range(1, n//6 + 1):
+            s += a * bernoulli(n - 6*j)
+            # Avoid computing each binomial coefficient from scratch
+            a *= _product(n - 6 - 6*j + 1, n - 6*j)
+            a //= _product(6*j + 4, 6*j + 9)
+        if n % 6 == 4:
+            s = -Rational(n + 3, 6) - s
+        else:
+            s = Rational(n + 3, 3) - s
+        return s / binomial(n + 3, n)
+
+    # We implement a specialized memoization scheme to handle each
+    # case modulo 6 separately
+    _cache = {0: S.One, 2: Rational(1, 6), 4: Rational(-1, 30)}
+    _highest = {0: 0, 2: 2, 4: 4}
+
+    @classmethod
+    def eval(cls, n, x=None):
+        if x is S.One:
+            return cls(n)
+        elif n.is_zero:
+            return S.One
+        elif n.is_integer is False or n.is_nonnegative is False:
+            if x is not None and x.is_Integer and x.is_nonpositive:
+                return S.NaN
+            return
+        # Bernoulli numbers
+        elif x is None:
+            if n is S.One:
+                return S.Half
+            elif n.is_odd and (n-1).is_positive:
+                return S.Zero
+            elif n.is_Number:
+                n = int(n)
+                # Use mpmath for enormous Bernoulli numbers
+                if n > 500:
+                    p, q = mp.bernfrac(n)
+                    return Rational(int(p), int(q))
+                case = n % 6
+                highest_cached = cls._highest[case]
+                if n <= highest_cached:
+                    return cls._cache[n]
+                # To avoid excessive recursion when, say, bernoulli(1000) is
+                # requested, calculate and cache the entire sequence ... B_988,
+                # B_994, B_1000 in increasing order
+                for i in range(highest_cached + 6, n + 6, 6):
+                    b = cls._calc_bernoulli(i)
+                    cls._cache[i] = b
+                    cls._highest[case] = i
+                return b
+        # Bernoulli polynomials
+        elif n.is_Number:
+            return bernoulli_poly(n, x)
+
+    def _eval_rewrite_as_zeta(self, n, x=1, **kwargs):
+        from sympy.functions.special.zeta_functions import zeta
+        return Piecewise((1, Eq(n, 0)), (-n * zeta(1-n, x), True))
+
+    def _eval_evalf(self, prec):
+        if not all(x.is_number for x in self.args):
+            return
+        n = self.args[0]._to_mpmath(prec)
+        x = (self.args[1] if len(self.args) > 1 else S.One)._to_mpmath(prec)
+        with workprec(prec):
+            if n == 0:
+                res = mp.mpf(1)
+            elif n == 1:
+                res = x - mp.mpf(0.5)
+            elif mp.isint(n) and n >= 0:
+                res = mp.bernoulli(n) if x == 1 else mp.bernpoly(n, x)
+            else:
+                res = -n * mp.zeta(1-n, x)
+        return Expr._from_mpmath(res, prec)
+
+
+#----------------------------------------------------------------------------#
+#                                                                            #
+#                                Bell numbers                                #
+#                                                                            #
+#----------------------------------------------------------------------------#
+
+
+class bell(Function):
+    r"""
+    Bell numbers / Bell polynomials
+
+    The Bell numbers satisfy `B_0 = 1` and
+
+    .. math:: B_n = \sum_{k=0}^{n-1} \binom{n-1}{k} B_k.
+
+    They are also given by:
+
+    .. math:: B_n = \frac{1}{e} \sum_{k=0}^{\infty} \frac{k^n}{k!}.
+
+    The Bell polynomials are given by `B_0(x) = 1` and
+
+    .. math:: B_n(x) = x \sum_{k=1}^{n-1} \binom{n-1}{k-1} B_{k-1}(x).
+
+    The second kind of Bell polynomials (are sometimes called "partial" Bell
+    polynomials or incomplete Bell polynomials) are defined as
+
+    .. math:: B_{n,k}(x_1, x_2,\dotsc x_{n-k+1}) =
+            \sum_{j_1+j_2+j_2+\dotsb=k \atop j_1+2j_2+3j_2+\dotsb=n}
+                \frac{n!}{j_1!j_2!\dotsb j_{n-k+1}!}
+                \left(\frac{x_1}{1!} \right)^{j_1}
+                \left(\frac{x_2}{2!} \right)^{j_2} \dotsb
+                \left(\frac{x_{n-k+1}}{(n-k+1)!} \right) ^{j_{n-k+1}}.
+
+    * ``bell(n)`` gives the `n^{th}` Bell number, `B_n`.
+    * ``bell(n, x)`` gives the `n^{th}` Bell polynomial, `B_n(x)`.
+    * ``bell(n, k, (x1, x2, ...))`` gives Bell polynomials of the second kind,
+      `B_{n,k}(x_1, x_2, \dotsc, x_{n-k+1})`.
+
+    Notes
+    =====
+
+    Not to be confused with Bernoulli numbers and Bernoulli polynomials,
+    which use the same notation.
+
+    Examples
+    ========
+
+    >>> from sympy import bell, Symbol, symbols
+
+    >>> [bell(n) for n in range(11)]
+    [1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975]
+    >>> bell(30)
+    846749014511809332450147
+    >>> bell(4, Symbol('t'))
+    t**4 + 6*t**3 + 7*t**2 + t
+    >>> bell(6, 2, symbols('x:6')[1:])
+    6*x1*x5 + 15*x2*x4 + 10*x3**2
+
+    See Also
+    ========
+
+    bernoulli, catalan, euler, fibonacci, harmonic, lucas, genocchi, partition, tribonacci
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Bell_number
+    .. [2] https://mathworld.wolfram.com/BellNumber.html
+    .. [3] https://mathworld.wolfram.com/BellPolynomial.html
+
+    """
+
+    @staticmethod
+    @recurrence_memo([1, 1])
+    def _bell(n, prev):
+        s = 1
+        a = 1
+        for k in range(1, n):
+            a = a * (n - k) // k
+            s += a * prev[k]
+        return s
+
+    @staticmethod
+    @recurrence_memo([S.One, _sym])
+    def _bell_poly(n, prev):
+        s = 1
+        a = 1
+        for k in range(2, n + 1):
+            a = a * (n - k + 1) // (k - 1)
+            s += a * prev[k - 1]
+        return expand_mul(_sym * s)
+
+    @staticmethod
+    def _bell_incomplete_poly(n, k, symbols):
+        r"""
+        The second kind of Bell polynomials (incomplete Bell polynomials).
+
+        Calculated by recurrence formula:
+
+        .. math:: B_{n,k}(x_1, x_2, \dotsc, x_{n-k+1}) =
+                \sum_{m=1}^{n-k+1}
+                \x_m \binom{n-1}{m-1} B_{n-m,k-1}(x_1, x_2, \dotsc, x_{n-m-k})
+
+        where
+            `B_{0,0} = 1;`
+            `B_{n,0} = 0; for n \ge 1`
+            `B_{0,k} = 0; for k \ge 1`
+
+        """
+        if (n == 0) and (k == 0):
+            return S.One
+        elif (n == 0) or (k == 0):
+            return S.Zero
+        s = S.Zero
+        a = S.One
+        for m in range(1, n - k + 2):
+            s += a * bell._bell_incomplete_poly(
+                n - m, k - 1, symbols) * symbols[m - 1]
+            a = a * (n - m) / m
+        return expand_mul(s)
+
+    @classmethod
+    def eval(cls, n, k_sym=None, symbols=None):
+        if n is S.Infinity:
+            if k_sym is None:
+                return S.Infinity
+            else:
+                raise ValueError("Bell polynomial is not defined")
+
+        if n.is_negative or n.is_integer is False:
+            raise ValueError("a non-negative integer expected")
+
+        if n.is_Integer and n.is_nonnegative:
+            if k_sym is None:
+                return Integer(cls._bell(int(n)))
+            elif symbols is None:
+                return cls._bell_poly(int(n)).subs(_sym, k_sym)
+            else:
+                r = cls._bell_incomplete_poly(int(n), int(k_sym), symbols)
+                return r
+
+    def _eval_rewrite_as_Sum(self, n, k_sym=None, symbols=None, **kwargs):
+        from sympy.concrete.summations import Sum
+        if (k_sym is not None) or (symbols is not None):
+            return self
+
+        # Dobinski's formula
+        if not n.is_nonnegative:
+            return self
+        k = Dummy('k', integer=True, nonnegative=True)
+        return 1 / E * Sum(k**n / factorial(k), (k, 0, S.Infinity))
+
+
+#----------------------------------------------------------------------------#
+#                                                                            #
+#                              Harmonic numbers                              #
+#                                                                            #
+#----------------------------------------------------------------------------#
+
+
+class harmonic(Function):
+    r"""
+    Harmonic numbers
+
+    The nth harmonic number is given by `\operatorname{H}_{n} =
+    1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n}`.
+
+    More generally:
+
+    .. math:: \operatorname{H}_{n,m} = \sum_{k=1}^{n} \frac{1}{k^m}
+
+    As `n \rightarrow \infty`, `\operatorname{H}_{n,m} \rightarrow \zeta(m)`,
+    the Riemann zeta function.
+
+    * ``harmonic(n)`` gives the nth harmonic number, `\operatorname{H}_n`
+
+    * ``harmonic(n, m)`` gives the nth generalized harmonic number
+      of order `m`, `\operatorname{H}_{n,m}`, where
+      ``harmonic(n) == harmonic(n, 1)``
+
+    This function can be extended to complex `n` and `m` where `n` is not a
+    negative integer or `m` is a nonpositive integer as
+
+    .. math:: \operatorname{H}_{n,m} = \begin{cases} \zeta(m) - \zeta(m, n+1)
+            & m \ne 1 \\ \psi(n+1) + \gamma & m = 1 \end{cases}
+
+    Examples
+    ========
+
+    >>> from sympy import harmonic, oo
+
+    >>> [harmonic(n) for n in range(6)]
+    [0, 1, 3/2, 11/6, 25/12, 137/60]
+    >>> [harmonic(n, 2) for n in range(6)]
+    [0, 1, 5/4, 49/36, 205/144, 5269/3600]
+    >>> harmonic(oo, 2)
+    pi**2/6
+
+    >>> from sympy import Symbol, Sum
+    >>> n = Symbol("n")
+
+    >>> harmonic(n).rewrite(Sum)
+    Sum(1/_k, (_k, 1, n))
+
+    We can evaluate harmonic numbers for all integral and positive
+    rational arguments:
+
+    >>> from sympy import S, expand_func, simplify
+    >>> harmonic(8)
+    761/280
+    >>> harmonic(11)
+    83711/27720
+
+    >>> H = harmonic(1/S(3))
+    >>> H
+    harmonic(1/3)
+    >>> He = expand_func(H)
+    >>> He
+    -log(6) - sqrt(3)*pi/6 + 2*Sum(log(sin(_k*pi/3))*cos(2*_k*pi/3), (_k, 1, 1))
+                           + 3*Sum(1/(3*_k + 1), (_k, 0, 0))
+    >>> He.doit()
+    -log(6) - sqrt(3)*pi/6 - log(sqrt(3)/2) + 3
+    >>> H = harmonic(25/S(7))
+    >>> He = simplify(expand_func(H).doit())
+    >>> He
+    log(sin(2*pi/7)**(2*cos(16*pi/7))/(14*sin(pi/7)**(2*cos(pi/7))*cos(pi/14)**(2*sin(pi/14)))) + pi*tan(pi/14)/2 + 30247/9900
+    >>> He.n(40)
+    1.983697455232980674869851942390639915940
+    >>> harmonic(25/S(7)).n(40)
+    1.983697455232980674869851942390639915940
+
+    We can rewrite harmonic numbers in terms of polygamma functions:
+
+    >>> from sympy import digamma, polygamma
+    >>> m = Symbol("m", integer=True, positive=True)
+
+    >>> harmonic(n).rewrite(digamma)
+    polygamma(0, n + 1) + EulerGamma
+
+    >>> harmonic(n).rewrite(polygamma)
+    polygamma(0, n + 1) + EulerGamma
+
+    >>> harmonic(n,3).rewrite(polygamma)
+    polygamma(2, n + 1)/2 + zeta(3)
+
+    >>> simplify(harmonic(n,m).rewrite(polygamma))
+    Piecewise((polygamma(0, n + 1) + EulerGamma, Eq(m, 1)),
+    (-(-1)**m*polygamma(m - 1, n + 1)/factorial(m - 1) + zeta(m), True))
+
+    Integer offsets in the argument can be pulled out:
+
+    >>> from sympy import expand_func
+
+    >>> expand_func(harmonic(n+4))
+    harmonic(n) + 1/(n + 4) + 1/(n + 3) + 1/(n + 2) + 1/(n + 1)
+
+    >>> expand_func(harmonic(n-4))
+    harmonic(n) - 1/(n - 1) - 1/(n - 2) - 1/(n - 3) - 1/n
+
+    Some limits can be computed as well:
+
+    >>> from sympy import limit, oo
+
+    >>> limit(harmonic(n), n, oo)
+    oo
+
+    >>> limit(harmonic(n, 2), n, oo)
+    pi**2/6
+
+    >>> limit(harmonic(n, 3), n, oo)
+    zeta(3)
+
+    For `m > 1`, `H_{n,m}` tends to `\zeta(m)` in the limit of infinite `n`:
+
+    >>> m = Symbol("m", positive=True)
+    >>> limit(harmonic(n, m+1), n, oo)
+    zeta(m + 1)
+
+    See Also
+    ========
+
+    bell, bernoulli, catalan, euler, fibonacci, lucas, genocchi, partition, tribonacci
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Harmonic_number
+    .. [2] https://functions.wolfram.com/GammaBetaErf/HarmonicNumber/
+    .. [3] https://functions.wolfram.com/GammaBetaErf/HarmonicNumber2/
+
+    """
+
+    @classmethod
+    def eval(cls, n, m=None):
+        from sympy.functions.special.zeta_functions import zeta
+        if m is S.One:
+            return cls(n)
+        if m is None:
+            m = S.One
+        if n.is_zero:
+            return S.Zero
+        elif m.is_zero:
+            return n
+        elif n is S.Infinity:
+            if m.is_negative:
+                return S.NaN
+            elif is_le(m, S.One):
+                return S.Infinity
+            elif is_gt(m, S.One):
+                return zeta(m)
+        elif m.is_Integer and m.is_nonpositive:
+            return (bernoulli(1-m, n+1) - bernoulli(1-m)) / (1-m)
+        elif n.is_Integer:
+            if n.is_negative and (m.is_integer is False or m.is_nonpositive is False):
+                return S.ComplexInfinity if m is S.One else S.NaN
+            if n.is_nonnegative:
+                return Add(*(k**(-m) for k in range(1, int(n)+1)))
+
+    def _eval_rewrite_as_polygamma(self, n, m=S.One, **kwargs):
+        from sympy.functions.special.gamma_functions import gamma, polygamma
+        if m.is_integer and m.is_positive:
+            return Piecewise((polygamma(0, n+1) + S.EulerGamma, Eq(m, 1)),
+                    (S.NegativeOne**m * (polygamma(m-1, 1) - polygamma(m-1, n+1)) /
+                    gamma(m), True))
+
+    def _eval_rewrite_as_digamma(self, n, m=1, **kwargs):
+        from sympy.functions.special.gamma_functions import polygamma
+        return self.rewrite(polygamma)
+
+    def _eval_rewrite_as_trigamma(self, n, m=1, **kwargs):
+        from sympy.functions.special.gamma_functions import polygamma
+        return self.rewrite(polygamma)
+
+    def _eval_rewrite_as_Sum(self, n, m=None, **kwargs):
+        from sympy.concrete.summations import Sum
+        k = Dummy("k", integer=True)
+        if m is None:
+            m = S.One
+        return Sum(k**(-m), (k, 1, n))
+
+    def _eval_rewrite_as_zeta(self, n, m=S.One, **kwargs):
+        from sympy.functions.special.zeta_functions import zeta
+        from sympy.functions.special.gamma_functions import digamma
+        return Piecewise((digamma(n + 1) + S.EulerGamma, Eq(m, 1)),
+                         (zeta(m) - zeta(m, n+1), True))
+
+    def _eval_expand_func(self, **hints):
+        from sympy.concrete.summations import Sum
+        n = self.args[0]
+        m = self.args[1] if len(self.args) == 2 else 1
+
+        if m == S.One:
+            if n.is_Add:
+                off = n.args[0]
+                nnew = n - off
+                if off.is_Integer and off.is_positive:
+                    result = [S.One/(nnew + i) for i in range(off, 0, -1)] + [harmonic(nnew)]
+                    return Add(*result)
+                elif off.is_Integer and off.is_negative:
+                    result = [-S.One/(nnew + i) for i in range(0, off, -1)] + [harmonic(nnew)]
+                    return Add(*result)
+
+            if n.is_Rational:
+                # Expansions for harmonic numbers at general rational arguments (u + p/q)
+                # Split n as u + p/q with p < q
+                p, q = n.as_numer_denom()
+                u = p // q
+                p = p - u * q
+                if u.is_nonnegative and p.is_positive and q.is_positive and p < q:
+                    from sympy.functions.elementary.exponential import log
+                    from sympy.functions.elementary.integers import floor
+                    from sympy.functions.elementary.trigonometric import sin, cos, cot
+                    k = Dummy("k")
+                    t1 = q * Sum(1 / (q * k + p), (k, 0, u))
+                    t2 = 2 * Sum(cos((2 * pi * p * k) / S(q)) *
+                                   log(sin((pi * k) / S(q))),
+                                   (k, 1, floor((q - 1) / S(2))))
+                    t3 = (pi / 2) * cot((pi * p) / q) + log(2 * q)
+                    return t1 + t2 - t3
+
+        return self
+
+    def _eval_rewrite_as_tractable(self, n, m=1, limitvar=None, **kwargs):
+        from sympy.functions.special.zeta_functions import zeta
+        from sympy.functions.special.gamma_functions import polygamma
+        pg = self.rewrite(polygamma)
+        if not isinstance(pg, harmonic):
+            return pg.rewrite("tractable", deep=True)
+        arg = m - S.One
+        if arg.is_nonzero:
+            return (zeta(m) - zeta(m, n+1)).rewrite("tractable", deep=True)
+
+    def _eval_evalf(self, prec):
+        if not all(x.is_number for x in self.args):
+            return
+        n = self.args[0]._to_mpmath(prec)
+        m = (self.args[1] if len(self.args) > 1 else S.One)._to_mpmath(prec)
+        if mp.isint(n) and n < 0:
+            return S.NaN
+        with workprec(prec):
+            if m == 1:
+                res = mp.harmonic(n)
+            else:
+                res = mp.zeta(m) - mp.zeta(m, n+1)
+        return Expr._from_mpmath(res, prec)
+
+    def fdiff(self, argindex=1):
+        from sympy.functions.special.zeta_functions import zeta
+        if len(self.args) == 2:
+            n, m = self.args
+        else:
+            n, m = self.args + (1,)
+        if argindex == 1:
+            return m * zeta(m+1, n+1)
+        else:
+            raise ArgumentIndexError
+
+
+#----------------------------------------------------------------------------#
+#                                                                            #
+#                           Euler numbers                                    #
+#                                                                            #
+#----------------------------------------------------------------------------#
+
+
+class euler(Function):
+    r"""
+    Euler numbers / Euler polynomials / Euler function
+
+    The Euler numbers are given by:
+
+    .. math:: E_{2n} = I \sum_{k=1}^{2n+1} \sum_{j=0}^k \binom{k}{j}
+        \frac{(-1)^j (k-2j)^{2n+1}}{2^k I^k k}
+
+    .. math:: E_{2n+1} = 0
+
+    Euler numbers and Euler polynomials are related by
+
+    .. math:: E_n = 2^n E_n\left(\frac{1}{2}\right).
+
+    We compute symbolic Euler polynomials using Appell sequences,
+    but numerical evaluation of the Euler polynomial is computed
+    more efficiently (and more accurately) using the mpmath library.
+
+    The Euler polynomials are special cases of the generalized Euler function,
+    related to the Genocchi function as
+
+    .. math:: \operatorname{E}(s, a) = -\frac{\operatorname{G}(s+1, a)}{s+1}
+
+    with the limit of `\psi\left(\frac{a+1}{2}\right) - \psi\left(\frac{a}{2}\right)`
+    being taken when `s = -1`. The (ordinary) Euler function interpolating
+    the Euler numbers is then obtained as
+    `\operatorname{E}(s) = 2^s \operatorname{E}\left(s, \frac{1}{2}\right)`.
+
+    * ``euler(n)`` gives the nth Euler number `E_n`.
+    * ``euler(s)`` gives the Euler function `\operatorname{E}(s)`.
+    * ``euler(n, x)`` gives the nth Euler polynomial `E_n(x)`.
+    * ``euler(s, a)`` gives the generalized Euler function `\operatorname{E}(s, a)`.
+
+    Examples
+    ========
+
+    >>> from sympy import euler, Symbol, S
+    >>> [euler(n) for n in range(10)]
+    [1, 0, -1, 0, 5, 0, -61, 0, 1385, 0]
+    >>> [2**n*euler(n,1) for n in range(10)]
+    [1, 1, 0, -2, 0, 16, 0, -272, 0, 7936]
+    >>> n = Symbol("n")
+    >>> euler(n + 2*n)
+    euler(3*n)
+
+    >>> x = Symbol("x")
+    >>> euler(n, x)
+    euler(n, x)
+
+    >>> euler(0, x)
+    1
+    >>> euler(1, x)
+    x - 1/2
+    >>> euler(2, x)
+    x**2 - x
+    >>> euler(3, x)
+    x**3 - 3*x**2/2 + 1/4
+    >>> euler(4, x)
+    x**4 - 2*x**3 + x
+
+    >>> euler(12, S.Half)
+    2702765/4096
+    >>> euler(12)
+    2702765
+
+    See Also
+    ========
+
+    andre, bell, bernoulli, catalan, fibonacci, harmonic, lucas, genocchi,
+    partition, tribonacci, sympy.polys.appellseqs.euler_poly
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Euler_numbers
+    .. [2] https://mathworld.wolfram.com/EulerNumber.html
+    .. [3] https://en.wikipedia.org/wiki/Alternating_permutation
+    .. [4] https://mathworld.wolfram.com/AlternatingPermutation.html
+
+    """
+
+    @classmethod
+    def eval(cls, n, x=None):
+        if n.is_zero:
+            return S.One
+        elif n is S.NegativeOne:
+            if x is None:
+                return S.Pi/2
+            from sympy.functions.special.gamma_functions import digamma
+            return digamma((x+1)/2) - digamma(x/2)
+        elif n.is_integer is False or n.is_nonnegative is False:
+            return
+        # Euler numbers
+        elif x is None:
+            if n.is_odd and n.is_positive:
+                return S.Zero
+            elif n.is_Number:
+                from mpmath import mp
+                n = n._to_mpmath(mp.prec)
+                res = mp.eulernum(n, exact=True)
+                return Integer(res)
+        # Euler polynomials
+        elif n.is_Number:
+            return euler_poly(n, x)
+
+    def _eval_rewrite_as_Sum(self, n, x=None, **kwargs):
+        from sympy.concrete.summations import Sum
+        if x is None and n.is_even:
+            k = Dummy("k", integer=True)
+            j = Dummy("j", integer=True)
+            n = n / 2
+            Em = (S.ImaginaryUnit * Sum(Sum(binomial(k, j) * (S.NegativeOne**j *
+                                                              (k - 2*j)**(2*n + 1)) /
+                  (2**k*S.ImaginaryUnit**k * k), (j, 0, k)), (k, 1, 2*n + 1)))
+            return Em
+        if x:
+            k = Dummy("k", integer=True)
+            return Sum(binomial(n, k)*euler(k)/2**k*(x - S.Half)**(n - k), (k, 0, n))
+
+    def _eval_rewrite_as_genocchi(self, n, x=None, **kwargs):
+        if x is None:
+            return Piecewise((S.Pi/2, Eq(n, -1)),
+                             (-2**n * genocchi(n+1, S.Half) / (n+1), True))
+        from sympy.functions.special.gamma_functions import digamma
+        return Piecewise((digamma((x+1)/2) - digamma(x/2), Eq(n, -1)),
+                         (-genocchi(n+1, x) / (n+1), True))
+
+    def _eval_evalf(self, prec):
+        if not all(i.is_number for i in self.args):
+            return
+        from mpmath import mp
+        m, x = (self.args[0], None) if len(self.args) == 1 else self.args
+        m = m._to_mpmath(prec)
+        if x is not None:
+            x = x._to_mpmath(prec)
+        with workprec(prec):
+            if mp.isint(m) and m >= 0:
+                res = mp.eulernum(m) if x is None else mp.eulerpoly(m, x)
+            else:
+                if m == -1:
+                    res = mp.pi if x is None else mp.digamma((x+1)/2) - mp.digamma(x/2)
+                else:
+                    y = 0.5 if x is None else x
+                    res = 2 * (mp.zeta(-m, y) - 2**(m+1) * mp.zeta(-m, (y+1)/2))
+                if x is None:
+                    res *= 2**m
+        return Expr._from_mpmath(res, prec)
+
+
+#----------------------------------------------------------------------------#
+#                                                                            #
+#                              Catalan numbers                               #
+#                                                                            #
+#----------------------------------------------------------------------------#
+
+
+class catalan(Function):
+    r"""
+    Catalan numbers
+
+    The `n^{th}` catalan number is given by:
+
+    .. math :: C_n = \frac{1}{n+1} \binom{2n}{n}
+
+    * ``catalan(n)`` gives the `n^{th}` Catalan number, `C_n`
+
+    Examples
+    ========
+
+    >>> from sympy import (Symbol, binomial, gamma, hyper,
+    ...     catalan, diff, combsimp, Rational, I)
+
+    >>> [catalan(i) for i in range(1,10)]
+    [1, 2, 5, 14, 42, 132, 429, 1430, 4862]
+
+    >>> n = Symbol("n", integer=True)
+
+    >>> catalan(n)
+    catalan(n)
+
+    Catalan numbers can be transformed into several other, identical
+    expressions involving other mathematical functions
+
+    >>> catalan(n).rewrite(binomial)
+    binomial(2*n, n)/(n + 1)
+
+    >>> catalan(n).rewrite(gamma)
+    4**n*gamma(n + 1/2)/(sqrt(pi)*gamma(n + 2))
+
+    >>> catalan(n).rewrite(hyper)
+    hyper((1 - n, -n), (2,), 1)
+
+    For some non-integer values of n we can get closed form
+    expressions by rewriting in terms of gamma functions:
+
+    >>> catalan(Rational(1, 2)).rewrite(gamma)
+    8/(3*pi)
+
+    We can differentiate the Catalan numbers C(n) interpreted as a
+    continuous real function in n:
+
+    >>> diff(catalan(n), n)
+    (polygamma(0, n + 1/2) - polygamma(0, n + 2) + log(4))*catalan(n)
+
+    As a more advanced example consider the following ratio
+    between consecutive numbers:
+
+    >>> combsimp((catalan(n + 1)/catalan(n)).rewrite(binomial))
+    2*(2*n + 1)/(n + 2)
+
+    The Catalan numbers can be generalized to complex numbers:
+
+    >>> catalan(I).rewrite(gamma)
+    4**I*gamma(1/2 + I)/(sqrt(pi)*gamma(2 + I))
+
+    and evaluated with arbitrary precision:
+
+    >>> catalan(I).evalf(20)
+    0.39764993382373624267 - 0.020884341620842555705*I
+
+    See Also
+    ========
+
+    andre, bell, bernoulli, euler, fibonacci, harmonic, lucas, genocchi,
+    partition, tribonacci, sympy.functions.combinatorial.factorials.binomial
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Catalan_number
+    .. [2] https://mathworld.wolfram.com/CatalanNumber.html
+    .. [3] https://functions.wolfram.com/GammaBetaErf/CatalanNumber/
+    .. [4] http://geometer.org/mathcircles/catalan.pdf
+
+    """
+
+    @classmethod
+    def eval(cls, n):
+        from sympy.functions.special.gamma_functions import gamma
+        if (n.is_Integer and n.is_nonnegative) or \
+           (n.is_noninteger and n.is_negative):
+            return 4**n*gamma(n + S.Half)/(gamma(S.Half)*gamma(n + 2))
+
+        if (n.is_integer and n.is_negative):
+            if (n + 1).is_negative:
+                return S.Zero
+            if (n + 1).is_zero:
+                return Rational(-1, 2)
+
+    def fdiff(self, argindex=1):
+        from sympy.functions.elementary.exponential import log
+        from sympy.functions.special.gamma_functions import polygamma
+        n = self.args[0]
+        return catalan(n)*(polygamma(0, n + S.Half) - polygamma(0, n + 2) + log(4))
+
+    def _eval_rewrite_as_binomial(self, n, **kwargs):
+        return binomial(2*n, n)/(n + 1)
+
+    def _eval_rewrite_as_factorial(self, n, **kwargs):
+        return factorial(2*n) / (factorial(n+1) * factorial(n))
+
+    def _eval_rewrite_as_gamma(self, n, piecewise=True, **kwargs):
+        from sympy.functions.special.gamma_functions import gamma
+        # The gamma function allows to generalize Catalan numbers to complex n
+        return 4**n*gamma(n + S.Half)/(gamma(S.Half)*gamma(n + 2))
+
+    def _eval_rewrite_as_hyper(self, n, **kwargs):
+        from sympy.functions.special.hyper import hyper
+        return hyper([1 - n, -n], [2], 1)
+
+    def _eval_rewrite_as_Product(self, n, **kwargs):
+        from sympy.concrete.products import Product
+        if not (n.is_integer and n.is_nonnegative):
+            return self
+        k = Dummy('k', integer=True, positive=True)
+        return Product((n + k) / k, (k, 2, n))
+
+    def _eval_is_integer(self):
+        if self.args[0].is_integer and self.args[0].is_nonnegative:
+            return True
+
+    def _eval_is_positive(self):
+        if self.args[0].is_nonnegative:
+            return True
+
+    def _eval_is_composite(self):
+        if self.args[0].is_integer and (self.args[0] - 3).is_positive:
+            return True
+
+    def _eval_evalf(self, prec):
+        from sympy.functions.special.gamma_functions import gamma
+        if self.args[0].is_number:
+            return self.rewrite(gamma)._eval_evalf(prec)
+
+
+#----------------------------------------------------------------------------#
+#                                                                            #
+#                           Genocchi numbers                                 #
+#                                                                            #
+#----------------------------------------------------------------------------#
+
+
+class genocchi(Function):
+    r"""
+    Genocchi numbers / Genocchi polynomials / Genocchi function
+
+    The Genocchi numbers are a sequence of integers `G_n` that satisfy the
+    relation:
+
+    .. math:: \frac{-2t}{1 + e^{-t}} = \sum_{n=0}^\infty \frac{G_n t^n}{n!}
+
+    They are related to the Bernoulli numbers by
+
+    .. math:: G_n = 2 (1 - 2^n) B_n
+
+    and generalize like the Bernoulli numbers to the Genocchi polynomials and
+    function as
+
+    .. math:: \operatorname{G}(s, a) = 2 \left(\operatorname{B}(s, a) -
+              2^s \operatorname{B}\left(s, \frac{a+1}{2}\right)\right)
+
+    .. versionchanged:: 1.12
+        ``genocchi(1)`` gives `-1` instead of `1`.
+
+    Examples
+    ========
+
+    >>> from sympy import genocchi, Symbol
+    >>> [genocchi(n) for n in range(9)]
+    [0, -1, -1, 0, 1, 0, -3, 0, 17]
+    >>> n = Symbol('n', integer=True, positive=True)
+    >>> genocchi(2*n + 1)
+    0
+    >>> x = Symbol('x')
+    >>> genocchi(4, x)
+    -4*x**3 + 6*x**2 - 1
+
+    See Also
+    ========
+
+    bell, bernoulli, catalan, euler, fibonacci, harmonic, lucas, partition, tribonacci
+    sympy.polys.appellseqs.genocchi_poly
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Genocchi_number
+    .. [2] https://mathworld.wolfram.com/GenocchiNumber.html
+    .. [3] Peter Luschny, "An introduction to the Bernoulli function",
+           https://arxiv.org/abs/2009.06743
+
+    """
+
+    @classmethod
+    def eval(cls, n, x=None):
+        if x is S.One:
+            return cls(n)
+        elif n.is_integer is False or n.is_nonnegative is False:
+            return
+        # Genocchi numbers
+        elif x is None:
+            if n.is_odd and (n-1).is_positive:
+                return S.Zero
+            elif n.is_Number:
+                return 2 * (1-S(2)**n) * bernoulli(n)
+        # Genocchi polynomials
+        elif n.is_Number:
+            return genocchi_poly(n, x)
+
+    def _eval_rewrite_as_bernoulli(self, n, x=1, **kwargs):
+        if x == 1 and n.is_integer and n.is_nonnegative:
+            return 2 * (1-S(2)**n) * bernoulli(n)
+        return 2 * (bernoulli(n, x) - 2**n * bernoulli(n, (x+1) / 2))
+
+    def _eval_rewrite_as_dirichlet_eta(self, n, x=1, **kwargs):
+        from sympy.functions.special.zeta_functions import dirichlet_eta
+        return -2*n * dirichlet_eta(1-n, x)
+
+    def _eval_is_integer(self):
+        if len(self.args) > 1 and self.args[1] != 1:
+            return
+        n = self.args[0]
+        if n.is_integer and n.is_nonnegative:
+            return True
+
+    def _eval_is_negative(self):
+        if len(self.args) > 1 and self.args[1] != 1:
+            return
+        n = self.args[0]
+        if n.is_integer and n.is_nonnegative:
+            if n.is_odd:
+                return fuzzy_not((n-1).is_positive)
+            return (n/2).is_odd
+
+    def _eval_is_positive(self):
+        if len(self.args) > 1 and self.args[1] != 1:
+            return
+        n = self.args[0]
+        if n.is_integer and n.is_nonnegative:
+            if n.is_zero or n.is_odd:
+                return False
+            return (n/2).is_even
+
+    def _eval_is_even(self):
+        if len(self.args) > 1 and self.args[1] != 1:
+            return
+        n = self.args[0]
+        if n.is_integer and n.is_nonnegative:
+            if n.is_even:
+                return n.is_zero
+            return (n-1).is_positive
+
+    def _eval_is_odd(self):
+        if len(self.args) > 1 and self.args[1] != 1:
+            return
+        n = self.args[0]
+        if n.is_integer and n.is_nonnegative:
+            if n.is_even:
+                return fuzzy_not(n.is_zero)
+            return fuzzy_not((n-1).is_positive)
+
+    def _eval_is_prime(self):
+        if len(self.args) > 1 and self.args[1] != 1:
+            return
+        n = self.args[0]
+        # only G_6 = -3 and G_8 = 17 are prime,
+        # but SymPy does not consider negatives as prime
+        # so only n=8 is tested
+        return (n-8).is_zero
+
+    def _eval_evalf(self, prec):
+        if all(i.is_number for i in self.args):
+            return self.rewrite(bernoulli)._eval_evalf(prec)
+
+
+#----------------------------------------------------------------------------#
+#                                                                            #
+#                              Andre numbers                                 #
+#                                                                            #
+#----------------------------------------------------------------------------#
+
+
+class andre(Function):
+    r"""
+    Andre numbers / Andre function
+
+    The Andre number `\mathcal{A}_n` is Luschny's name for half the number of
+    *alternating permutations* on `n` elements, where a permutation is alternating
+    if adjacent elements alternately compare "greater" and "smaller" going from
+    left to right. For example, `2 < 3 > 1 < 4` is an alternating permutation.
+
+    This sequence is A000111 in the OEIS, which assigns the names *up/down numbers*
+    and *Euler zigzag numbers*. It satisfies a recurrence relation similar to that
+    for the Catalan numbers, with `\mathcal{A}_0 = 1` and
+
+    .. math:: 2 \mathcal{A}_{n+1} = \sum_{k=0}^n \binom{n}{k} \mathcal{A}_k \mathcal{A}_{n-k}
+
+    The Bernoulli and Euler numbers are signed transformations of the odd- and
+    even-indexed elements of this sequence respectively:
+
+    .. math :: \operatorname{B}_{2k} = \frac{2k \mathcal{A}_{2k-1}}{(-4)^k - (-16)^k}
+
+    .. math :: \operatorname{E}_{2k} = (-1)^k \mathcal{A}_{2k}
+
+    Like the Bernoulli and Euler numbers, the Andre numbers are interpolated by the
+    entire Andre function:
+
+    .. math :: \mathcal{A}(s) = (-i)^{s+1} \operatorname{Li}_{-s}(i) +
+            i^{s+1} \operatorname{Li}_{-s}(-i) = \\ \frac{2 \Gamma(s+1)}{(2\pi)^{s+1}}
+            (\zeta(s+1, 1/4) - \zeta(s+1, 3/4) \cos{\pi s})
+
+    Examples
+    ========
+
+    >>> from sympy import andre, euler, bernoulli
+    >>> [andre(n) for n in range(11)]
+    [1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, 50521]
+    >>> [(-1)**k * andre(2*k) for k in range(7)]
+    [1, -1, 5, -61, 1385, -50521, 2702765]
+    >>> [euler(2*k) for k in range(7)]
+    [1, -1, 5, -61, 1385, -50521, 2702765]
+    >>> [andre(2*k-1) * (2*k) / ((-4)**k - (-16)**k) for k in range(1, 8)]
+    [1/6, -1/30, 1/42, -1/30, 5/66, -691/2730, 7/6]
+    >>> [bernoulli(2*k) for k in range(1, 8)]
+    [1/6, -1/30, 1/42, -1/30, 5/66, -691/2730, 7/6]
+
+    See Also
+    ========
+
+    bernoulli, catalan, euler, sympy.polys.appellseqs.andre_poly
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Alternating_permutation
+    .. [2] https://mathworld.wolfram.com/EulerZigzagNumber.html
+    .. [3] Peter Luschny, "An introduction to the Bernoulli function",
+           https://arxiv.org/abs/2009.06743
+    """
+
+    @classmethod
+    def eval(cls, n):
+        if n is S.NaN:
+            return S.NaN
+        elif n is S.Infinity:
+            return S.Infinity
+        if n.is_zero:
+            return S.One
+        elif n == -1:
+            return -log(2)
+        elif n == -2:
+            return -2*S.Catalan
+        elif n.is_Integer:
+            if n.is_nonnegative and n.is_even:
+                return abs(euler(n))
+            elif n.is_odd:
+                from sympy.functions.special.zeta_functions import zeta
+                m = -n-1
+                return I**m * Rational(1-2**m, 4**m) * zeta(-n)
+
+    def _eval_rewrite_as_zeta(self, s, **kwargs):
+        from sympy.functions.elementary.trigonometric import cos
+        from sympy.functions.special.gamma_functions import gamma
+        from sympy.functions.special.zeta_functions import zeta
+        return 2 * gamma(s+1) / (2*pi)**(s+1) * \
+                (zeta(s+1, S.One/4) - cos(pi*s) * zeta(s+1, S(3)/4))
+
+    def _eval_rewrite_as_polylog(self, s, **kwargs):
+        from sympy.functions.special.zeta_functions import polylog
+        return (-I)**(s+1) * polylog(-s, I) + I**(s+1) * polylog(-s, -I)
+
+    def _eval_is_integer(self):
+        n = self.args[0]
+        if n.is_integer and n.is_nonnegative:
+            return True
+
+    def _eval_is_positive(self):
+        if self.args[0].is_nonnegative:
+            return True
+
+    def _eval_evalf(self, prec):
+        if not self.args[0].is_number:
+            return
+        s = self.args[0]._to_mpmath(prec+12)
+        with workprec(prec+12):
+            sp, cp = mp.sinpi(s/2), mp.cospi(s/2)
+            res = 2*mp.dirichlet(-s, (-sp, cp, sp, -cp))
+        return Expr._from_mpmath(res, prec)
+
+
+#----------------------------------------------------------------------------#
+#                                                                            #
+#                           Partition numbers                                #
+#                                                                            #
+#----------------------------------------------------------------------------#
+
+_npartition = [1, 1]
+class partition(Function):
+    r"""
+    Partition numbers
+
+    The Partition numbers are a sequence of integers `p_n` that represent the
+    number of distinct ways of representing `n` as a sum of natural numbers
+    (with order irrelevant). The generating function for `p_n` is given by:
+
+    .. math:: \sum_{n=0}^\infty p_n x^n = \prod_{k=1}^\infty (1 - x^k)^{-1}
+
+    Examples
+    ========
+
+    >>> from sympy import partition, Symbol
+    >>> [partition(n) for n in range(9)]
+    [1, 1, 2, 3, 5, 7, 11, 15, 22]
+    >>> n = Symbol('n', integer=True, negative=True)
+    >>> partition(n)
+    0
+
+    See Also
+    ========
+
+    bell, bernoulli, catalan, euler, fibonacci, harmonic, lucas, genocchi, tribonacci
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Partition_(number_theory%29
+    .. [2] https://en.wikipedia.org/wiki/Pentagonal_number_theorem
+
+    """
+
+    @staticmethod
+    def _partition(n):
+        L = len(_npartition)
+        if n < L:
+            return _npartition[n]
+        # lengthen cache
+        for _n in range(L, n + 1):
+            v, p, i = 0, 0, 0
+            while 1:
+                s = 0
+                p += 3*i + 1  # p = pentagonal number: 1, 5, 12, ...
+                if _n >= p:
+                    s += _npartition[_n - p]
+                i += 1
+                gp = p + i  # gp = generalized pentagonal: 2, 7, 15, ...
+                if _n >= gp:
+                    s += _npartition[_n - gp]
+                if s == 0:
+                    break
+                else:
+                    v += s if i%2 == 1 else -s
+            _npartition.append(v)
+        return v
+
+    @classmethod
+    def eval(cls, n):
+        is_int = n.is_integer
+        if is_int == False:
+            raise ValueError("Partition numbers are defined only for "
+                             "integers")
+        elif is_int:
+            if n.is_negative:
+                return S.Zero
+
+            if n.is_zero or (n - 1).is_zero:
+                return S.One
+
+            if n.is_Integer:
+                return Integer(cls._partition(n))
+
+
+    def _eval_is_integer(self):
+        if self.args[0].is_integer:
+            return True
+
+    def _eval_is_negative(self):
+        if self.args[0].is_integer:
+            return False
+
+    def _eval_is_positive(self):
+        n = self.args[0]
+        if n.is_nonnegative and n.is_integer:
+            return True
+
+
+#######################################################################
+###
+### Functions for enumerating partitions, permutations and combinations
+###
+#######################################################################
+
+
+class _MultisetHistogram(tuple):
+    pass
+
+
+_N = -1
+_ITEMS = -2
+_M = slice(None, _ITEMS)
+
+
+def _multiset_histogram(n):
+    """Return tuple used in permutation and combination counting. Input
+    is a dictionary giving items with counts as values or a sequence of
+    items (which need not be sorted).
+
+    The data is stored in a class deriving from tuple so it is easily
+    recognized and so it can be converted easily to a list.
+    """
+    if isinstance(n, dict):  # item: count
+        if not all(isinstance(v, int) and v >= 0 for v in n.values()):
+            raise ValueError
+        tot = sum(n.values())
+        items = sum(1 for k in n if n[k] > 0)
+        return _MultisetHistogram([n[k] for k in n if n[k] > 0] + [items, tot])
+    else:
+        n = list(n)
+        s = set(n)
+        lens = len(s)
+        lenn = len(n)
+        if lens == lenn:
+            n = [1]*lenn + [lenn, lenn]
+            return _MultisetHistogram(n)
+        m = dict(zip(s, range(lens)))
+        d = dict(zip(range(lens), (0,)*lens))
+        for i in n:
+            d[m[i]] += 1
+        return _multiset_histogram(d)
+
+
+def nP(n, k=None, replacement=False):
+    """Return the number of permutations of ``n`` items taken ``k`` at a time.
+
+    Possible values for ``n``:
+
+        integer - set of length ``n``
+
+        sequence - converted to a multiset internally
+
+        multiset - {element: multiplicity}
+
+    If ``k`` is None then the total of all permutations of length 0
+    through the number of items represented by ``n`` will be returned.
+
+    If ``replacement`` is True then a given item can appear more than once
+    in the ``k`` items. (For example, for 'ab' permutations of 2 would
+    include 'aa', 'ab', 'ba' and 'bb'.) The multiplicity of elements in
+    ``n`` is ignored when ``replacement`` is True but the total number
+    of elements is considered since no element can appear more times than
+    the number of elements in ``n``.
+
+    Examples
+    ========
+
+    >>> from sympy.functions.combinatorial.numbers import nP
+    >>> from sympy.utilities.iterables import multiset_permutations, multiset
+    >>> nP(3, 2)
+    6
+    >>> nP('abc', 2) == nP(multiset('abc'), 2) == 6
+    True
+    >>> nP('aab', 2)
+    3
+    >>> nP([1, 2, 2], 2)
+    3
+    >>> [nP(3, i) for i in range(4)]
+    [1, 3, 6, 6]
+    >>> nP(3) == sum(_)
+    True
+
+    When ``replacement`` is True, each item can have multiplicity
+    equal to the length represented by ``n``:
+
+    >>> nP('aabc', replacement=True)
+    121
+    >>> [len(list(multiset_permutations('aaaabbbbcccc', i))) for i in range(5)]
+    [1, 3, 9, 27, 81]
+    >>> sum(_)
+    121
+
+    See Also
+    ========
+    sympy.utilities.iterables.multiset_permutations
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Permutation
+
+    """
+    try:
+        n = as_int(n)
+    except ValueError:
+        return Integer(_nP(_multiset_histogram(n), k, replacement))
+    return Integer(_nP(n, k, replacement))
+
+
+@cacheit
+def _nP(n, k=None, replacement=False):
+
+    if k == 0:
+        return 1
+    if isinstance(n, SYMPY_INTS):  # n different items
+        # assert n >= 0
+        if k is None:
+            return sum(_nP(n, i, replacement) for i in range(n + 1))
+        elif replacement:
+            return n**k
+        elif k > n:
+            return 0
+        elif k == n:
+            return factorial(k)
+        elif k == 1:
+            return n
+        else:
+            # assert k >= 0
+            return _product(n - k + 1, n)
+    elif isinstance(n, _MultisetHistogram):
+        if k is None:
+            return sum(_nP(n, i, replacement) for i in range(n[_N] + 1))
+        elif replacement:
+            return n[_ITEMS]**k
+        elif k == n[_N]:
+            return factorial(k)/prod([factorial(i) for i in n[_M] if i > 1])
+        elif k > n[_N]:
+            return 0
+        elif k == 1:
+            return n[_ITEMS]
+        else:
+            # assert k >= 0
+            tot = 0
+            n = list(n)
+            for i in range(len(n[_M])):
+                if not n[i]:
+                    continue
+                n[_N] -= 1
+                if n[i] == 1:
+                    n[i] = 0
+                    n[_ITEMS] -= 1
+                    tot += _nP(_MultisetHistogram(n), k - 1)
+                    n[_ITEMS] += 1
+                    n[i] = 1
+                else:
+                    n[i] -= 1
+                    tot += _nP(_MultisetHistogram(n), k - 1)
+                    n[i] += 1
+                n[_N] += 1
+            return tot
+
+
+@cacheit
+def _AOP_product(n):
+    """for n = (m1, m2, .., mk) return the coefficients of the polynomial,
+    prod(sum(x**i for i in range(nj + 1)) for nj in n); i.e. the coefficients
+    of the product of AOPs (all-one polynomials) or order given in n.  The
+    resulting coefficient corresponding to x**r is the number of r-length
+    combinations of sum(n) elements with multiplicities given in n.
+    The coefficients are given as a default dictionary (so if a query is made
+    for a key that is not present, 0 will be returned).
+
+    Examples
+    ========
+
+    >>> from sympy.functions.combinatorial.numbers import _AOP_product
+    >>> from sympy.abc import x
+    >>> n = (2, 2, 3)  # e.g. aabbccc
+    >>> prod = ((x**2 + x + 1)*(x**2 + x + 1)*(x**3 + x**2 + x + 1)).expand()
+    >>> c = _AOP_product(n); dict(c)
+    {0: 1, 1: 3, 2: 6, 3: 8, 4: 8, 5: 6, 6: 3, 7: 1}
+    >>> [c[i] for i in range(8)] == [prod.coeff(x, i) for i in range(8)]
+    True
+
+    The generating poly used here is the same as that listed in
+    https://tinyurl.com/cep849r, but in a refactored form.
+
+    """
+
+    n = list(n)
+    ord = sum(n)
+    need = (ord + 2)//2
+    rv = [1]*(n.pop() + 1)
+    rv.extend((0,) * (need - len(rv)))
+    rv = rv[:need]
+    while n:
+        ni = n.pop()
+        N = ni + 1
+        was = rv[:]
+        for i in range(1, min(N, len(rv))):
+            rv[i] += rv[i - 1]
+        for i in range(N, need):
+            rv[i] += rv[i - 1] - was[i - N]
+    rev = list(reversed(rv))
+    if ord % 2:
+        rv = rv + rev
+    else:
+        rv[-1:] = rev
+    d = defaultdict(int)
+    for i, r in enumerate(rv):
+        d[i] = r
+    return d
+
+
+def nC(n, k=None, replacement=False):
+    """Return the number of combinations of ``n`` items taken ``k`` at a time.
+
+    Possible values for ``n``:
+
+        integer - set of length ``n``
+
+        sequence - converted to a multiset internally
+
+        multiset - {element: multiplicity}
+
+    If ``k`` is None then the total of all combinations of length 0
+    through the number of items represented in ``n`` will be returned.
+
+    If ``replacement`` is True then a given item can appear more than once
+    in the ``k`` items. (For example, for 'ab' sets of 2 would include 'aa',
+    'ab', and 'bb'.) The multiplicity of elements in ``n`` is ignored when
+    ``replacement`` is True but the total number of elements is considered
+    since no element can appear more times than the number of elements in
+    ``n``.
+
+    Examples
+    ========
+
+    >>> from sympy.functions.combinatorial.numbers import nC
+    >>> from sympy.utilities.iterables import multiset_combinations
+    >>> nC(3, 2)
+    3
+    >>> nC('abc', 2)
+    3
+    >>> nC('aab', 2)
+    2
+
+    When ``replacement`` is True, each item can have multiplicity
+    equal to the length represented by ``n``:
+
+    >>> nC('aabc', replacement=True)
+    35
+    >>> [len(list(multiset_combinations('aaaabbbbcccc', i))) for i in range(5)]
+    [1, 3, 6, 10, 15]
+    >>> sum(_)
+    35
+
+    If there are ``k`` items with multiplicities ``m_1, m_2, ..., m_k``
+    then the total of all combinations of length 0 through ``k`` is the
+    product, ``(m_1 + 1)*(m_2 + 1)*...*(m_k + 1)``. When the multiplicity
+    of each item is 1 (i.e., k unique items) then there are 2**k
+    combinations. For example, if there are 4 unique items, the total number
+    of combinations is 16:
+
+    >>> sum(nC(4, i) for i in range(5))
+    16
+
+    See Also
+    ========
+
+    sympy.utilities.iterables.multiset_combinations
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Combination
+    .. [2] https://tinyurl.com/cep849r
+
+    """
+
+    if isinstance(n, SYMPY_INTS):
+        if k is None:
+            if not replacement:
+                return 2**n
+            return sum(nC(n, i, replacement) for i in range(n + 1))
+        if k < 0:
+            raise ValueError("k cannot be negative")
+        if replacement:
+            return binomial(n + k - 1, k)
+        return binomial(n, k)
+    if isinstance(n, _MultisetHistogram):
+        N = n[_N]
+        if k is None:
+            if not replacement:
+                return prod(m + 1 for m in n[_M])
+            return sum(nC(n, i, replacement) for i in range(N + 1))
+        elif replacement:
+            return nC(n[_ITEMS], k, replacement)
+        # assert k >= 0
+        elif k in (1, N - 1):
+            return n[_ITEMS]
+        elif k in (0, N):
+            return 1
+        return _AOP_product(tuple(n[_M]))[k]
+    else:
+        return nC(_multiset_histogram(n), k, replacement)
+
+
+def _eval_stirling1(n, k):
+    if n == k == 0:
+        return S.One
+    if 0 in (n, k):
+        return S.Zero
+
+    # some special values
+    if n == k:
+        return S.One
+    elif k == n - 1:
+        return binomial(n, 2)
+    elif k == n - 2:
+        return (3*n - 1)*binomial(n, 3)/4
+    elif k == n - 3:
+        return binomial(n, 2)*binomial(n, 4)
+
+    return _stirling1(n, k)
+
+
+@cacheit
+def _stirling1(n, k):
+    row = [0, 1]+[0]*(k-1) # for n = 1
+    for i in range(2, n+1):
+        for j in range(min(k,i), 0, -1):
+            row[j] = (i-1) * row[j] + row[j-1]
+    return Integer(row[k])
+
+
+def _eval_stirling2(n, k):
+    if n == k == 0:
+        return S.One
+    if 0 in (n, k):
+        return S.Zero
+
+    # some special values
+    if n == k:
+        return S.One
+    elif k == n - 1:
+        return binomial(n, 2)
+    elif k == 1:
+        return S.One
+    elif k == 2:
+        return Integer(2**(n - 1) - 1)
+
+    return _stirling2(n, k)
+
+
+@cacheit
+def _stirling2(n, k):
+    row = [0, 1]+[0]*(k-1) # for n = 1
+    for i in range(2, n+1):
+        for j in range(min(k,i), 0, -1):
+            row[j] = j * row[j] + row[j-1]
+    return Integer(row[k])
+
+
+def stirling(n, k, d=None, kind=2, signed=False):
+    r"""Return Stirling number $S(n, k)$ of the first or second (default) kind.
+
+    The sum of all Stirling numbers of the second kind for $k = 1$
+    through $n$ is ``bell(n)``. The recurrence relationship for these numbers
+    is:
+
+    .. math :: {0 \brace 0} = 1; {n \brace 0} = {0 \brace k} = 0;
+
+    .. math :: {{n+1} \brace k} = j {n \brace k} + {n \brace {k-1}}
+
+    where $j$ is:
+        $n$ for Stirling numbers of the first kind,
+        $-n$ for signed Stirling numbers of the first kind,
+        $k$ for Stirling numbers of the second kind.
+
+    The first kind of Stirling number counts the number of permutations of
+    ``n`` distinct items that have ``k`` cycles; the second kind counts the
+    ways in which ``n`` distinct items can be partitioned into ``k`` parts.
+    If ``d`` is given, the "reduced Stirling number of the second kind" is
+    returned: $S^{d}(n, k) = S(n - d + 1, k - d + 1)$ with $n \ge k \ge d$.
+    (This counts the ways to partition $n$ consecutive integers into $k$
+    groups with no pairwise difference less than $d$. See example below.)
+
+    To obtain the signed Stirling numbers of the first kind, use keyword
+    ``signed=True``. Using this keyword automatically sets ``kind`` to 1.
+
+    Examples
+    ========
+
+    >>> from sympy.functions.combinatorial.numbers import stirling, bell
+    >>> from sympy.combinatorics import Permutation
+    >>> from sympy.utilities.iterables import multiset_partitions, permutations
+
+    First kind (unsigned by default):
+
+    >>> [stirling(6, i, kind=1) for i in range(7)]
+    [0, 120, 274, 225, 85, 15, 1]
+    >>> perms = list(permutations(range(4)))
+    >>> [sum(Permutation(p).cycles == i for p in perms) for i in range(5)]
+    [0, 6, 11, 6, 1]
+    >>> [stirling(4, i, kind=1) for i in range(5)]
+    [0, 6, 11, 6, 1]
+
+    First kind (signed):
+
+    >>> [stirling(4, i, signed=True) for i in range(5)]
+    [0, -6, 11, -6, 1]
+
+    Second kind:
+
+    >>> [stirling(10, i) for i in range(12)]
+    [0, 1, 511, 9330, 34105, 42525, 22827, 5880, 750, 45, 1, 0]
+    >>> sum(_) == bell(10)
+    True
+    >>> len(list(multiset_partitions(range(4), 2))) == stirling(4, 2)
+    True
+
+    Reduced second kind:
+
+    >>> from sympy import subsets, oo
+    >>> def delta(p):
+    ...    if len(p) == 1:
+    ...        return oo
+    ...    return min(abs(i[0] - i[1]) for i in subsets(p, 2))
+    >>> parts = multiset_partitions(range(5), 3)
+    >>> d = 2
+    >>> sum(1 for p in parts if all(delta(i) >= d for i in p))
+    7
+    >>> stirling(5, 3, 2)
+    7
+
+    See Also
+    ========
+    sympy.utilities.iterables.multiset_partitions
+
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Stirling_numbers_of_the_first_kind
+    .. [2] https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind
+
+    """
+    # TODO: make this a class like bell()
+
+    n = as_int(n)
+    k = as_int(k)
+    if n < 0:
+        raise ValueError('n must be nonnegative')
+    if k > n:
+        return S.Zero
+    if d:
+        # assert k >= d
+        # kind is ignored -- only kind=2 is supported
+        return _eval_stirling2(n - d + 1, k - d + 1)
+    elif signed:
+        # kind is ignored -- only kind=1 is supported
+        return S.NegativeOne**(n - k)*_eval_stirling1(n, k)
+
+    if kind == 1:
+        return _eval_stirling1(n, k)
+    elif kind == 2:
+        return _eval_stirling2(n, k)
+    else:
+        raise ValueError('kind must be 1 or 2, not %s' % k)
+
+
+@cacheit
+def _nT(n, k):
+    """Return the partitions of ``n`` items into ``k`` parts. This
+    is used by ``nT`` for the case when ``n`` is an integer."""
+    # really quick exits
+    if k > n or k < 0:
+        return 0
+    if k in (1, n):
+        return 1
+    if k == 0:
+        return 0
+    # exits that could be done below but this is quicker
+    if k == 2:
+        return n//2
+    d = n - k
+    if d <= 3:
+        return d
+    # quick exit
+    if 3*k >= n:  # or, equivalently, 2*k >= d
+        # all the information needed in this case
+        # will be in the cache needed to calculate
+        # partition(d), so...
+        # update cache
+        tot = partition._partition(d)
+        # and correct for values not needed
+        if d - k > 0:
+            tot -= sum(_npartition[:d - k])
+        return tot
+    # regular exit
+    # nT(n, k) = Sum(nT(n - k, m), (m, 1, k));
+    # calculate needed nT(i, j) values
+    p = [1]*d
+    for i in range(2, k + 1):
+        for m  in range(i + 1, d):
+            p[m] += p[m - i]
+        d -= 1
+    # if p[0] were appended to the end of p then the last
+    # k values of p are the nT(n, j) values for 0 < j < k in reverse
+    # order p[-1] = nT(n, 1), p[-2] = nT(n, 2), etc.... Instead of
+    # putting the 1 from p[0] there, however, it is simply added to
+    # the sum below which is valid for 1 < k <= n//2
+    return (1 + sum(p[1 - k:]))
+
+
+def nT(n, k=None):
+    """Return the number of ``k``-sized partitions of ``n`` items.
+
+    Possible values for ``n``:
+
+        integer - ``n`` identical items
+
+        sequence - converted to a multiset internally
+
+        multiset - {element: multiplicity}
+
+    Note: the convention for ``nT`` is different than that of ``nC`` and
+    ``nP`` in that
+    here an integer indicates ``n`` *identical* items instead of a set of
+    length ``n``; this is in keeping with the ``partitions`` function which
+    treats its integer-``n`` input like a list of ``n`` 1s. One can use
+    ``range(n)`` for ``n`` to indicate ``n`` distinct items.
+
+    If ``k`` is None then the total number of ways to partition the elements
+    represented in ``n`` will be returned.
+
+    Examples
+    ========
+
+    >>> from sympy.functions.combinatorial.numbers import nT
+
+    Partitions of the given multiset:
+
+    >>> [nT('aabbc', i) for i in range(1, 7)]
+    [1, 8, 11, 5, 1, 0]
+    >>> nT('aabbc') == sum(_)
+    True
+
+    >>> [nT("mississippi", i) for i in range(1, 12)]
+    [1, 74, 609, 1521, 1768, 1224, 579, 197, 50, 9, 1]
+
+    Partitions when all items are identical:
+
+    >>> [nT(5, i) for i in range(1, 6)]
+    [1, 2, 2, 1, 1]
+    >>> nT('1'*5) == sum(_)
+    True
+
+    When all items are different:
+
+    >>> [nT(range(5), i) for i in range(1, 6)]
+    [1, 15, 25, 10, 1]
+    >>> nT(range(5)) == sum(_)
+    True
+
+    Partitions of an integer expressed as a sum of positive integers:
+
+    >>> from sympy import partition
+    >>> partition(4)
+    5
+    >>> nT(4, 1) + nT(4, 2) + nT(4, 3) + nT(4, 4)
+    5
+    >>> nT('1'*4)
+    5
+
+    See Also
+    ========
+    sympy.utilities.iterables.partitions
+    sympy.utilities.iterables.multiset_partitions
+    sympy.functions.combinatorial.numbers.partition
+
+    References
+    ==========
+
+    .. [1] https://web.archive.org/web/20210507012732/https://teaching.csse.uwa.edu.au/units/CITS7209/partition.pdf
+
+    """
+
+    if isinstance(n, SYMPY_INTS):
+        # n identical items
+        if k is None:
+            return partition(n)
+        if isinstance(k, SYMPY_INTS):
+            n = as_int(n)
+            k = as_int(k)
+            return Integer(_nT(n, k))
+    if not isinstance(n, _MultisetHistogram):
+        try:
+            # if n contains hashable items there is some
+            # quick handling that can be done
+            u = len(set(n))
+            if u <= 1:
+                return nT(len(n), k)
+            elif u == len(n):
+                n = range(u)
+            raise TypeError
+        except TypeError:
+            n = _multiset_histogram(n)
+    N = n[_N]
+    if k is None and N == 1:
+        return 1
+    if k in (1, N):
+        return 1
+    if k == 2 or N == 2 and k is None:
+        m, r = divmod(N, 2)
+        rv = sum(nC(n, i) for i in range(1, m + 1))
+        if not r:
+            rv -= nC(n, m)//2
+        if k is None:
+            rv += 1  # for k == 1
+        return rv
+    if N == n[_ITEMS]:
+        # all distinct
+        if k is None:
+            return bell(N)
+        return stirling(N, k)
+    m = MultisetPartitionTraverser()
+    if k is None:
+        return m.count_partitions(n[_M])
+    # MultisetPartitionTraverser does not have a range-limited count
+    # method, so need to enumerate and count
+    tot = 0
+    for discard in m.enum_range(n[_M], k-1, k):
+        tot += 1
+    return tot
+
+
+#-----------------------------------------------------------------------------#
+#                                                                             #
+#                          Motzkin numbers                                    #
+#                                                                             #
+#-----------------------------------------------------------------------------#
+
+
+class motzkin(Function):
+    """
+    The nth Motzkin number is the number
+    of ways of drawing non-intersecting chords
+    between n points on a circle (not necessarily touching
+    every point by a chord). The Motzkin numbers are named
+    after Theodore Motzkin and have diverse applications
+    in geometry, combinatorics and number theory.
+
+    Motzkin numbers are the integer sequence defined by the
+    initial terms `M_0 = 1`, `M_1 = 1` and the two-term recurrence relation
+    `M_n = \frac{2*n + 1}{n + 2} * M_{n-1} + \frac{3n - 3}{n + 2} * M_{n-2}`.
+
+
+    Examples
+    ========
+
+    >>> from sympy import motzkin
+
+    >>> motzkin.is_motzkin(5)
+    False
+    >>> motzkin.find_motzkin_numbers_in_range(2,300)
+    [2, 4, 9, 21, 51, 127]
+    >>> motzkin.find_motzkin_numbers_in_range(2,900)
+    [2, 4, 9, 21, 51, 127, 323, 835]
+    >>> motzkin.find_first_n_motzkins(10)
+    [1, 1, 2, 4, 9, 21, 51, 127, 323, 835]
+
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Motzkin_number
+    .. [2] https://mathworld.wolfram.com/MotzkinNumber.html
+
+    """
+
+    @staticmethod
+    def is_motzkin(n):
+        try:
+            n = as_int(n)
+        except ValueError:
+            return False
+        if n > 0:
+             if n in (1, 2):
+                return True
+
+             tn1 = 1
+             tn = 2
+             i = 3
+             while tn < n:
+                 a = ((2*i + 1)*tn + (3*i - 3)*tn1)/(i + 2)
+                 i += 1
+                 tn1 = tn
+                 tn = a
+
+             if tn == n:
+                 return True
+             else:
+                 return False
+
+        else:
+            return False
+
+    @staticmethod
+    def find_motzkin_numbers_in_range(x, y):
+        if 0 <= x <= y:
+            motzkins = []
+            if x <= 1 <= y:
+                motzkins.append(1)
+            tn1 = 1
+            tn = 2
+            i = 3
+            while tn <= y:
+                if tn >= x:
+                    motzkins.append(tn)
+                a = ((2*i + 1)*tn + (3*i - 3)*tn1)/(i + 2)
+                i += 1
+                tn1 = tn
+                tn = int(a)
+
+            return motzkins
+
+        else:
+            raise ValueError('The provided range is not valid. This condition should satisfy x <= y')
+
+    @staticmethod
+    def find_first_n_motzkins(n):
+        try:
+            n = as_int(n)
+        except ValueError:
+            raise ValueError('The provided number must be a positive integer')
+        if n < 0:
+            raise ValueError('The provided number must be a positive integer')
+        motzkins = [1]
+        if n >= 1:
+            motzkins.append(1)
+        tn1 = 1
+        tn = 2
+        i = 3
+        while i <= n:
+            motzkins.append(tn)
+            a = ((2*i + 1)*tn + (3*i - 3)*tn1)/(i + 2)
+            i += 1
+            tn1 = tn
+            tn = int(a)
+
+        return motzkins
+
+    @staticmethod
+    @recurrence_memo([S.One, S.One])
+    def _motzkin(n, prev):
+        return ((2*n + 1)*prev[-1] + (3*n - 3)*prev[-2]) // (n + 2)
+
+    @classmethod
+    def eval(cls, n):
+        try:
+            n = as_int(n)
+        except ValueError:
+            raise ValueError('The provided number must be a positive integer')
+        if n < 0:
+            raise ValueError('The provided number must be a positive integer')
+        return Integer(cls._motzkin(n - 1))
+
+
+def nD(i=None, brute=None, *, n=None, m=None):
+    """return the number of derangements for: ``n`` unique items, ``i``
+    items (as a sequence or multiset), or multiplicities, ``m`` given
+    as a sequence or multiset.
+
+    Examples
+    ========
+
+    >>> from sympy.utilities.iterables import generate_derangements as enum
+    >>> from sympy.functions.combinatorial.numbers import nD
+
+    A derangement ``d`` of sequence ``s`` has all ``d[i] != s[i]``:
+
+    >>> set([''.join(i) for i in enum('abc')])
+    {'bca', 'cab'}
+    >>> nD('abc')
+    2
+
+    Input as iterable or dictionary (multiset form) is accepted:
+
+    >>> assert nD([1, 2, 2, 3, 3, 3]) == nD({1: 1, 2: 2, 3: 3})
+
+    By default, a brute-force enumeration and count of multiset permutations
+    is only done if there are fewer than 9 elements. There may be cases when
+    there is high multiplicity with few unique elements that will benefit
+    from a brute-force enumeration, too. For this reason, the `brute`
+    keyword (default None) is provided. When False, the brute-force
+    enumeration will never be used. When True, it will always be used.
+
+    >>> nD('1111222233', brute=True)
+    44
+
+    For convenience, one may specify ``n`` distinct items using the
+    ``n`` keyword:
+
+    >>> assert nD(n=3) == nD('abc') == 2
+
+    Since the number of derangments depends on the multiplicity of the
+    elements and not the elements themselves, it may be more convenient
+    to give a list or multiset of multiplicities using keyword ``m``:
+
+    >>> assert nD('abc') == nD(m=(1,1,1)) == nD(m={1:3}) == 2
+
+    """
+    from sympy.integrals.integrals import integrate
+    from sympy.functions.special.polynomials import laguerre
+    from sympy.abc import x
+    def ok(x):
+        if not isinstance(x, SYMPY_INTS):
+            raise TypeError('expecting integer values')
+        if x < 0:
+            raise ValueError('value must not be negative')
+        return True
+
+    if (i, n, m).count(None) != 2:
+        raise ValueError('enter only 1 of i, n, or m')
+    if i is not None:
+        if isinstance(i, SYMPY_INTS):
+            raise TypeError('items must be a list or dictionary')
+        if not i:
+            return S.Zero
+        if type(i) is not dict:
+            s = list(i)
+            ms = multiset(s)
+        elif type(i) is dict:
+            all(ok(_) for _ in i.values())
+            ms = {k: v for k, v in i.items() if v}
+            s = None
+        if not ms:
+            return S.Zero
+        N = sum(ms.values())
+        counts = multiset(ms.values())
+        nkey = len(ms)
+    elif n is not None:
+        ok(n)
+        if not n:
+            return S.Zero
+        return subfactorial(n)
+    elif m is not None:
+        if isinstance(m, dict):
+            all(ok(i) and ok(j) for i, j in m.items())
+            counts = {k: v for k, v in m.items() if k*v}
+        elif iterable(m) or isinstance(m, str):
+            m = list(m)
+            all(ok(i) for i in m)
+            counts = multiset([i for i in m if i])
+        else:
+            raise TypeError('expecting iterable')
+        if not counts:
+            return S.Zero
+        N = sum(k*v for k, v in counts.items())
+        nkey = sum(counts.values())
+        s = None
+    big = int(max(counts))
+    if big == 1:  # no repetition
+        return subfactorial(nkey)
+    nval = len(counts)
+    if big*2 > N:
+        return S.Zero
+    if big*2 == N:
+        if nkey == 2 and nval == 1:
+            return S.One  # aaabbb
+        if nkey - 1 == big:  # one element repeated
+            return factorial(big)  # e.g. abc part of abcddd
+    if N < 9 and brute is None or brute:
+        # for all possibilities, this was found to be faster
+        if s is None:
+            s = []
+            i = 0
+            for m, v in counts.items():
+                for j in range(v):
+                    s.extend([i]*m)
+                    i += 1
+        return Integer(sum(1 for i in multiset_derangements(s)))
+    from sympy.functions.elementary.exponential import exp
+    return Integer(abs(integrate(exp(-x)*Mul(*[
+        laguerre(i, x)**m for i, m in counts.items()]), (x, 0, oo))))

llmeval-env/lib/python3.10/site-packages/sympy/functions/combinatorial/tests/test_comb_factorials.py

@@ -0,0 +1,650 @@
+from sympy.concrete.products import Product
+from sympy.core.function import expand_func
+from sympy.core.mod import Mod
+from sympy.core.mul import Mul
+from sympy.core import EulerGamma
+from sympy.core.numbers import (Float, I, Rational, nan, oo, pi, zoo)
+from sympy.core.relational import Eq
+from sympy.core.singleton import S
+from sympy.core.symbol import (Dummy, Symbol, symbols)
+from sympy.functions.combinatorial.factorials import (ff, rf, binomial, factorial, factorial2)
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.piecewise import Piecewise
+from sympy.functions.special.gamma_functions import (gamma, polygamma)
+from sympy.polys.polytools import Poly
+from sympy.series.order import O
+from sympy.simplify.simplify import simplify
+from sympy.core.expr import unchanged
+from sympy.core.function import ArgumentIndexError
+from sympy.functions.combinatorial.factorials import subfactorial
+from sympy.functions.special.gamma_functions import uppergamma
+from sympy.testing.pytest import XFAIL, raises, slow
+
+#Solves and Fixes Issue #10388 - This is the updated test for the same solved issue
+
+def test_rf_eval_apply():
+    x, y = symbols('x,y')
+    n, k = symbols('n k', integer=True)
+    m = Symbol('m', integer=True, nonnegative=True)
+
+    assert rf(nan, y) is nan
+    assert rf(x, nan) is nan
+
+    assert unchanged(rf, x, y)
+
+    assert rf(oo, 0) == 1
+    assert rf(-oo, 0) == 1
+
+    assert rf(oo, 6) is oo
+    assert rf(-oo, 7) is -oo
+    assert rf(-oo, 6) is oo
+
+    assert rf(oo, -6) is oo
+    assert rf(-oo, -7) is oo
+
+    assert rf(-1, pi) == 0
+    assert rf(-5, 1 + I) == 0
+
+    assert unchanged(rf, -3, k)
+    assert unchanged(rf, x, Symbol('k', integer=False))
+    assert rf(-3, Symbol('k', integer=False)) == 0
+    assert rf(Symbol('x', negative=True, integer=True), Symbol('k', integer=False)) == 0
+
+    assert rf(x, 0) == 1
+    assert rf(x, 1) == x
+    assert rf(x, 2) == x*(x + 1)
+    assert rf(x, 3) == x*(x + 1)*(x + 2)
+    assert rf(x, 5) == x*(x + 1)*(x + 2)*(x + 3)*(x + 4)
+
+    assert rf(x, -1) == 1/(x - 1)
+    assert rf(x, -2) == 1/((x - 1)*(x - 2))
+    assert rf(x, -3) == 1/((x - 1)*(x - 2)*(x - 3))
+
+    assert rf(1, 100) == factorial(100)
+
+    assert rf(x**2 + 3*x, 2) == (x**2 + 3*x)*(x**2 + 3*x + 1)
+    assert isinstance(rf(x**2 + 3*x, 2), Mul)
+    assert rf(x**3 + x, -2) == 1/((x**3 + x - 1)*(x**3 + x - 2))
+
+    assert rf(Poly(x**2 + 3*x, x), 2) == Poly(x**4 + 8*x**3 + 19*x**2 + 12*x, x)
+    assert isinstance(rf(Poly(x**2 + 3*x, x), 2), Poly)
+    raises(ValueError, lambda: rf(Poly(x**2 + 3*x, x, y), 2))
+    assert rf(Poly(x**3 + x, x), -2) == 1/(x**6 - 9*x**5 + 35*x**4 - 75*x**3 + 94*x**2 - 66*x + 20)
+    raises(ValueError, lambda: rf(Poly(x**3 + x, x, y), -2))
+
+    assert rf(x, m).is_integer is None
+    assert rf(n, k).is_integer is None
+    assert rf(n, m).is_integer is True
+    assert rf(n, k + pi).is_integer is False
+    assert rf(n, m + pi).is_integer is False
+    assert rf(pi, m).is_integer is False
+
+    def check(x, k, o, n):
+        a, b = Dummy(), Dummy()
+        r = lambda x, k: o(a, b).rewrite(n).subs({a:x,b:k})
+        for i in range(-5,5):
+          for j in range(-5,5):
+              assert o(i, j) == r(i, j), (o, n, i, j)
+    check(x, k, rf, ff)
+    check(x, k, rf, binomial)
+    check(n, k, rf, factorial)
+    check(x, y, rf, factorial)
+    check(x, y, rf, binomial)
+
+    assert rf(x, k).rewrite(ff) == ff(x + k - 1, k)
+    assert rf(x, k).rewrite(gamma) == Piecewise(
+        (gamma(k + x)/gamma(x), x > 0),
+        ((-1)**k*gamma(1 - x)/gamma(-k - x + 1), True))
+    assert rf(5, k).rewrite(gamma) == gamma(k + 5)/24
+    assert rf(x, k).rewrite(binomial) == factorial(k)*binomial(x + k - 1, k)
+    assert rf(n, k).rewrite(factorial) == Piecewise(
+        (factorial(k + n - 1)/factorial(n - 1), n > 0),
+        ((-1)**k*factorial(-n)/factorial(-k - n), True))
+    assert rf(5, k).rewrite(factorial) == factorial(k + 4)/24
+    assert rf(x, y).rewrite(factorial) == rf(x, y)
+    assert rf(x, y).rewrite(binomial) == rf(x, y)
+
+    import random
+    from mpmath import rf as mpmath_rf
+    for i in range(100):
+        x = -500 + 500 * random.random()
+        k = -500 + 500 * random.random()
+        assert (abs(mpmath_rf(x, k) - rf(x, k)) < 10**(-15))
+
+
+def test_ff_eval_apply():
+    x, y = symbols('x,y')
+    n, k = symbols('n k', integer=True)
+    m = Symbol('m', integer=True, nonnegative=True)
+
+    assert ff(nan, y) is nan
+    assert ff(x, nan) is nan
+
+    assert unchanged(ff, x, y)
+
+    assert ff(oo, 0) == 1
+    assert ff(-oo, 0) == 1
+
+    assert ff(oo, 6) is oo
+    assert ff(-oo, 7) is -oo
+    assert ff(-oo, 6) is oo
+
+    assert ff(oo, -6) is oo
+    assert ff(-oo, -7) is oo
+
+    assert ff(x, 0) == 1
+    assert ff(x, 1) == x
+    assert ff(x, 2) == x*(x - 1)
+    assert ff(x, 3) == x*(x - 1)*(x - 2)
+    assert ff(x, 5) == x*(x - 1)*(x - 2)*(x - 3)*(x - 4)
+
+    assert ff(x, -1) == 1/(x + 1)
+    assert ff(x, -2) == 1/((x + 1)*(x + 2))
+    assert ff(x, -3) == 1/((x + 1)*(x + 2)*(x + 3))
+
+    assert ff(100, 100) == factorial(100)
+
+    assert ff(2*x**2 - 5*x, 2) == (2*x**2  - 5*x)*(2*x**2 - 5*x - 1)
+    assert isinstance(ff(2*x**2 - 5*x, 2), Mul)
+    assert ff(x**2 + 3*x, -2) == 1/((x**2 + 3*x + 1)*(x**2 + 3*x + 2))
+
+    assert ff(Poly(2*x**2 - 5*x, x), 2) == Poly(4*x**4 - 28*x**3 + 59*x**2 - 35*x, x)
+    assert isinstance(ff(Poly(2*x**2 - 5*x, x), 2), Poly)
+    raises(ValueError, lambda: ff(Poly(2*x**2 - 5*x, x, y), 2))
+    assert ff(Poly(x**2 + 3*x, x), -2) == 1/(x**4 + 12*x**3 + 49*x**2 + 78*x + 40)
+    raises(ValueError, lambda: ff(Poly(x**2 + 3*x, x, y), -2))
+
+
+    assert ff(x, m).is_integer is None
+    assert ff(n, k).is_integer is None
+    assert ff(n, m).is_integer is True
+    assert ff(n, k + pi).is_integer is False
+    assert ff(n, m + pi).is_integer is False
+    assert ff(pi, m).is_integer is False
+
+    assert isinstance(ff(x, x), ff)
+    assert ff(n, n) == factorial(n)
+
+    def check(x, k, o, n):
+        a, b = Dummy(), Dummy()
+        r = lambda x, k: o(a, b).rewrite(n).subs({a:x,b:k})
+        for i in range(-5,5):
+          for j in range(-5,5):
+              assert o(i, j) == r(i, j), (o, n)
+    check(x, k, ff, rf)
+    check(x, k, ff, gamma)
+    check(n, k, ff, factorial)
+    check(x, k, ff, binomial)
+    check(x, y, ff, factorial)
+    check(x, y, ff, binomial)
+
+    assert ff(x, k).rewrite(rf) == rf(x - k + 1, k)
+    assert ff(x, k).rewrite(gamma) == Piecewise(
+        (gamma(x + 1)/gamma(-k + x + 1), x >= 0),
+        ((-1)**k*gamma(k - x)/gamma(-x), True))
+    assert ff(5, k).rewrite(gamma) == 120/gamma(6 - k)
+    assert ff(n, k).rewrite(factorial) == Piecewise(
+        (factorial(n)/factorial(-k + n), n >= 0),
+        ((-1)**k*factorial(k - n - 1)/factorial(-n - 1), True))
+    assert ff(5, k).rewrite(factorial) == 120/factorial(5 - k)
+    assert ff(x, k).rewrite(binomial) == factorial(k) * binomial(x, k)
+    assert ff(x, y).rewrite(factorial) == ff(x, y)
+    assert ff(x, y).rewrite(binomial) == ff(x, y)
+
+    import random
+    from mpmath import ff as mpmath_ff
+    for i in range(100):
+        x = -500 + 500 * random.random()
+        k = -500 + 500 * random.random()
+        a = mpmath_ff(x, k)
+        b = ff(x, k)
+        assert (abs(a - b) < abs(a) * 10**(-15))
+
+
+def test_rf_ff_eval_hiprec():
+    maple = Float('6.9109401292234329956525265438452')
+    us = ff(18, Rational(2, 3)).evalf(32)
+    assert abs(us - maple)/us < 1e-31
+
+    maple = Float('6.8261540131125511557924466355367')
+    us = rf(18, Rational(2, 3)).evalf(32)
+    assert abs(us - maple)/us < 1e-31
+
+    maple = Float('34.007346127440197150854651814225')
+    us = rf(Float('4.4', 32), Float('2.2', 32));
+    assert abs(us - maple)/us < 1e-31
+
+
+def test_rf_lambdify_mpmath():
+    from sympy.utilities.lambdify import lambdify
+    x, y = symbols('x,y')
+    f = lambdify((x,y), rf(x, y), 'mpmath')
+    maple = Float('34.007346127440197')
+    us = f(4.4, 2.2)
+    assert abs(us - maple)/us < 1e-15
+
+
+def test_factorial():
+    x = Symbol('x')
+    n = Symbol('n', integer=True)
+    k = Symbol('k', integer=True, nonnegative=True)
+    r = Symbol('r', integer=False)
+    s = Symbol('s', integer=False, negative=True)
+    t = Symbol('t', nonnegative=True)
+    u = Symbol('u', noninteger=True)
+
+    assert factorial(-2) is zoo
+    assert factorial(0) == 1
+    assert factorial(7) == 5040
+    assert factorial(19) == 121645100408832000
+    assert factorial(31) == 8222838654177922817725562880000000
+    assert factorial(n).func == factorial
+    assert factorial(2*n).func == factorial
+
+    assert factorial(x).is_integer is None
+    assert factorial(n).is_integer is None
+    assert factorial(k).is_integer
+    assert factorial(r).is_integer is None
+
+    assert factorial(n).is_positive is None
+    assert factorial(k).is_positive
+
+    assert factorial(x).is_real is None
+    assert factorial(n).is_real is None
+    assert factorial(k).is_real is True
+    assert factorial(r).is_real is None
+    assert factorial(s).is_real is True
+    assert factorial(t).is_real is True
+    assert factorial(u).is_real is True
+
+    assert factorial(x).is_composite is None
+    assert factorial(n).is_composite is None
+    assert factorial(k).is_composite is None
+    assert factorial(k + 3).is_composite is True
+    assert factorial(r).is_composite is None
+    assert factorial(s).is_composite is None
+    assert factorial(t).is_composite is None
+    assert factorial(u).is_composite is None
+
+    assert factorial(oo) is oo
+
+
+def test_factorial_Mod():
+    pr = Symbol('pr', prime=True)
+    p, q = 10**9 + 9, 10**9 + 33 # prime modulo
+    r, s = 10**7 + 5, 33333333 # composite modulo
+    assert Mod(factorial(pr - 1), pr) == pr - 1
+    assert Mod(factorial(pr - 1), -pr) == -1
+    assert Mod(factorial(r - 1, evaluate=False), r) == 0
+    assert Mod(factorial(s - 1, evaluate=False), s) == 0
+    assert Mod(factorial(p - 1, evaluate=False), p) == p - 1
+    assert Mod(factorial(q - 1, evaluate=False), q) == q - 1
+    assert Mod(factorial(p - 50, evaluate=False), p) == 854928834
+    assert Mod(factorial(q - 1800, evaluate=False), q) == 905504050
+    assert Mod(factorial(153, evaluate=False), r) == Mod(factorial(153), r)
+    assert Mod(factorial(255, evaluate=False), s) == Mod(factorial(255), s)
+    assert Mod(factorial(4, evaluate=False), 3) == S.Zero
+    assert Mod(factorial(5, evaluate=False), 6) == S.Zero
+
+
+def test_factorial_diff():
+    n = Symbol('n', integer=True)
+
+    assert factorial(n).diff(n) == \
+        gamma(1 + n)*polygamma(0, 1 + n)
+    assert factorial(n**2).diff(n) == \
+        2*n*gamma(1 + n**2)*polygamma(0, 1 + n**2)
+    raises(ArgumentIndexError, lambda: factorial(n**2).fdiff(2))
+
+
+def test_factorial_series():
+    n = Symbol('n', integer=True)
+
+    assert factorial(n).series(n, 0, 3) == \
+        1 - n*EulerGamma + n**2*(EulerGamma**2/2 + pi**2/12) + O(n**3)
+
+
+def test_factorial_rewrite():
+    n = Symbol('n', integer=True)
+    k = Symbol('k', integer=True, nonnegative=True)
+
+    assert factorial(n).rewrite(gamma) == gamma(n + 1)
+    _i = Dummy('i')
+    assert factorial(k).rewrite(Product).dummy_eq(Product(_i, (_i, 1, k)))
+    assert factorial(n).rewrite(Product) == factorial(n)
+
+
+def test_factorial2():
+    n = Symbol('n', integer=True)
+
+    assert factorial2(-1) == 1
+    assert factorial2(0) == 1
+    assert factorial2(7) == 105
+    assert factorial2(8) == 384
+
+    # The following is exhaustive
+    tt = Symbol('tt', integer=True, nonnegative=True)
+    tte = Symbol('tte', even=True, nonnegative=True)
+    tpe = Symbol('tpe', even=True, positive=True)
+    tto = Symbol('tto', odd=True, nonnegative=True)
+    tf = Symbol('tf', integer=True, nonnegative=False)
+    tfe = Symbol('tfe', even=True, nonnegative=False)
+    tfo = Symbol('tfo', odd=True, nonnegative=False)
+    ft = Symbol('ft', integer=False, nonnegative=True)
+    ff = Symbol('ff', integer=False, nonnegative=False)
+    fn = Symbol('fn', integer=False)
+    nt = Symbol('nt', nonnegative=True)
+    nf = Symbol('nf', nonnegative=False)
+    nn = Symbol('nn')
+    z = Symbol('z', zero=True)
+    #Solves and Fixes Issue #10388 - This is the updated test for the same solved issue
+    raises(ValueError, lambda: factorial2(oo))
+    raises(ValueError, lambda: factorial2(Rational(5, 2)))
+    raises(ValueError, lambda: factorial2(-4))
+    assert factorial2(n).is_integer is None
+    assert factorial2(tt - 1).is_integer
+    assert factorial2(tte - 1).is_integer
+    assert factorial2(tpe - 3).is_integer
+    assert factorial2(tto - 4).is_integer
+    assert factorial2(tto - 2).is_integer
+    assert factorial2(tf).is_integer is None
+    assert factorial2(tfe).is_integer is None
+    assert factorial2(tfo).is_integer is None
+    assert factorial2(ft).is_integer is None
+    assert factorial2(ff).is_integer is None
+    assert factorial2(fn).is_integer is None
+    assert factorial2(nt).is_integer is None
+    assert factorial2(nf).is_integer is None
+    assert factorial2(nn).is_integer is None
+
+    assert factorial2(n).is_positive is None
+    assert factorial2(tt - 1).is_positive is True
+    assert factorial2(tte - 1).is_positive is True
+    assert factorial2(tpe - 3).is_positive is True
+    assert factorial2(tpe - 1).is_positive is True
+    assert factorial2(tto - 2).is_positive is True
+    assert factorial2(tto - 1).is_positive is True
+    assert factorial2(tf).is_positive is None
+    assert factorial2(tfe).is_positive is None
+    assert factorial2(tfo).is_positive is None
+    assert factorial2(ft).is_positive is None
+    assert factorial2(ff).is_positive is None
+    assert factorial2(fn).is_positive is None
+    assert factorial2(nt).is_positive is None
+    assert factorial2(nf).is_positive is None
+    assert factorial2(nn).is_positive is None
+
+    assert factorial2(tt).is_even is None
+    assert factorial2(tt).is_odd is None
+    assert factorial2(tte).is_even is None
+    assert factorial2(tte).is_odd is None
+    assert factorial2(tte + 2).is_even is True
+    assert factorial2(tpe).is_even is True
+    assert factorial2(tpe).is_odd is False
+    assert factorial2(tto).is_odd is True
+    assert factorial2(tf).is_even is None
+    assert factorial2(tf).is_odd is None
+    assert factorial2(tfe).is_even is None
+    assert factorial2(tfe).is_odd is None
+    assert factorial2(tfo).is_even is False
+    assert factorial2(tfo).is_odd is None
+    assert factorial2(z).is_even is False
+    assert factorial2(z).is_odd is True
+
+
+def test_factorial2_rewrite():
+    n = Symbol('n', integer=True)
+    assert factorial2(n).rewrite(gamma) == \
+        2**(n/2)*Piecewise((1, Eq(Mod(n, 2), 0)), (sqrt(2)/sqrt(pi), Eq(Mod(n, 2), 1)))*gamma(n/2 + 1)
+    assert factorial2(2*n).rewrite(gamma) == 2**n*gamma(n + 1)
+    assert factorial2(2*n + 1).rewrite(gamma) == \
+        sqrt(2)*2**(n + S.Half)*gamma(n + Rational(3, 2))/sqrt(pi)
+
+
+def test_binomial():
+    x = Symbol('x')
+    n = Symbol('n', integer=True)
+    nz = Symbol('nz', integer=True, nonzero=True)
+    k = Symbol('k', integer=True)
+    kp = Symbol('kp', integer=True, positive=True)
+    kn = Symbol('kn', integer=True, negative=True)
+    u = Symbol('u', negative=True)
+    v = Symbol('v', nonnegative=True)
+    p = Symbol('p', positive=True)
+    z = Symbol('z', zero=True)
+    nt = Symbol('nt', integer=False)
+    kt = Symbol('kt', integer=False)
+    a = Symbol('a', integer=True, nonnegative=True)
+    b = Symbol('b', integer=True, nonnegative=True)
+
+    assert binomial(0, 0) == 1
+    assert binomial(1, 1) == 1
+    assert binomial(10, 10) == 1
+    assert binomial(n, z) == 1
+    assert binomial(1, 2) == 0
+    assert binomial(-1, 2) == 1
+    assert binomial(1, -1) == 0
+    assert binomial(-1, 1) == -1
+    assert binomial(-1, -1) == 0
+    assert binomial(S.Half, S.Half) == 1
+    assert binomial(-10, 1) == -10
+    assert binomial(-10, 7) == -11440
+    assert binomial(n, -1) == 0 # holds for all integers (negative, zero, positive)
+    assert binomial(kp, -1) == 0
+    assert binomial(nz, 0) == 1
+    assert expand_func(binomial(n, 1)) == n
+    assert expand_func(binomial(n, 2)) == n*(n - 1)/2
+    assert expand_func(binomial(n, n - 2)) == n*(n - 1)/2
+    assert expand_func(binomial(n, n - 1)) == n
+    assert binomial(n, 3).func == binomial
+    assert binomial(n, 3).expand(func=True) ==  n**3/6 - n**2/2 + n/3
+    assert expand_func(binomial(n, 3)) ==  n*(n - 2)*(n - 1)/6
+    assert binomial(n, n).func == binomial # e.g. (-1, -1) == 0, (2, 2) == 1
+    assert binomial(n, n + 1).func == binomial  # e.g. (-1, 0) == 1
+    assert binomial(kp, kp + 1) == 0
+    assert binomial(kn, kn) == 0 # issue #14529
+    assert binomial(n, u).func == binomial
+    assert binomial(kp, u).func == binomial
+    assert binomial(n, p).func == binomial
+    assert binomial(n, k).func == binomial
+    assert binomial(n, n + p).func == binomial
+    assert binomial(kp, kp + p).func == binomial
+
+    assert expand_func(binomial(n, n - 3)) == n*(n - 2)*(n - 1)/6
+
+    assert binomial(n, k).is_integer
+    assert binomial(nt, k).is_integer is None
+    assert binomial(x, nt).is_integer is False
+
+    assert binomial(gamma(25), 6) == 79232165267303928292058750056084441948572511312165380965440075720159859792344339983120618959044048198214221915637090855535036339620413440000
+    assert binomial(1324, 47) == 906266255662694632984994480774946083064699457235920708992926525848438478406790323869952
+    assert binomial(1735, 43) == 190910140420204130794758005450919715396159959034348676124678207874195064798202216379800
+    assert binomial(2512, 53) == 213894469313832631145798303740098720367984955243020898718979538096223399813295457822575338958939834177325304000
+    assert binomial(3383, 52) == 27922807788818096863529701501764372757272890613101645521813434902890007725667814813832027795881839396839287659777235
+    assert binomial(4321, 51) == 124595639629264868916081001263541480185227731958274383287107643816863897851139048158022599533438936036467601690983780576
+
+    assert binomial(a, b).is_nonnegative is True
+    assert binomial(-1, 2, evaluate=False).is_nonnegative is True
+    assert binomial(10, 5, evaluate=False).is_nonnegative is True
+    assert binomial(10, -3, evaluate=False).is_nonnegative is True
+    assert binomial(-10, -3, evaluate=False).is_nonnegative is True
+    assert binomial(-10, 2, evaluate=False).is_nonnegative is True
+    assert binomial(-10, 1, evaluate=False).is_nonnegative is False
+    assert binomial(-10, 7, evaluate=False).is_nonnegative is False
+
+    # issue #14625
+    for _ in (pi, -pi, nt, v, a):
+        assert binomial(_, _) == 1
+        assert binomial(_, _ - 1) == _
+    assert isinstance(binomial(u, u), binomial)
+    assert isinstance(binomial(u, u - 1), binomial)
+    assert isinstance(binomial(x, x), binomial)
+    assert isinstance(binomial(x, x - 1), binomial)
+
+    #issue #18802
+    assert expand_func(binomial(x + 1, x)) == x + 1
+    assert expand_func(binomial(x, x - 1)) == x
+    assert expand_func(binomial(x + 1, x - 1)) == x*(x + 1)/2
+    assert expand_func(binomial(x**2 + 1, x**2)) == x**2 + 1
+
+    # issue #13980 and #13981
+    assert binomial(-7, -5) == 0
+    assert binomial(-23, -12) == 0
+    assert binomial(Rational(13, 2), -10) == 0
+    assert binomial(-49, -51) == 0
+
+    assert binomial(19, Rational(-7, 2)) == S(-68719476736)/(911337863661225*pi)
+    assert binomial(0, Rational(3, 2)) == S(-2)/(3*pi)
+    assert binomial(-3, Rational(-7, 2)) is zoo
+    assert binomial(kn, kt) is zoo
+
+    assert binomial(nt, kt).func == binomial
+    assert binomial(nt, Rational(15, 6)) == 8*gamma(nt + 1)/(15*sqrt(pi)*gamma(nt - Rational(3, 2)))
+    assert binomial(Rational(20, 3), Rational(-10, 8)) == gamma(Rational(23, 3))/(gamma(Rational(-1, 4))*gamma(Rational(107, 12)))
+    assert binomial(Rational(19, 2), Rational(-7, 2)) == Rational(-1615, 8388608)
+    assert binomial(Rational(-13, 5), Rational(-7, 8)) == gamma(Rational(-8, 5))/(gamma(Rational(-29, 40))*gamma(Rational(1, 8)))
+    assert binomial(Rational(-19, 8), Rational(-13, 5)) == gamma(Rational(-11, 8))/(gamma(Rational(-8, 5))*gamma(Rational(49, 40)))
+
+    # binomial for complexes
+    assert binomial(I, Rational(-89, 8)) == gamma(1 + I)/(gamma(Rational(-81, 8))*gamma(Rational(97, 8) + I))
+    assert binomial(I, 2*I) == gamma(1 + I)/(gamma(1 - I)*gamma(1 + 2*I))
+    assert binomial(-7, I) is zoo
+    assert binomial(Rational(-7, 6), I) == gamma(Rational(-1, 6))/(gamma(Rational(-1, 6) - I)*gamma(1 + I))
+    assert binomial((1+2*I), (1+3*I)) == gamma(2 + 2*I)/(gamma(1 - I)*gamma(2 + 3*I))
+    assert binomial(I, 5) == Rational(1, 3) - I/S(12)
+    assert binomial((2*I + 3), 7) == -13*I/S(63)
+    assert isinstance(binomial(I, n), binomial)
+    assert expand_func(binomial(3, 2, evaluate=False)) == 3
+    assert expand_func(binomial(n, 0, evaluate=False)) == 1
+    assert expand_func(binomial(n, -2, evaluate=False)) == 0
+    assert expand_func(binomial(n, k)) == binomial(n, k)
+
+
+def test_binomial_Mod():
+    p, q = 10**5 + 3, 10**9 + 33 # prime modulo
+    r = 10**7 + 5 # composite modulo
+
+    # A few tests to get coverage
+    # Lucas Theorem
+    assert Mod(binomial(156675, 4433, evaluate=False), p) == Mod(binomial(156675, 4433), p)
+
+    # factorial Mod
+    assert Mod(binomial(1234, 432, evaluate=False), q) == Mod(binomial(1234, 432), q)
+
+    # binomial factorize
+    assert Mod(binomial(253, 113, evaluate=False), r) == Mod(binomial(253, 113), r)
+
+
+@slow
+def test_binomial_Mod_slow():
+    p, q = 10**5 + 3, 10**9 + 33 # prime modulo
+    r, s = 10**7 + 5, 33333333 # composite modulo
+
+    n, k, m = symbols('n k m')
+    assert (binomial(n, k) % q).subs({n: s, k: p}) == Mod(binomial(s, p), q)
+    assert (binomial(n, k) % m).subs({n: 8, k: 5, m: 13}) == 4
+    assert (binomial(9, k) % 7).subs(k, 2) == 1
+
+    # Lucas Theorem
+    assert Mod(binomial(123456, 43253, evaluate=False), p) == Mod(binomial(123456, 43253), p)
+    assert Mod(binomial(-178911, 237, evaluate=False), p) == Mod(-binomial(178911 + 237 - 1, 237), p)
+    assert Mod(binomial(-178911, 238, evaluate=False), p) == Mod(binomial(178911 + 238 - 1, 238), p)
+
+    # factorial Mod
+    assert Mod(binomial(9734, 451, evaluate=False), q) == Mod(binomial(9734, 451), q)
+    assert Mod(binomial(-10733, 4459, evaluate=False), q) == Mod(binomial(-10733, 4459), q)
+    assert Mod(binomial(-15733, 4458, evaluate=False), q) == Mod(binomial(-15733, 4458), q)
+    assert Mod(binomial(23, -38, evaluate=False), q) is S.Zero
+    assert Mod(binomial(23, 38, evaluate=False), q) is S.Zero
+
+    # binomial factorize
+    assert Mod(binomial(753, 119, evaluate=False), r) == Mod(binomial(753, 119), r)
+    assert Mod(binomial(3781, 948, evaluate=False), s) == Mod(binomial(3781, 948), s)
+    assert Mod(binomial(25773, 1793, evaluate=False), s) == Mod(binomial(25773, 1793), s)
+    assert Mod(binomial(-753, 118, evaluate=False), r) == Mod(binomial(-753, 118), r)
+    assert Mod(binomial(-25773, 1793, evaluate=False), s) == Mod(binomial(-25773, 1793), s)
+
+
+def test_binomial_diff():
+    n = Symbol('n', integer=True)
+    k = Symbol('k', integer=True)
+
+    assert binomial(n, k).diff(n) == \
+        (-polygamma(0, 1 + n - k) + polygamma(0, 1 + n))*binomial(n, k)
+    assert binomial(n**2, k**3).diff(n) == \
+        2*n*(-polygamma(
+            0, 1 + n**2 - k**3) + polygamma(0, 1 + n**2))*binomial(n**2, k**3)
+
+    assert binomial(n, k).diff(k) == \
+        (-polygamma(0, 1 + k) + polygamma(0, 1 + n - k))*binomial(n, k)
+    assert binomial(n**2, k**3).diff(k) == \
+        3*k**2*(-polygamma(
+            0, 1 + k**3) + polygamma(0, 1 + n**2 - k**3))*binomial(n**2, k**3)
+    raises(ArgumentIndexError, lambda: binomial(n, k).fdiff(3))
+
+
+def test_binomial_rewrite():
+    n = Symbol('n', integer=True)
+    k = Symbol('k', integer=True)
+    x = Symbol('x')
+
+    assert binomial(n, k).rewrite(
+        factorial) == factorial(n)/(factorial(k)*factorial(n - k))
+    assert binomial(
+        n, k).rewrite(gamma) == gamma(n + 1)/(gamma(k + 1)*gamma(n - k + 1))
+    assert binomial(n, k).rewrite(ff) == ff(n, k) / factorial(k)
+    assert binomial(n, x).rewrite(ff) == binomial(n, x)
+
+
+@XFAIL
+def test_factorial_simplify_fail():
+    # simplify(factorial(x + 1).diff(x) - ((x + 1)*factorial(x)).diff(x))) == 0
+    from sympy.abc import x
+    assert simplify(x*polygamma(0, x + 1) - x*polygamma(0, x + 2) +
+                    polygamma(0, x + 1) - polygamma(0, x + 2) + 1) == 0
+
+
+def test_subfactorial():
+    assert all(subfactorial(i) == ans for i, ans in enumerate(
+        [1, 0, 1, 2, 9, 44, 265, 1854, 14833, 133496]))
+    assert subfactorial(oo) is oo
+    assert subfactorial(nan) is nan
+    assert subfactorial(23) == 9510425471055777937262
+    assert unchanged(subfactorial, 2.2)
+
+    x = Symbol('x')
+    assert subfactorial(x).rewrite(uppergamma) == uppergamma(x + 1, -1)/S.Exp1
+
+    tt = Symbol('tt', integer=True, nonnegative=True)
+    tf = Symbol('tf', integer=True, nonnegative=False)
+    tn = Symbol('tf', integer=True)
+    ft = Symbol('ft', integer=False, nonnegative=True)
+    ff = Symbol('ff', integer=False, nonnegative=False)
+    fn = Symbol('ff', integer=False)
+    nt = Symbol('nt', nonnegative=True)
+    nf = Symbol('nf', nonnegative=False)
+    nn = Symbol('nf')
+    te = Symbol('te', even=True, nonnegative=True)
+    to = Symbol('to', odd=True, nonnegative=True)
+    assert subfactorial(tt).is_integer
+    assert subfactorial(tf).is_integer is None
+    assert subfactorial(tn).is_integer is None
+    assert subfactorial(ft).is_integer is None
+    assert subfactorial(ff).is_integer is None
+    assert subfactorial(fn).is_integer is None
+    assert subfactorial(nt).is_integer is None
+    assert subfactorial(nf).is_integer is None
+    assert subfactorial(nn).is_integer is None
+    assert subfactorial(tt).is_nonnegative
+    assert subfactorial(tf).is_nonnegative is None
+    assert subfactorial(tn).is_nonnegative is None
+    assert subfactorial(ft).is_nonnegative is None
+    assert subfactorial(ff).is_nonnegative is None
+    assert subfactorial(fn).is_nonnegative is None
+    assert subfactorial(nt).is_nonnegative is None
+    assert subfactorial(nf).is_nonnegative is None
+    assert subfactorial(nn).is_nonnegative is None
+    assert subfactorial(tt).is_even is None
+    assert subfactorial(tt).is_odd is None
+    assert subfactorial(te).is_odd is True
+    assert subfactorial(to).is_even is True

llmeval-env/lib/python3.10/site-packages/sympy/functions/combinatorial/tests/test_comb_numbers.py

@@ -0,0 +1,852 @@
+import string
+
+from sympy.concrete.products import Product
+from sympy.concrete.summations import Sum
+from sympy.core.function import (diff, expand_func)
+from sympy.core import (EulerGamma, TribonacciConstant)
+from sympy.core.numbers import (Float, I, Rational, oo, pi)
+from sympy.core.singleton import S
+from sympy.core.symbol import (Dummy, Symbol, symbols)
+from sympy.functions.combinatorial.numbers import carmichael
+from sympy.functions.elementary.complexes import (im, re)
+from sympy.functions.elementary.integers import floor
+from sympy.polys.polytools import cancel
+from sympy.series.limits import limit, Limit
+from sympy.series.order import O
+from sympy.functions import (
+    bernoulli, harmonic, bell, fibonacci, tribonacci, lucas, euler, catalan,
+    genocchi, andre, partition, motzkin, binomial, gamma, sqrt, cbrt, hyper, log, digamma,
+    trigamma, polygamma, factorial, sin, cos, cot, polylog, zeta, dirichlet_eta)
+from sympy.functions.combinatorial.numbers import _nT
+
+from sympy.core.expr import unchanged
+from sympy.core.numbers import GoldenRatio, Integer
+
+from sympy.testing.pytest import raises, nocache_fail, warns_deprecated_sympy
+from sympy.abc import x
+
+
+def test_carmichael():
+    assert carmichael.find_carmichael_numbers_in_range(0, 561) == []
+    assert carmichael.find_carmichael_numbers_in_range(561, 562) == [561]
+    assert carmichael.find_carmichael_numbers_in_range(561, 1105) == carmichael.find_carmichael_numbers_in_range(561,
+                                                                                                                 562)
+    assert carmichael.find_first_n_carmichaels(5) == [561, 1105, 1729, 2465, 2821]
+    raises(ValueError, lambda: carmichael.is_carmichael(-2))
+    raises(ValueError, lambda: carmichael.find_carmichael_numbers_in_range(-2, 2))
+    raises(ValueError, lambda: carmichael.find_carmichael_numbers_in_range(22, 2))
+    with warns_deprecated_sympy():
+        assert carmichael.is_prime(2821) == False
+
+
+def test_bernoulli():
+    assert bernoulli(0) == 1
+    assert bernoulli(1) == Rational(1, 2)
+    assert bernoulli(2) == Rational(1, 6)
+    assert bernoulli(3) == 0
+    assert bernoulli(4) == Rational(-1, 30)
+    assert bernoulli(5) == 0
+    assert bernoulli(6) == Rational(1, 42)
+    assert bernoulli(7) == 0
+    assert bernoulli(8) == Rational(-1, 30)
+    assert bernoulli(10) == Rational(5, 66)
+    assert bernoulli(1000001) == 0
+
+    assert bernoulli(0, x) == 1
+    assert bernoulli(1, x) == x - S.Half
+    assert bernoulli(2, x) == x**2 - x + Rational(1, 6)
+    assert bernoulli(3, x) == x**3 - (3*x**2)/2 + x/2
+
+    # Should be fast; computed with mpmath
+    b = bernoulli(1000)
+    assert b.p % 10**10 == 7950421099
+    assert b.q == 342999030
+
+    b = bernoulli(10**6, evaluate=False).evalf()
+    assert str(b) == '-2.23799235765713e+4767529'
+
+    # Issue #8527
+    l = Symbol('l', integer=True)
+    m = Symbol('m', integer=True, nonnegative=True)
+    n = Symbol('n', integer=True, positive=True)
+    assert isinstance(bernoulli(2 * l + 1), bernoulli)
+    assert isinstance(bernoulli(2 * m + 1), bernoulli)
+    assert bernoulli(2 * n + 1) == 0
+
+    assert bernoulli(x, 1) == bernoulli(x)
+
+    assert str(bernoulli(0.0, 2.3).evalf(n=10)) == '1.000000000'
+    assert str(bernoulli(1.0).evalf(n=10)) == '0.5000000000'
+    assert str(bernoulli(1.2).evalf(n=10)) == '0.4195995367'
+    assert str(bernoulli(1.2, 0.8).evalf(n=10)) == '0.2144830348'
+    assert str(bernoulli(1.2, -0.8).evalf(n=10)) == '-1.158865646 - 0.6745558744*I'
+    assert str(bernoulli(3.0, 1j).evalf(n=10)) == '1.5 - 0.5*I'
+    assert str(bernoulli(I).evalf(n=10)) == '0.9268485643 - 0.5821580598*I'
+    assert str(bernoulli(I, I).evalf(n=10)) == '0.1267792071 + 0.01947413152*I'
+    assert bernoulli(x).evalf() == bernoulli(x)
+
+
+def test_bernoulli_rewrite():
+    from sympy.functions.elementary.piecewise import Piecewise
+    n = Symbol('n', integer=True, nonnegative=True)
+
+    assert bernoulli(-1).rewrite(zeta) == pi**2/6
+    assert bernoulli(-2).rewrite(zeta) == 2*zeta(3)
+    assert not bernoulli(n, -3).rewrite(zeta).has(harmonic)
+    assert bernoulli(-4, x).rewrite(zeta) == 4*zeta(5, x)
+    assert isinstance(bernoulli(n, x).rewrite(zeta), Piecewise)
+    assert bernoulli(n+1, x).rewrite(zeta) == -(n+1) * zeta(-n, x)
+
+
+def test_fibonacci():
+    assert [fibonacci(n) for n in range(-3, 5)] == [2, -1, 1, 0, 1, 1, 2, 3]
+    assert fibonacci(100) == 354224848179261915075
+    assert [lucas(n) for n in range(-3, 5)] == [-4, 3, -1, 2, 1, 3, 4, 7]
+    assert lucas(100) == 792070839848372253127
+
+    assert fibonacci(1, x) == 1
+    assert fibonacci(2, x) == x
+    assert fibonacci(3, x) == x**2 + 1
+    assert fibonacci(4, x) == x**3 + 2*x
+
+    # issue #8800
+    n = Dummy('n')
+    assert fibonacci(n).limit(n, S.Infinity) is S.Infinity
+    assert lucas(n).limit(n, S.Infinity) is S.Infinity
+
+    assert fibonacci(n).rewrite(sqrt) == \
+        2**(-n)*sqrt(5)*((1 + sqrt(5))**n - (-sqrt(5) + 1)**n) / 5
+    assert fibonacci(n).rewrite(sqrt).subs(n, 10).expand() == fibonacci(10)
+    assert fibonacci(n).rewrite(GoldenRatio).subs(n,10).evalf() == \
+        Float(fibonacci(10))
+    assert lucas(n).rewrite(sqrt) == \
+        (fibonacci(n-1).rewrite(sqrt) + fibonacci(n+1).rewrite(sqrt)).simplify()
+    assert lucas(n).rewrite(sqrt).subs(n, 10).expand() == lucas(10)
+    raises(ValueError, lambda: fibonacci(-3, x))
+
+
+def test_tribonacci():
+    assert [tribonacci(n) for n in range(8)] == [0, 1, 1, 2, 4, 7, 13, 24]
+    assert tribonacci(100) == 98079530178586034536500564
+
+    assert tribonacci(0, x) == 0
+    assert tribonacci(1, x) == 1
+    assert tribonacci(2, x) == x**2
+    assert tribonacci(3, x) == x**4 + x
+    assert tribonacci(4, x) == x**6 + 2*x**3 + 1
+    assert tribonacci(5, x) == x**8 + 3*x**5 + 3*x**2
+
+    n = Dummy('n')
+    assert tribonacci(n).limit(n, S.Infinity) is S.Infinity
+
+    w = (-1 + S.ImaginaryUnit * sqrt(3)) / 2
+    a = (1 + cbrt(19 + 3*sqrt(33)) + cbrt(19 - 3*sqrt(33))) / 3
+    b = (1 + w*cbrt(19 + 3*sqrt(33)) + w**2*cbrt(19 - 3*sqrt(33))) / 3
+    c = (1 + w**2*cbrt(19 + 3*sqrt(33)) + w*cbrt(19 - 3*sqrt(33))) / 3
+    assert tribonacci(n).rewrite(sqrt) == \
+      (a**(n + 1)/((a - b)*(a - c))
+      + b**(n + 1)/((b - a)*(b - c))
+      + c**(n + 1)/((c - a)*(c - b)))
+    assert tribonacci(n).rewrite(sqrt).subs(n, 4).simplify() == tribonacci(4)
+    assert tribonacci(n).rewrite(GoldenRatio).subs(n,10).evalf() == \
+        Float(tribonacci(10))
+    assert tribonacci(n).rewrite(TribonacciConstant) == floor(
+            3*TribonacciConstant**n*(102*sqrt(33) + 586)**Rational(1, 3)/
+            (-2*(102*sqrt(33) + 586)**Rational(1, 3) + 4 + (102*sqrt(33)
+            + 586)**Rational(2, 3)) + S.Half)
+    raises(ValueError, lambda: tribonacci(-1, x))
+
+
+@nocache_fail
+def test_bell():
+    assert [bell(n) for n in range(8)] == [1, 1, 2, 5, 15, 52, 203, 877]
+
+    assert bell(0, x) == 1
+    assert bell(1, x) == x
+    assert bell(2, x) == x**2 + x
+    assert bell(5, x) == x**5 + 10*x**4 + 25*x**3 + 15*x**2 + x
+    assert bell(oo) is S.Infinity
+    raises(ValueError, lambda: bell(oo, x))
+
+    raises(ValueError, lambda: bell(-1))
+    raises(ValueError, lambda: bell(S.Half))
+
+    X = symbols('x:6')
+    # X = (x0, x1, .. x5)
+    # at the same time: X[1] = x1, X[2] = x2 for standard readablity.
+    # but we must supply zero-based indexed object X[1:] = (x1, .. x5)
+
+    assert bell(6, 2, X[1:]) == 6*X[5]*X[1] + 15*X[4]*X[2] + 10*X[3]**2
+    assert bell(
+        6, 3, X[1:]) == 15*X[4]*X[1]**2 + 60*X[3]*X[2]*X[1] + 15*X[2]**3
+
+    X = (1, 10, 100, 1000, 10000)
+    assert bell(6, 2, X) == (6 + 15 + 10)*10000
+
+    X = (1, 2, 3, 3, 5)
+    assert bell(6, 2, X) == 6*5 + 15*3*2 + 10*3**2
+
+    X = (1, 2, 3, 5)
+    assert bell(6, 3, X) == 15*5 + 60*3*2 + 15*2**3
+
+    # Dobinski's formula
+    n = Symbol('n', integer=True, nonnegative=True)
+    # For large numbers, this is too slow
+    # For nonintegers, there are significant precision errors
+    for i in [0, 2, 3, 7, 13, 42, 55]:
+        # Running without the cache this is either very slow or goes into an
+        # infinite loop.
+        assert bell(i).evalf() == bell(n).rewrite(Sum).evalf(subs={n: i})
+
+    m = Symbol("m")
+    assert bell(m).rewrite(Sum) == bell(m)
+    assert bell(n, m).rewrite(Sum) == bell(n, m)
+    # issue 9184
+    n = Dummy('n')
+    assert bell(n).limit(n, S.Infinity) is S.Infinity
+
+
+def test_harmonic():
+    n = Symbol("n")
+    m = Symbol("m")
+
+    assert harmonic(n, 0) == n
+    assert harmonic(n).evalf() == harmonic(n)
+    assert harmonic(n, 1) == harmonic(n)
+    assert harmonic(1, n) == 1
+
+    assert harmonic(0, 1) == 0
+    assert harmonic(1, 1) == 1
+    assert harmonic(2, 1) == Rational(3, 2)
+    assert harmonic(3, 1) == Rational(11, 6)
+    assert harmonic(4, 1) == Rational(25, 12)
+    assert harmonic(0, 2) == 0
+    assert harmonic(1, 2) == 1
+    assert harmonic(2, 2) == Rational(5, 4)
+    assert harmonic(3, 2) == Rational(49, 36)
+    assert harmonic(4, 2) == Rational(205, 144)
+    assert harmonic(0, 3) == 0
+    assert harmonic(1, 3) == 1
+    assert harmonic(2, 3) == Rational(9, 8)
+    assert harmonic(3, 3) == Rational(251, 216)
+    assert harmonic(4, 3) == Rational(2035, 1728)
+
+    assert harmonic(oo, -1) is S.NaN
+    assert harmonic(oo, 0) is oo
+    assert harmonic(oo, S.Half) is oo
+    assert harmonic(oo, 1) is oo
+    assert harmonic(oo, 2) == (pi**2)/6
+    assert harmonic(oo, 3) == zeta(3)
+    assert harmonic(oo, Dummy(negative=True)) is S.NaN
+    ip = Dummy(integer=True, positive=True)
+    if (1/ip <= 1) is True:  #---------------------------------+
+        assert None, 'delete this if-block and the next line' #|
+    ip = Dummy(even=True, positive=True)  #--------------------+
+    assert harmonic(oo, 1/ip) is oo
+    assert harmonic(oo, 1 + ip) is zeta(1 + ip)
+
+    assert harmonic(0, m) == 0
+    assert harmonic(-1, -1) == 0
+    assert harmonic(-1, 0) == -1
+    assert harmonic(-1, 1) is S.ComplexInfinity
+    assert harmonic(-1, 2) is S.NaN
+    assert harmonic(-3, -2) == -5
+    assert harmonic(-3, -3) == 9
+
+
+def test_harmonic_rational():
+    ne = S(6)
+    no = S(5)
+    pe = S(8)
+    po = S(9)
+    qe = S(10)
+    qo = S(13)
+
+    Heee = harmonic(ne + pe/qe)
+    Aeee = (-log(10) + 2*(Rational(-1, 4) + sqrt(5)/4)*log(sqrt(-sqrt(5)/8 + Rational(5, 8)))
+             + 2*(-sqrt(5)/4 - Rational(1, 4))*log(sqrt(sqrt(5)/8 + Rational(5, 8)))
+             + pi*sqrt(2*sqrt(5)/5 + 1)/2 + Rational(13944145, 4720968))
+
+    Heeo = harmonic(ne + pe/qo)
+    Aeeo = (-log(26) + 2*log(sin(pi*Rational(3, 13)))*cos(pi*Rational(4, 13)) + 2*log(sin(pi*Rational(2, 13)))*cos(pi*Rational(32, 13))
+             + 2*log(sin(pi*Rational(5, 13)))*cos(pi*Rational(80, 13)) - 2*log(sin(pi*Rational(6, 13)))*cos(pi*Rational(5, 13))
+             - 2*log(sin(pi*Rational(4, 13)))*cos(pi/13) + pi*cot(pi*Rational(5, 13))/2 - 2*log(sin(pi/13))*cos(pi*Rational(3, 13))
+             + Rational(2422020029, 702257080))
+
+    Heoe = harmonic(ne + po/qe)
+    Aeoe = (-log(20) + 2*(Rational(1, 4) + sqrt(5)/4)*log(Rational(-1, 4) + sqrt(5)/4)
+             + 2*(Rational(-1, 4) + sqrt(5)/4)*log(sqrt(-sqrt(5)/8 + Rational(5, 8)))
+             + 2*(-sqrt(5)/4 - Rational(1, 4))*log(sqrt(sqrt(5)/8 + Rational(5, 8)))
+             + 2*(-sqrt(5)/4 + Rational(1, 4))*log(Rational(1, 4) + sqrt(5)/4)
+             + Rational(11818877030, 4286604231) + pi*sqrt(2*sqrt(5) + 5)/2)
+
+    Heoo = harmonic(ne + po/qo)
+    Aeoo = (-log(26) + 2*log(sin(pi*Rational(3, 13)))*cos(pi*Rational(54, 13)) + 2*log(sin(pi*Rational(4, 13)))*cos(pi*Rational(6, 13))
+             + 2*log(sin(pi*Rational(6, 13)))*cos(pi*Rational(108, 13)) - 2*log(sin(pi*Rational(5, 13)))*cos(pi/13)
+             - 2*log(sin(pi/13))*cos(pi*Rational(5, 13)) + pi*cot(pi*Rational(4, 13))/2
+             - 2*log(sin(pi*Rational(2, 13)))*cos(pi*Rational(3, 13)) + Rational(11669332571, 3628714320))
+
+    Hoee = harmonic(no + pe/qe)
+    Aoee = (-log(10) + 2*(Rational(-1, 4) + sqrt(5)/4)*log(sqrt(-sqrt(5)/8 + Rational(5, 8)))
+             + 2*(-sqrt(5)/4 - Rational(1, 4))*log(sqrt(sqrt(5)/8 + Rational(5, 8)))
+             + pi*sqrt(2*sqrt(5)/5 + 1)/2 + Rational(779405, 277704))
+
+    Hoeo = harmonic(no + pe/qo)
+    Aoeo = (-log(26) + 2*log(sin(pi*Rational(3, 13)))*cos(pi*Rational(4, 13)) + 2*log(sin(pi*Rational(2, 13)))*cos(pi*Rational(32, 13))
+             + 2*log(sin(pi*Rational(5, 13)))*cos(pi*Rational(80, 13)) - 2*log(sin(pi*Rational(6, 13)))*cos(pi*Rational(5, 13))
+             - 2*log(sin(pi*Rational(4, 13)))*cos(pi/13) + pi*cot(pi*Rational(5, 13))/2
+             - 2*log(sin(pi/13))*cos(pi*Rational(3, 13)) + Rational(53857323, 16331560))
+
+    Hooe = harmonic(no + po/qe)
+    Aooe = (-log(20) + 2*(Rational(1, 4) + sqrt(5)/4)*log(Rational(-1, 4) + sqrt(5)/4)
+             + 2*(Rational(-1, 4) + sqrt(5)/4)*log(sqrt(-sqrt(5)/8 + Rational(5, 8)))
+             + 2*(-sqrt(5)/4 - Rational(1, 4))*log(sqrt(sqrt(5)/8 + Rational(5, 8)))
+             + 2*(-sqrt(5)/4 + Rational(1, 4))*log(Rational(1, 4) + sqrt(5)/4)
+             + Rational(486853480, 186374097) + pi*sqrt(2*sqrt(5) + 5)/2)
+
+    Hooo = harmonic(no + po/qo)
+    Aooo = (-log(26) + 2*log(sin(pi*Rational(3, 13)))*cos(pi*Rational(54, 13)) + 2*log(sin(pi*Rational(4, 13)))*cos(pi*Rational(6, 13))
+             + 2*log(sin(pi*Rational(6, 13)))*cos(pi*Rational(108, 13)) - 2*log(sin(pi*Rational(5, 13)))*cos(pi/13)
+             - 2*log(sin(pi/13))*cos(pi*Rational(5, 13)) + pi*cot(pi*Rational(4, 13))/2
+             - 2*log(sin(pi*Rational(2, 13)))*cos(3*pi/13) + Rational(383693479, 125128080))
+
+    H = [Heee, Heeo, Heoe, Heoo, Hoee, Hoeo, Hooe, Hooo]
+    A = [Aeee, Aeeo, Aeoe, Aeoo, Aoee, Aoeo, Aooe, Aooo]
+    for h, a in zip(H, A):
+        e = expand_func(h).doit()
+        assert cancel(e/a) == 1
+        assert abs(h.n() - a.n()) < 1e-12
+
+
+def test_harmonic_evalf():
+    assert str(harmonic(1.5).evalf(n=10)) == '1.280372306'
+    assert str(harmonic(1.5, 2).evalf(n=10)) == '1.154576311'  # issue 7443
+    assert str(harmonic(4.0, -3).evalf(n=10)) == '100.0000000'
+    assert str(harmonic(7.0, 1.0).evalf(n=10)) == '2.592857143'
+    assert str(harmonic(1, pi).evalf(n=10)) == '1.000000000'
+    assert str(harmonic(2, pi).evalf(n=10)) == '1.113314732'
+    assert str(harmonic(1000.0, pi).evalf(n=10)) == '1.176241563'
+    assert str(harmonic(I).evalf(n=10)) == '0.6718659855 + 1.076674047*I'
+    assert str(harmonic(I, I).evalf(n=10)) == '-0.3970915266 + 1.9629689*I'
+
+    assert harmonic(-1.0, 1).evalf() is S.NaN
+    assert harmonic(-2.0, 2.0).evalf() is S.NaN
+
+def test_harmonic_rewrite():
+    from sympy.functions.elementary.piecewise import Piecewise
+    n = Symbol("n")
+    m = Symbol("m", integer=True, positive=True)
+    x1 = Symbol("x1", positive=True)
+    x2 = Symbol("x2", negative=True)
+
+    assert harmonic(n).rewrite(digamma) == polygamma(0, n + 1) + EulerGamma
+    assert harmonic(n).rewrite(trigamma) == polygamma(0, n + 1) + EulerGamma
+    assert harmonic(n).rewrite(polygamma) == polygamma(0, n + 1) + EulerGamma
+
+    assert harmonic(n,3).rewrite(polygamma) == polygamma(2, n + 1)/2 - polygamma(2, 1)/2
+    assert isinstance(harmonic(n,m).rewrite(polygamma), Piecewise)
+
+    assert expand_func(harmonic(n+4)) == harmonic(n) + 1/(n + 4) + 1/(n + 3) + 1/(n + 2) + 1/(n + 1)
+    assert expand_func(harmonic(n-4)) == harmonic(n) - 1/(n - 1) - 1/(n - 2) - 1/(n - 3) - 1/n
+
+    assert harmonic(n, m).rewrite("tractable") == harmonic(n, m).rewrite(polygamma)
+    assert harmonic(n, x1).rewrite("tractable") == harmonic(n, x1)
+    assert harmonic(n, x1 + 1).rewrite("tractable") == zeta(x1 + 1) - zeta(x1 + 1, n + 1)
+    assert harmonic(n, x2).rewrite("tractable") == zeta(x2) - zeta(x2, n + 1)
+
+    _k = Dummy("k")
+    assert harmonic(n).rewrite(Sum).dummy_eq(Sum(1/_k, (_k, 1, n)))
+    assert harmonic(n, m).rewrite(Sum).dummy_eq(Sum(_k**(-m), (_k, 1, n)))
+
+
+def test_harmonic_calculus():
+    y = Symbol("y", positive=True)
+    z = Symbol("z", negative=True)
+    assert harmonic(x, 1).limit(x, 0) == 0
+    assert harmonic(x, y).limit(x, 0) == 0
+    assert harmonic(x, 1).series(x, y, 2) == \
+            harmonic(y) + (x - y)*zeta(2, y + 1) + O((x - y)**2, (x, y))
+    assert limit(harmonic(x, y), x, oo) == harmonic(oo, y)
+    assert limit(harmonic(x, y + 1), x, oo) == zeta(y + 1)
+    assert limit(harmonic(x, y - 1), x, oo) == harmonic(oo, y - 1)
+    assert limit(harmonic(x, z), x, oo) == Limit(harmonic(x, z), x, oo, dir='-')
+    assert limit(harmonic(x, z + 1), x, oo) == oo
+    assert limit(harmonic(x, z + 2), x, oo) == harmonic(oo, z + 2)
+    assert limit(harmonic(x, z - 1), x, oo) == Limit(harmonic(x, z - 1), x, oo, dir='-')
+
+
+def test_euler():
+    assert euler(0) == 1
+    assert euler(1) == 0
+    assert euler(2) == -1
+    assert euler(3) == 0
+    assert euler(4) == 5
+    assert euler(6) == -61
+    assert euler(8) == 1385
+
+    assert euler(20, evaluate=False) != 370371188237525
+
+    n = Symbol('n', integer=True)
+    assert euler(n) != -1
+    assert euler(n).subs(n, 2) == -1
+
+    assert euler(-1) == S.Pi / 2
+    assert euler(-1, 1) == 2*log(2)
+    assert euler(-2).evalf() == (2*S.Catalan).evalf()
+    assert euler(-3).evalf() == (S.Pi**3 / 16).evalf()
+    assert str(euler(2.3).evalf(n=10)) == '-1.052850274'
+    assert str(euler(1.2, 3.4).evalf(n=10)) == '3.575613489'
+    assert str(euler(I).evalf(n=10)) == '1.248446443 - 0.7675445124*I'
+    assert str(euler(I, I).evalf(n=10)) == '0.04812930469 + 0.01052411008*I'
+
+    assert euler(20).evalf() == 370371188237525.0
+    assert euler(20, evaluate=False).evalf() == 370371188237525.0
+
+    assert euler(n).rewrite(Sum) == euler(n)
+    n = Symbol('n', integer=True, nonnegative=True)
+    assert euler(2*n + 1).rewrite(Sum) == 0
+    _j = Dummy('j')
+    _k = Dummy('k')
+    assert euler(2*n).rewrite(Sum).dummy_eq(
+            I*Sum((-1)**_j*2**(-_k)*I**(-_k)*(-2*_j + _k)**(2*n + 1)*
+                  binomial(_k, _j)/_k, (_j, 0, _k), (_k, 1, 2*n + 1)))
+
+
+def test_euler_odd():
+    n = Symbol('n', odd=True, positive=True)
+    assert euler(n) == 0
+    n = Symbol('n', odd=True)
+    assert euler(n) != 0
+
+
+def test_euler_polynomials():
+    assert euler(0, x) == 1
+    assert euler(1, x) == x - S.Half
+    assert euler(2, x) == x**2 - x
+    assert euler(3, x) == x**3 - (3*x**2)/2 + Rational(1, 4)
+    m = Symbol('m')
+    assert isinstance(euler(m, x), euler)
+    from sympy.core.numbers import Float
+    A = Float('-0.46237208575048694923364757452876131e8')  # from Maple
+    B = euler(19, S.Pi).evalf(32)
+    assert abs((A - B)/A) < 1e-31
+
+
+def test_euler_polynomial_rewrite():
+    m = Symbol('m')
+    A = euler(m, x).rewrite('Sum');
+    assert A.subs({m:3, x:5}).doit() == euler(3, 5)
+
+
+def test_catalan():
+    n = Symbol('n', integer=True)
+    m = Symbol('m', integer=True, positive=True)
+    k = Symbol('k', integer=True, nonnegative=True)
+    p = Symbol('p', nonnegative=True)
+
+    catalans = [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786]
+    for i, c in enumerate(catalans):
+        assert catalan(i) == c
+        assert catalan(n).rewrite(factorial).subs(n, i) == c
+        assert catalan(n).rewrite(Product).subs(n, i).doit() == c
+
+    assert unchanged(catalan, x)
+    assert catalan(2*x).rewrite(binomial) == binomial(4*x, 2*x)/(2*x + 1)
+    assert catalan(S.Half).rewrite(gamma) == 8/(3*pi)
+    assert catalan(S.Half).rewrite(factorial).rewrite(gamma) ==\
+        8 / (3 * pi)
+    assert catalan(3*x).rewrite(gamma) == 4**(
+        3*x)*gamma(3*x + S.Half)/(sqrt(pi)*gamma(3*x + 2))
+    assert catalan(x).rewrite(hyper) == hyper((-x + 1, -x), (2,), 1)
+
+    assert catalan(n).rewrite(factorial) == factorial(2*n) / (factorial(n + 1)
+                                                              * factorial(n))
+    assert isinstance(catalan(n).rewrite(Product), catalan)
+    assert isinstance(catalan(m).rewrite(Product), Product)
+
+    assert diff(catalan(x), x) == (polygamma(
+        0, x + S.Half) - polygamma(0, x + 2) + log(4))*catalan(x)
+
+    assert catalan(x).evalf() == catalan(x)
+    c = catalan(S.Half).evalf()
+    assert str(c) == '0.848826363156775'
+    c = catalan(I).evalf(3)
+    assert str((re(c), im(c))) == '(0.398, -0.0209)'
+
+    # Assumptions
+    assert catalan(p).is_positive is True
+    assert catalan(k).is_integer is True
+    assert catalan(m+3).is_composite is True
+
+
+def test_genocchi():
+    genocchis = [0, -1, -1, 0, 1, 0, -3, 0, 17]
+    for n, g in enumerate(genocchis):
+        assert genocchi(n) == g
+
+    m = Symbol('m', integer=True)
+    n = Symbol('n', integer=True, positive=True)
+    assert unchanged(genocchi, m)
+    assert genocchi(2*n + 1) == 0
+    gn = 2 * (1 - 2**n) * bernoulli(n)
+    assert genocchi(n).rewrite(bernoulli).factor() == gn.factor()
+    gnx = 2 * (bernoulli(n, x) - 2**n * bernoulli(n, (x+1) / 2))
+    assert genocchi(n, x).rewrite(bernoulli).factor() == gnx.factor()
+    assert genocchi(2 * n).is_odd
+    assert genocchi(2 * n).is_even is False
+    assert genocchi(2 * n + 1).is_even
+    assert genocchi(n).is_integer
+    assert genocchi(4 * n).is_positive
+    # these are the only 2 prime Genocchi numbers
+    assert genocchi(6, evaluate=False).is_prime == S(-3).is_prime
+    assert genocchi(8, evaluate=False).is_prime
+    assert genocchi(4 * n + 2).is_negative
+    assert genocchi(4 * n + 1).is_negative is False
+    assert genocchi(4 * n - 2).is_negative
+
+    g0 = genocchi(0, evaluate=False)
+    assert g0.is_positive is False
+    assert g0.is_negative is False
+    assert g0.is_even is True
+    assert g0.is_odd is False
+
+    assert genocchi(0, x) == 0
+    assert genocchi(1, x) == -1
+    assert genocchi(2, x) == 1 - 2*x
+    assert genocchi(3, x) == 3*x - 3*x**2
+    assert genocchi(4, x) == -1 + 6*x**2 - 4*x**3
+    y = Symbol("y")
+    assert genocchi(5, (x+y)**100) == -5*(x+y)**400 + 10*(x+y)**300 - 5*(x+y)**100
+
+    assert str(genocchi(5.0, 4.0).evalf(n=10)) == '-660.0000000'
+    assert str(genocchi(Rational(5, 4)).evalf(n=10)) == '-1.104286457'
+    assert str(genocchi(-2).evalf(n=10)) == '3.606170709'
+    assert str(genocchi(1.3, 3.7).evalf(n=10)) == '-1.847375373'
+    assert str(genocchi(I, 1.0).evalf(n=10)) == '-0.3161917278 - 1.45311955*I'
+
+    n = Symbol('n')
+    assert genocchi(n, x).rewrite(dirichlet_eta) == -2*n * dirichlet_eta(1-n, x)
+
+
+def test_andre():
+    nums = [1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, 50521]
+    for n, a in enumerate(nums):
+        assert andre(n) == a
+    assert andre(S.Infinity) == S.Infinity
+    assert andre(-1) == -log(2)
+    assert andre(-2) == -2*S.Catalan
+    assert andre(-3) == 3*zeta(3)/16
+    assert andre(-5) == -15*zeta(5)/256
+    # In fact andre(-2*n) is related to the Dirichlet *beta* function
+    # at 2*n, but SymPy doesn't implement that (or general L-functions)
+    assert unchanged(andre, -4)
+
+    n = Symbol('n', integer=True, nonnegative=True)
+    assert unchanged(andre, n)
+    assert andre(n).is_integer is True
+    assert andre(n).is_positive is True
+
+    assert str(andre(10, evaluate=False).evalf(n=10)) == '50521.00000'
+    assert str(andre(-1, evaluate=False).evalf(n=10)) == '-0.6931471806'
+    assert str(andre(-2, evaluate=False).evalf(n=10)) == '-1.831931188'
+    assert str(andre(-4, evaluate=False).evalf(n=10)) == '1.977889103'
+    assert str(andre(I, evaluate=False).evalf(n=10)) == '2.378417833 + 0.6343322845*I'
+
+    assert andre(x).rewrite(polylog) == \
+            (-I)**(x+1) * polylog(-x, I) + I**(x+1) * polylog(-x, -I)
+    assert andre(x).rewrite(zeta) == \
+            2 * gamma(x+1) / (2*pi)**(x+1) * \
+            (zeta(x+1, Rational(1,4)) - cos(pi*x) * zeta(x+1, Rational(3,4)))
+
+
+@nocache_fail
+def test_partition():
+    partition_nums = [1, 1, 2, 3, 5, 7, 11, 15, 22]
+    for n, p in enumerate(partition_nums):
+        assert partition(n) == p
+
+    x = Symbol('x')
+    y = Symbol('y', real=True)
+    m = Symbol('m', integer=True)
+    n = Symbol('n', integer=True, negative=True)
+    p = Symbol('p', integer=True, nonnegative=True)
+    assert partition(m).is_integer
+    assert not partition(m).is_negative
+    assert partition(m).is_nonnegative
+    assert partition(n).is_zero
+    assert partition(p).is_positive
+    assert partition(x).subs(x, 7) == 15
+    assert partition(y).subs(y, 8) == 22
+    raises(ValueError, lambda: partition(Rational(5, 4)))
+
+
+def test__nT():
+       assert [_nT(i, j) for i in range(5) for j in range(i + 2)] == [
+    1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 2, 1, 1, 0]
+       check = [_nT(10, i) for i in range(11)]
+       assert check == [0, 1, 5, 8, 9, 7, 5, 3, 2, 1, 1]
+       assert all(type(i) is int for i in check)
+       assert _nT(10, 5) == 7
+       assert _nT(100, 98) == 2
+       assert _nT(100, 100) == 1
+       assert _nT(10, 3) == 8
+
+
+def test_nC_nP_nT():
+    from sympy.utilities.iterables import (
+        multiset_permutations, multiset_combinations, multiset_partitions,
+        partitions, subsets, permutations)
+    from sympy.functions.combinatorial.numbers import (
+        nP, nC, nT, stirling, _stirling1, _stirling2, _multiset_histogram, _AOP_product)
+
+    from sympy.combinatorics.permutations import Permutation
+    from sympy.core.random import choice
+
+    c = string.ascii_lowercase
+    for i in range(100):
+        s = ''.join(choice(c) for i in range(7))
+        u = len(s) == len(set(s))
+        try:
+            tot = 0
+            for i in range(8):
+                check = nP(s, i)
+                tot += check
+                assert len(list(multiset_permutations(s, i))) == check
+                if u:
+                    assert nP(len(s), i) == check
+            assert nP(s) == tot
+        except AssertionError:
+            print(s, i, 'failed perm test')
+            raise ValueError()
+
+    for i in range(100):
+        s = ''.join(choice(c) for i in range(7))
+        u = len(s) == len(set(s))
+        try:
+            tot = 0
+            for i in range(8):
+                check = nC(s, i)
+                tot += check
+                assert len(list(multiset_combinations(s, i))) == check
+                if u:
+                    assert nC(len(s), i) == check
+            assert nC(s) == tot
+            if u:
+                assert nC(len(s)) == tot
+        except AssertionError:
+            print(s, i, 'failed combo test')
+            raise ValueError()
+
+    for i in range(1, 10):
+        tot = 0
+        for j in range(1, i + 2):
+            check = nT(i, j)
+            assert check.is_Integer
+            tot += check
+            assert sum(1 for p in partitions(i, j, size=True) if p[0] == j) == check
+        assert nT(i) == tot
+
+    for i in range(1, 10):
+        tot = 0
+        for j in range(1, i + 2):
+            check = nT(range(i), j)
+            tot += check
+            assert len(list(multiset_partitions(list(range(i)), j))) == check
+        assert nT(range(i)) == tot
+
+    for i in range(100):
+        s = ''.join(choice(c) for i in range(7))
+        u = len(s) == len(set(s))
+        try:
+            tot = 0
+            for i in range(1, 8):
+                check = nT(s, i)
+                tot += check
+                assert len(list(multiset_partitions(s, i))) == check
+                if u:
+                    assert nT(range(len(s)), i) == check
+            if u:
+                assert nT(range(len(s))) == tot
+            assert nT(s) == tot
+        except AssertionError:
+            print(s, i, 'failed partition test')
+            raise ValueError()
+
+    # tests for Stirling numbers of the first kind that are not tested in the
+    # above
+    assert [stirling(9, i, kind=1) for i in range(11)] == [
+        0, 40320, 109584, 118124, 67284, 22449, 4536, 546, 36, 1, 0]
+    perms = list(permutations(range(4)))
+    assert [sum(1 for p in perms if Permutation(p).cycles == i)
+            for i in range(5)] == [0, 6, 11, 6, 1] == [
+            stirling(4, i, kind=1) for i in range(5)]
+    # http://oeis.org/A008275
+    assert [stirling(n, k, signed=1)
+        for n in range(10) for k in range(1, n + 1)] == [
+            1, -1,
+            1, 2, -3,
+            1, -6, 11, -6,
+            1, 24, -50, 35, -10,
+            1, -120, 274, -225, 85, -15,
+            1, 720, -1764, 1624, -735, 175, -21,
+            1, -5040, 13068, -13132, 6769, -1960, 322, -28,
+            1, 40320, -109584, 118124, -67284, 22449, -4536, 546, -36, 1]
+    # https://en.wikipedia.org/wiki/Stirling_numbers_of_the_first_kind
+    assert  [stirling(n, k, kind=1)
+        for n in range(10) for k in range(n+1)] == [
+            1,
+            0, 1,
+            0, 1, 1,
+            0, 2, 3, 1,
+            0, 6, 11, 6, 1,
+            0, 24, 50, 35, 10, 1,
+            0, 120, 274, 225, 85, 15, 1,
+            0, 720, 1764, 1624, 735, 175, 21, 1,
+            0, 5040, 13068, 13132, 6769, 1960, 322, 28, 1,
+            0, 40320, 109584, 118124, 67284, 22449, 4536, 546, 36, 1]
+    # https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind
+    assert [stirling(n, k, kind=2)
+        for n in range(10) for k in range(n+1)] == [
+            1,
+            0, 1,
+            0, 1, 1,
+            0, 1, 3, 1,
+            0, 1, 7, 6, 1,
+            0, 1, 15, 25, 10, 1,
+            0, 1, 31, 90, 65, 15, 1,
+            0, 1, 63, 301, 350, 140, 21, 1,
+            0, 1, 127, 966, 1701, 1050, 266, 28, 1,
+            0, 1, 255, 3025, 7770, 6951, 2646, 462, 36, 1]
+    assert stirling(3, 4, kind=1) == stirling(3, 4, kind=1) == 0
+    raises(ValueError, lambda: stirling(-2, 2))
+
+    # Assertion that the return type is SymPy Integer.
+    assert isinstance(_stirling1(6, 3), Integer)
+    assert isinstance(_stirling2(6, 3), Integer)
+
+    def delta(p):
+        if len(p) == 1:
+            return oo
+        return min(abs(i[0] - i[1]) for i in subsets(p, 2))
+    parts = multiset_partitions(range(5), 3)
+    d = 2
+    assert (sum(1 for p in parts if all(delta(i) >= d for i in p)) ==
+            stirling(5, 3, d=d) == 7)
+
+    # other coverage tests
+    assert nC('abb', 2) == nC('aab', 2) == 2
+    assert nP(3, 3, replacement=True) == nP('aabc', 3, replacement=True) == 27
+    assert nP(3, 4) == 0
+    assert nP('aabc', 5) == 0
+    assert nC(4, 2, replacement=True) == nC('abcdd', 2, replacement=True) == \
+        len(list(multiset_combinations('aabbccdd', 2))) == 10
+    assert nC('abcdd') == sum(nC('abcdd', i) for i in range(6)) == 24
+    assert nC(list('abcdd'), 4) == 4
+    assert nT('aaaa') == nT(4) == len(list(partitions(4))) == 5
+    assert nT('aaab') == len(list(multiset_partitions('aaab'))) == 7
+    assert nC('aabb'*3, 3) == 4  # aaa, bbb, abb, baa
+    assert dict(_AOP_product((4,1,1,1))) == {
+        0: 1, 1: 4, 2: 7, 3: 8, 4: 8, 5: 7, 6: 4, 7: 1}
+    # the following was the first t that showed a problem in a previous form of
+    # the function, so it's not as random as it may appear
+    t = (3, 9, 4, 6, 6, 5, 5, 2, 10, 4)
+    assert sum(_AOP_product(t)[i] for i in range(55)) == 58212000
+    raises(ValueError, lambda: _multiset_histogram({1:'a'}))
+
+
+def test_PR_14617():
+    from sympy.functions.combinatorial.numbers import nT
+    for n in (0, []):
+        for k in (-1, 0, 1):
+            if k == 0:
+                assert nT(n, k) == 1
+            else:
+                assert nT(n, k) == 0
+
+
+def test_issue_8496():
+    n = Symbol("n")
+    k = Symbol("k")
+
+    raises(TypeError, lambda: catalan(n, k))
+
+
+def test_issue_8601():
+    n = Symbol('n', integer=True, negative=True)
+
+    assert catalan(n - 1) is S.Zero
+    assert catalan(Rational(-1, 2)) is S.ComplexInfinity
+    assert catalan(-S.One) == Rational(-1, 2)
+    c1 = catalan(-5.6).evalf()
+    assert str(c1) == '6.93334070531408e-5'
+    c2 = catalan(-35.4).evalf()
+    assert str(c2) == '-4.14189164517449e-24'
+
+
+def test_motzkin():
+    assert motzkin.is_motzkin(4) == True
+    assert motzkin.is_motzkin(9) == True
+    assert motzkin.is_motzkin(10) == False
+    assert motzkin.find_motzkin_numbers_in_range(10,200) == [21, 51, 127]
+    assert motzkin.find_motzkin_numbers_in_range(10,400) == [21, 51, 127, 323]
+    assert motzkin.find_motzkin_numbers_in_range(10,1600) == [21, 51, 127, 323, 835]
+    assert motzkin.find_first_n_motzkins(5) == [1, 1, 2, 4, 9]
+    assert motzkin.find_first_n_motzkins(7) == [1, 1, 2, 4, 9, 21, 51]
+    assert motzkin.find_first_n_motzkins(10) == [1, 1, 2, 4, 9, 21, 51, 127, 323, 835]
+    raises(ValueError, lambda: motzkin.eval(77.58))
+    raises(ValueError, lambda: motzkin.eval(-8))
+    raises(ValueError, lambda: motzkin.find_motzkin_numbers_in_range(-2,7))
+    raises(ValueError, lambda: motzkin.find_motzkin_numbers_in_range(13,7))
+    raises(ValueError, lambda: motzkin.find_first_n_motzkins(112.8))
+
+
+def test_nD_derangements():
+    from sympy.utilities.iterables import (partitions, multiset,
+        multiset_derangements, multiset_permutations)
+    from sympy.functions.combinatorial.numbers import nD
+
+    got = []
+    for i in partitions(8, k=4):
+        s = []
+        it = 0
+        for k, v in i.items():
+            for i in range(v):
+                s.extend([it]*k)
+                it += 1
+        ms = multiset(s)
+        c1 = sum(1 for i in multiset_permutations(s) if
+            all(i != j for i, j in zip(i, s)))
+        assert c1 == nD(ms) == nD(ms, 0) == nD(ms, 1)
+        v = [tuple(i) for i in multiset_derangements(s)]
+        c2 = len(v)
+        assert c2 == len(set(v))
+        assert c1 == c2
+        got.append(c1)
+    assert got == [1, 4, 6, 12, 24, 24, 61, 126, 315, 780, 297, 772,
+        2033, 5430, 14833]
+
+    assert nD('1112233456', brute=True) == nD('1112233456') == 16356
+    assert nD('') == nD([]) == nD({}) == 0
+    assert nD({1: 0}) == 0
+    raises(ValueError, lambda: nD({1: -1}))
+    assert nD('112') == 0
+    assert nD(i='112') == 0
+    assert [nD(n=i) for i in range(6)] == [0, 0, 1, 2, 9, 44]
+    assert nD((i for i in range(4))) == nD('0123') == 9
+    assert nD(m=(i for i in range(4))) == 3
+    assert nD(m={0: 1, 1: 1, 2: 1, 3: 1}) == 3
+    assert nD(m=[0, 1, 2, 3]) == 3
+    raises(TypeError, lambda: nD(m=0))
+    raises(TypeError, lambda: nD(-1))
+    assert nD({-1: 1, -2: 1}) == 1
+    assert nD(m={0: 3}) == 0
+    raises(ValueError, lambda: nD(i='123', n=3))
+    raises(ValueError, lambda: nD(i='123', m=(1,2)))
+    raises(ValueError, lambda: nD(n=0, m=(1,2)))
+    raises(ValueError, lambda: nD({1: -1}))
+    raises(ValueError, lambda: nD(m={-1: 1, 2: 1}))
+    raises(ValueError, lambda: nD(m={1: -1, 2: 1}))
+    raises(ValueError, lambda: nD(m=[-1, 2]))
+    raises(TypeError, lambda: nD({1: x}))
+    raises(TypeError, lambda: nD(m={1: x}))
+    raises(TypeError, lambda: nD(m={x: 1}))

llmeval-env/lib/python3.10/site-packages/sympy/functions/elementary/benchmarks/bench_exp.py

@@ -0,0 +1,11 @@
+from sympy.core.symbol import symbols
+from sympy.functions.elementary.exponential import exp
+
+x, y = symbols('x,y')
+
+e = exp(2*x)
+q = exp(3*x)
+
+
+def timeit_exp_subs():
+    e.subs(q, y)

llmeval-env/lib/python3.10/site-packages/sympy/functions/elementary/tests/test_complexes.py

@@ -0,0 +1,1018 @@
+from sympy.core.expr import Expr
+from sympy.core.function import (Derivative, Function, Lambda, expand)
+from sympy.core.numbers import (E, I, Rational, comp, nan, oo, pi, zoo)
+from sympy.core.relational import Eq
+from sympy.core.singleton import S
+from sympy.core.symbol import (Symbol, symbols)
+from sympy.functions.elementary.complexes import (Abs, adjoint, arg, conjugate, im, re, sign, transpose)
+from sympy.functions.elementary.exponential import (exp, exp_polar, log)
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.piecewise import Piecewise
+from sympy.functions.elementary.trigonometric import (acos, atan, atan2, cos, sin)
+from sympy.functions.special.delta_functions import (DiracDelta, Heaviside)
+from sympy.integrals.integrals import Integral
+from sympy.matrices.dense import Matrix
+from sympy.matrices.expressions.funcmatrix import FunctionMatrix
+from sympy.matrices.expressions.matexpr import MatrixSymbol
+from sympy.matrices.immutable import (ImmutableMatrix, ImmutableSparseMatrix)
+from sympy.matrices import SparseMatrix
+from sympy.sets.sets import Interval
+from sympy.core.expr import unchanged
+from sympy.core.function import ArgumentIndexError
+from sympy.testing.pytest import XFAIL, raises, _both_exp_pow
+
+
+def N_equals(a, b):
+    """Check whether two complex numbers are numerically close"""
+    return comp(a.n(), b.n(), 1.e-6)
+
+
+def test_re():
+    x, y = symbols('x,y')
+    a, b = symbols('a,b', real=True)
+
+    r = Symbol('r', real=True)
+    i = Symbol('i', imaginary=True)
+
+    assert re(nan) is nan
+
+    assert re(oo) is oo
+    assert re(-oo) is -oo
+
+    assert re(0) == 0
+
+    assert re(1) == 1
+    assert re(-1) == -1
+
+    assert re(E) == E
+    assert re(-E) == -E
+
+    assert unchanged(re, x)
+    assert re(x*I) == -im(x)
+    assert re(r*I) == 0
+    assert re(r) == r
+    assert re(i*I) == I * i
+    assert re(i) == 0
+
+    assert re(x + y) == re(x) + re(y)
+    assert re(x + r) == re(x) + r
+
+    assert re(re(x)) == re(x)
+
+    assert re(2 + I) == 2
+    assert re(x + I) == re(x)
+
+    assert re(x + y*I) == re(x) - im(y)
+    assert re(x + r*I) == re(x)
+
+    assert re(log(2*I)) == log(2)
+
+    assert re((2 + I)**2).expand(complex=True) == 3
+
+    assert re(conjugate(x)) == re(x)
+    assert conjugate(re(x)) == re(x)
+
+    assert re(x).as_real_imag() == (re(x), 0)
+
+    assert re(i*r*x).diff(r) == re(i*x)
+    assert re(i*r*x).diff(i) == I*r*im(x)
+
+    assert re(
+        sqrt(a + b*I)) == (a**2 + b**2)**Rational(1, 4)*cos(atan2(b, a)/2)
+    assert re(a * (2 + b*I)) == 2*a
+
+    assert re((1 + sqrt(a + b*I))/2) == \
+        (a**2 + b**2)**Rational(1, 4)*cos(atan2(b, a)/2)/2 + S.Half
+
+    assert re(x).rewrite(im) == x - S.ImaginaryUnit*im(x)
+    assert (x + re(y)).rewrite(re, im) == x + y - S.ImaginaryUnit*im(y)
+
+    a = Symbol('a', algebraic=True)
+    t = Symbol('t', transcendental=True)
+    x = Symbol('x')
+    assert re(a).is_algebraic
+    assert re(x).is_algebraic is None
+    assert re(t).is_algebraic is False
+
+    assert re(S.ComplexInfinity) is S.NaN
+
+    n, m, l = symbols('n m l')
+    A = MatrixSymbol('A',n,m)
+    assert re(A) == (S.Half) * (A + conjugate(A))
+
+    A = Matrix([[1 + 4*I,2],[0, -3*I]])
+    assert re(A) == Matrix([[1, 2],[0, 0]])
+
+    A = ImmutableMatrix([[1 + 3*I, 3-2*I],[0, 2*I]])
+    assert re(A) == ImmutableMatrix([[1, 3],[0, 0]])
+
+    X = SparseMatrix([[2*j + i*I for i in range(5)] for j in range(5)])
+    assert re(X) - Matrix([[0, 0, 0, 0, 0],
+                           [2, 2, 2, 2, 2],
+                           [4, 4, 4, 4, 4],
+                           [6, 6, 6, 6, 6],
+                           [8, 8, 8, 8, 8]]) == Matrix.zeros(5)
+
+    assert im(X) - Matrix([[0, 1, 2, 3, 4],
+                           [0, 1, 2, 3, 4],
+                           [0, 1, 2, 3, 4],
+                           [0, 1, 2, 3, 4],
+                           [0, 1, 2, 3, 4]]) == Matrix.zeros(5)
+
+    X = FunctionMatrix(3, 3, Lambda((n, m), n + m*I))
+    assert re(X) == Matrix([[0, 0, 0], [1, 1, 1], [2, 2, 2]])
+
+
+def test_im():
+    x, y = symbols('x,y')
+    a, b = symbols('a,b', real=True)
+
+    r = Symbol('r', real=True)
+    i = Symbol('i', imaginary=True)
+
+    assert im(nan) is nan
+
+    assert im(oo*I) is oo
+    assert im(-oo*I) is -oo
+
+    assert im(0) == 0
+
+    assert im(1) == 0
+    assert im(-1) == 0
+
+    assert im(E*I) == E
+    assert im(-E*I) == -E
+
+    assert unchanged(im, x)
+    assert im(x*I) == re(x)
+    assert im(r*I) == r
+    assert im(r) == 0
+    assert im(i*I) == 0
+    assert im(i) == -I * i
+
+    assert im(x + y) == im(x) + im(y)
+    assert im(x + r) == im(x)
+    assert im(x + r*I) == im(x) + r
+
+    assert im(im(x)*I) == im(x)
+
+    assert im(2 + I) == 1
+    assert im(x + I) == im(x) + 1
+
+    assert im(x + y*I) == im(x) + re(y)
+    assert im(x + r*I) == im(x) + r
+
+    assert im(log(2*I)) == pi/2
+
+    assert im((2 + I)**2).expand(complex=True) == 4
+
+    assert im(conjugate(x)) == -im(x)
+    assert conjugate(im(x)) == im(x)
+
+    assert im(x).as_real_imag() == (im(x), 0)
+
+    assert im(i*r*x).diff(r) == im(i*x)
+    assert im(i*r*x).diff(i) == -I * re(r*x)
+
+    assert im(
+        sqrt(a + b*I)) == (a**2 + b**2)**Rational(1, 4)*sin(atan2(b, a)/2)
+    assert im(a * (2 + b*I)) == a*b
+
+    assert im((1 + sqrt(a + b*I))/2) == \
+        (a**2 + b**2)**Rational(1, 4)*sin(atan2(b, a)/2)/2
+
+    assert im(x).rewrite(re) == -S.ImaginaryUnit * (x - re(x))
+    assert (x + im(y)).rewrite(im, re) == x - S.ImaginaryUnit * (y - re(y))
+
+    a = Symbol('a', algebraic=True)
+    t = Symbol('t', transcendental=True)
+    x = Symbol('x')
+    assert re(a).is_algebraic
+    assert re(x).is_algebraic is None
+    assert re(t).is_algebraic is False
+
+    assert im(S.ComplexInfinity) is S.NaN
+
+    n, m, l = symbols('n m l')
+    A = MatrixSymbol('A',n,m)
+
+    assert im(A) == (S.One/(2*I)) * (A - conjugate(A))
+
+    A = Matrix([[1 + 4*I, 2],[0, -3*I]])
+    assert im(A) == Matrix([[4, 0],[0, -3]])
+
+    A = ImmutableMatrix([[1 + 3*I, 3-2*I],[0, 2*I]])
+    assert im(A) == ImmutableMatrix([[3, -2],[0, 2]])
+
+    X = ImmutableSparseMatrix(
+            [[i*I + i for i in range(5)] for i in range(5)])
+    Y = SparseMatrix([list(range(5)) for i in range(5)])
+    assert im(X).as_immutable() == Y
+
+    X = FunctionMatrix(3, 3, Lambda((n, m), n + m*I))
+    assert im(X) == Matrix([[0, 1, 2], [0, 1, 2], [0, 1, 2]])
+
+def test_sign():
+    assert sign(1.2) == 1
+    assert sign(-1.2) == -1
+    assert sign(3*I) == I
+    assert sign(-3*I) == -I
+    assert sign(0) == 0
+    assert sign(0, evaluate=False).doit() == 0
+    assert sign(oo, evaluate=False).doit() == 1
+    assert sign(nan) is nan
+    assert sign(2 + 2*I).doit() == sqrt(2)*(2 + 2*I)/4
+    assert sign(2 + 3*I).simplify() == sign(2 + 3*I)
+    assert sign(2 + 2*I).simplify() == sign(1 + I)
+    assert sign(im(sqrt(1 - sqrt(3)))) == 1
+    assert sign(sqrt(1 - sqrt(3))) == I
+
+    x = Symbol('x')
+    assert sign(x).is_finite is True
+    assert sign(x).is_complex is True
+    assert sign(x).is_imaginary is None
+    assert sign(x).is_integer is None
+    assert sign(x).is_real is None
+    assert sign(x).is_zero is None
+    assert sign(x).doit() == sign(x)
+    assert sign(1.2*x) == sign(x)
+    assert sign(2*x) == sign(x)
+    assert sign(I*x) == I*sign(x)
+    assert sign(-2*I*x) == -I*sign(x)
+    assert sign(conjugate(x)) == conjugate(sign(x))
+
+    p = Symbol('p', positive=True)
+    n = Symbol('n', negative=True)
+    m = Symbol('m', negative=True)
+    assert sign(2*p*x) == sign(x)
+    assert sign(n*x) == -sign(x)
+    assert sign(n*m*x) == sign(x)
+
+    x = Symbol('x', imaginary=True)
+    assert sign(x).is_imaginary is True
+    assert sign(x).is_integer is False
+    assert sign(x).is_real is False
+    assert sign(x).is_zero is False
+    assert sign(x).diff(x) == 2*DiracDelta(-I*x)
+    assert sign(x).doit() == x / Abs(x)
+    assert conjugate(sign(x)) == -sign(x)
+
+    x = Symbol('x', real=True)
+    assert sign(x).is_imaginary is False
+    assert sign(x).is_integer is True
+    assert sign(x).is_real is True
+    assert sign(x).is_zero is None
+    assert sign(x).diff(x) == 2*DiracDelta(x)
+    assert sign(x).doit() == sign(x)
+    assert conjugate(sign(x)) == sign(x)
+
+    x = Symbol('x', nonzero=True)
+    assert sign(x).is_imaginary is False
+    assert sign(x).is_integer is True
+    assert sign(x).is_real is True
+    assert sign(x).is_zero is False
+    assert sign(x).doit() == x / Abs(x)
+    assert sign(Abs(x)) == 1
+    assert Abs(sign(x)) == 1
+
+    x = Symbol('x', positive=True)
+    assert sign(x).is_imaginary is False
+    assert sign(x).is_integer is True
+    assert sign(x).is_real is True
+    assert sign(x).is_zero is False
+    assert sign(x).doit() == x / Abs(x)
+    assert sign(Abs(x)) == 1
+    assert Abs(sign(x)) == 1
+
+    x = 0
+    assert sign(x).is_imaginary is False
+    assert sign(x).is_integer is True
+    assert sign(x).is_real is True
+    assert sign(x).is_zero is True
+    assert sign(x).doit() == 0
+    assert sign(Abs(x)) == 0
+    assert Abs(sign(x)) == 0
+
+    nz = Symbol('nz', nonzero=True, integer=True)
+    assert sign(nz).is_imaginary is False
+    assert sign(nz).is_integer is True
+    assert sign(nz).is_real is True
+    assert sign(nz).is_zero is False
+    assert sign(nz)**2 == 1
+    assert (sign(nz)**3).args == (sign(nz), 3)
+
+    assert sign(Symbol('x', nonnegative=True)).is_nonnegative
+    assert sign(Symbol('x', nonnegative=True)).is_nonpositive is None
+    assert sign(Symbol('x', nonpositive=True)).is_nonnegative is None
+    assert sign(Symbol('x', nonpositive=True)).is_nonpositive
+    assert sign(Symbol('x', real=True)).is_nonnegative is None
+    assert sign(Symbol('x', real=True)).is_nonpositive is None
+    assert sign(Symbol('x', real=True, zero=False)).is_nonpositive is None
+
+    x, y = Symbol('x', real=True), Symbol('y')
+    f = Function('f')
+    assert sign(x).rewrite(Piecewise) == \
+        Piecewise((1, x > 0), (-1, x < 0), (0, True))
+    assert sign(y).rewrite(Piecewise) == sign(y)
+    assert sign(x).rewrite(Heaviside) == 2*Heaviside(x, H0=S(1)/2) - 1
+    assert sign(y).rewrite(Heaviside) == sign(y)
+    assert sign(y).rewrite(Abs) == Piecewise((0, Eq(y, 0)), (y/Abs(y), True))
+    assert sign(f(y)).rewrite(Abs) == Piecewise((0, Eq(f(y), 0)), (f(y)/Abs(f(y)), True))
+
+    # evaluate what can be evaluated
+    assert sign(exp_polar(I*pi)*pi) is S.NegativeOne
+
+    eq = -sqrt(10 + 6*sqrt(3)) + sqrt(1 + sqrt(3)) + sqrt(3 + 3*sqrt(3))
+    # if there is a fast way to know when and when you cannot prove an
+    # expression like this is zero then the equality to zero is ok
+    assert sign(eq).func is sign or sign(eq) == 0
+    # but sometimes it's hard to do this so it's better not to load
+    # abs down with tests that will be very slow
+    q = 1 + sqrt(2) - 2*sqrt(3) + 1331*sqrt(6)
+    p = expand(q**3)**Rational(1, 3)
+    d = p - q
+    assert sign(d).func is sign or sign(d) == 0
+
+
+def test_as_real_imag():
+    n = pi**1000
+    # the special code for working out the real
+    # and complex parts of a power with Integer exponent
+    # should not run if there is no imaginary part, hence
+    # this should not hang
+    assert n.as_real_imag() == (n, 0)
+
+    # issue 6261
+    x = Symbol('x')
+    assert sqrt(x).as_real_imag() == \
+        ((re(x)**2 + im(x)**2)**Rational(1, 4)*cos(atan2(im(x), re(x))/2),
+     (re(x)**2 + im(x)**2)**Rational(1, 4)*sin(atan2(im(x), re(x))/2))
+
+    # issue 3853
+    a, b = symbols('a,b', real=True)
+    assert ((1 + sqrt(a + b*I))/2).as_real_imag() == \
+           (
+               (a**2 + b**2)**Rational(
+                   1, 4)*cos(atan2(b, a)/2)/2 + S.Half,
+               (a**2 + b**2)**Rational(1, 4)*sin(atan2(b, a)/2)/2)
+
+    assert sqrt(a**2).as_real_imag() == (sqrt(a**2), 0)
+    i = symbols('i', imaginary=True)
+    assert sqrt(i**2).as_real_imag() == (0, abs(i))
+
+    assert ((1 + I)/(1 - I)).as_real_imag() == (0, 1)
+    assert ((1 + I)**3/(1 - I)).as_real_imag() == (-2, 0)
+
+
+@XFAIL
+def test_sign_issue_3068():
+    n = pi**1000
+    i = int(n)
+    x = Symbol('x')
+    assert (n - i).round() == 1  # doesn't hang
+    assert sign(n - i) == 1
+    # perhaps it's not possible to get the sign right when
+    # only 1 digit is being requested for this situation;
+    # 2 digits works
+    assert (n - x).n(1, subs={x: i}) > 0
+    assert (n - x).n(2, subs={x: i}) > 0
+
+
+def test_Abs():
+    raises(TypeError, lambda: Abs(Interval(2, 3)))  # issue 8717
+
+    x, y = symbols('x,y')
+    assert sign(sign(x)) == sign(x)
+    assert sign(x*y).func is sign
+    assert Abs(0) == 0
+    assert Abs(1) == 1
+    assert Abs(-1) == 1
+    assert Abs(I) == 1
+    assert Abs(-I) == 1
+    assert Abs(nan) is nan
+    assert Abs(zoo) is oo
+    assert Abs(I * pi) == pi
+    assert Abs(-I * pi) == pi
+    assert Abs(I * x) == Abs(x)
+    assert Abs(-I * x) == Abs(x)
+    assert Abs(-2*x) == 2*Abs(x)
+    assert Abs(-2.0*x) == 2.0*Abs(x)
+    assert Abs(2*pi*x*y) == 2*pi*Abs(x*y)
+    assert Abs(conjugate(x)) == Abs(x)
+    assert conjugate(Abs(x)) == Abs(x)
+    assert Abs(x).expand(complex=True) == sqrt(re(x)**2 + im(x)**2)
+
+    a = Symbol('a', positive=True)
+    assert Abs(2*pi*x*a) == 2*pi*a*Abs(x)
+    assert Abs(2*pi*I*x*a) == 2*pi*a*Abs(x)
+
+    x = Symbol('x', real=True)
+    n = Symbol('n', integer=True)
+    assert Abs((-1)**n) == 1
+    assert x**(2*n) == Abs(x)**(2*n)
+    assert Abs(x).diff(x) == sign(x)
+    assert abs(x) == Abs(x)  # Python built-in
+    assert Abs(x)**3 == x**2*Abs(x)
+    assert Abs(x)**4 == x**4
+    assert (
+        Abs(x)**(3*n)).args == (Abs(x), 3*n)  # leave symbolic odd unchanged
+    assert (1/Abs(x)).args == (Abs(x), -1)
+    assert 1/Abs(x)**3 == 1/(x**2*Abs(x))
+    assert Abs(x)**-3 == Abs(x)/(x**4)
+    assert Abs(x**3) == x**2*Abs(x)
+    assert Abs(I**I) == exp(-pi/2)
+    assert Abs((4 + 5*I)**(6 + 7*I)) == 68921*exp(-7*atan(Rational(5, 4)))
+    y = Symbol('y', real=True)
+    assert Abs(I**y) == 1
+    y = Symbol('y')
+    assert Abs(I**y) == exp(-pi*im(y)/2)
+
+    x = Symbol('x', imaginary=True)
+    assert Abs(x).diff(x) == -sign(x)
+
+    eq = -sqrt(10 + 6*sqrt(3)) + sqrt(1 + sqrt(3)) + sqrt(3 + 3*sqrt(3))
+    # if there is a fast way to know when you can and when you cannot prove an
+    # expression like this is zero then the equality to zero is ok
+    assert abs(eq).func is Abs or abs(eq) == 0
+    # but sometimes it's hard to do this so it's better not to load
+    # abs down with tests that will be very slow
+    q = 1 + sqrt(2) - 2*sqrt(3) + 1331*sqrt(6)
+    p = expand(q**3)**Rational(1, 3)
+    d = p - q
+    assert abs(d).func is Abs or abs(d) == 0
+
+    assert Abs(4*exp(pi*I/4)) == 4
+    assert Abs(3**(2 + I)) == 9
+    assert Abs((-3)**(1 - I)) == 3*exp(pi)
+
+    assert Abs(oo) is oo
+    assert Abs(-oo) is oo
+    assert Abs(oo + I) is oo
+    assert Abs(oo + I*oo) is oo
+
+    a = Symbol('a', algebraic=True)
+    t = Symbol('t', transcendental=True)
+    x = Symbol('x')
+    assert re(a).is_algebraic
+    assert re(x).is_algebraic is None
+    assert re(t).is_algebraic is False
+    assert Abs(x).fdiff() == sign(x)
+    raises(ArgumentIndexError, lambda: Abs(x).fdiff(2))
+
+    # doesn't have recursion error
+    arg = sqrt(acos(1 - I)*acos(1 + I))
+    assert abs(arg) == arg
+
+    # special handling to put Abs in denom
+    assert abs(1/x) == 1/Abs(x)
+    e = abs(2/x**2)
+    assert e.is_Mul and e == 2/Abs(x**2)
+    assert unchanged(Abs, y/x)
+    assert unchanged(Abs, x/(x + 1))
+    assert unchanged(Abs, x*y)
+    p = Symbol('p', positive=True)
+    assert abs(x/p) == abs(x)/p
+
+    # coverage
+    assert unchanged(Abs, Symbol('x', real=True)**y)
+    # issue 19627
+    f = Function('f', positive=True)
+    assert sqrt(f(x)**2) == f(x)
+    # issue 21625
+    assert unchanged(Abs, S("im(acos(-i + acosh(-g + i)))"))
+
+
+def test_Abs_rewrite():
+    x = Symbol('x', real=True)
+    a = Abs(x).rewrite(Heaviside).expand()
+    assert a == x*Heaviside(x) - x*Heaviside(-x)
+    for i in [-2, -1, 0, 1, 2]:
+        assert a.subs(x, i) == abs(i)
+    y = Symbol('y')
+    assert Abs(y).rewrite(Heaviside) == Abs(y)
+
+    x, y = Symbol('x', real=True), Symbol('y')
+    assert Abs(x).rewrite(Piecewise) == Piecewise((x, x >= 0), (-x, True))
+    assert Abs(y).rewrite(Piecewise) == Abs(y)
+    assert Abs(y).rewrite(sign) == y/sign(y)
+
+    i = Symbol('i', imaginary=True)
+    assert abs(i).rewrite(Piecewise) == Piecewise((I*i, I*i >= 0), (-I*i, True))
+
+
+    assert Abs(y).rewrite(conjugate) == sqrt(y*conjugate(y))
+    assert Abs(i).rewrite(conjugate) == sqrt(-i**2) #  == -I*i
+
+    y = Symbol('y', extended_real=True)
+    assert  (Abs(exp(-I*x)-exp(-I*y))**2).rewrite(conjugate) == \
+        -exp(I*x)*exp(-I*y) + 2 - exp(-I*x)*exp(I*y)
+
+
+def test_Abs_real():
+    # test some properties of abs that only apply
+    # to real numbers
+    x = Symbol('x', complex=True)
+    assert sqrt(x**2) != Abs(x)
+    assert Abs(x**2) != x**2
+
+    x = Symbol('x', real=True)
+    assert sqrt(x**2) == Abs(x)
+    assert Abs(x**2) == x**2
+
+    # if the symbol is zero, the following will still apply
+    nn = Symbol('nn', nonnegative=True, real=True)
+    np = Symbol('np', nonpositive=True, real=True)
+    assert Abs(nn) == nn
+    assert Abs(np) == -np
+
+
+def test_Abs_properties():
+    x = Symbol('x')
+    assert Abs(x).is_real is None
+    assert Abs(x).is_extended_real is True
+    assert Abs(x).is_rational is None
+    assert Abs(x).is_positive is None
+    assert Abs(x).is_nonnegative is None
+    assert Abs(x).is_extended_positive is None
+    assert Abs(x).is_extended_nonnegative is True
+
+    f = Symbol('x', finite=True)
+    assert Abs(f).is_real is True
+    assert Abs(f).is_extended_real is True
+    assert Abs(f).is_rational is None
+    assert Abs(f).is_positive is None
+    assert Abs(f).is_nonnegative is True
+    assert Abs(f).is_extended_positive is None
+    assert Abs(f).is_extended_nonnegative is True
+
+    z = Symbol('z', complex=True, zero=False)
+    assert Abs(z).is_real is True # since complex implies finite
+    assert Abs(z).is_extended_real is True
+    assert Abs(z).is_rational is None
+    assert Abs(z).is_positive is True
+    assert Abs(z).is_extended_positive is True
+    assert Abs(z).is_zero is False
+
+    p = Symbol('p', positive=True)
+    assert Abs(p).is_real is True
+    assert Abs(p).is_extended_real is True
+    assert Abs(p).is_rational is None
+    assert Abs(p).is_positive is True
+    assert Abs(p).is_zero is False
+
+    q = Symbol('q', rational=True)
+    assert Abs(q).is_real is True
+    assert Abs(q).is_rational is True
+    assert Abs(q).is_integer is None
+    assert Abs(q).is_positive is None
+    assert Abs(q).is_nonnegative is True
+
+    i = Symbol('i', integer=True)
+    assert Abs(i).is_real is True
+    assert Abs(i).is_integer is True
+    assert Abs(i).is_positive is None
+    assert Abs(i).is_nonnegative is True
+
+    e = Symbol('n', even=True)
+    ne = Symbol('ne', real=True, even=False)
+    assert Abs(e).is_even is True
+    assert Abs(ne).is_even is False
+    assert Abs(i).is_even is None
+
+    o = Symbol('n', odd=True)
+    no = Symbol('no', real=True, odd=False)
+    assert Abs(o).is_odd is True
+    assert Abs(no).is_odd is False
+    assert Abs(i).is_odd is None
+
+
+def test_abs():
+    # this tests that abs calls Abs; don't rename to
+    # test_Abs since that test is already above
+    a = Symbol('a', positive=True)
+    assert abs(I*(1 + a)**2) == (1 + a)**2
+
+
+def test_arg():
+    assert arg(0) is nan
+    assert arg(1) == 0
+    assert arg(-1) == pi
+    assert arg(I) == pi/2
+    assert arg(-I) == -pi/2
+    assert arg(1 + I) == pi/4
+    assert arg(-1 + I) == pi*Rational(3, 4)
+    assert arg(1 - I) == -pi/4
+    assert arg(exp_polar(4*pi*I)) == 4*pi
+    assert arg(exp_polar(-7*pi*I)) == -7*pi
+    assert arg(exp_polar(5 - 3*pi*I/4)) == pi*Rational(-3, 4)
+    f = Function('f')
+    assert not arg(f(0) + I*f(1)).atoms(re)
+
+    # check nesting
+    x = Symbol('x')
+    assert arg(arg(arg(x))) is not S.NaN
+    assert arg(arg(arg(arg(x)))) is S.NaN
+    r = Symbol('r', extended_real=True)
+    assert arg(arg(r)) is not S.NaN
+    assert arg(arg(arg(r))) is S.NaN
+
+    p = Function('p', extended_positive=True)
+    assert arg(p(x)) == 0
+    assert arg((3 + I)*p(x)) == arg(3  + I)
+
+    p = Symbol('p', positive=True)
+    assert arg(p) == 0
+    assert arg(p*I) == pi/2
+
+    n = Symbol('n', negative=True)
+    assert arg(n) == pi
+    assert arg(n*I) == -pi/2
+
+    x = Symbol('x')
+    assert conjugate(arg(x)) == arg(x)
+
+    e = p + I*p**2
+    assert arg(e) == arg(1 + p*I)
+    # make sure sign doesn't swap
+    e = -2*p + 4*I*p**2
+    assert arg(e) == arg(-1 + 2*p*I)
+    # make sure sign isn't lost
+    x = symbols('x', real=True)  # could be zero
+    e = x + I*x
+    assert arg(e) == arg(x*(1 + I))
+    assert arg(e/p) == arg(x*(1 + I))
+    e = p*cos(p) + I*log(p)*exp(p)
+    assert arg(e).args[0] == e
+    # keep it simple -- let the user do more advanced cancellation
+    e = (p + 1) + I*(p**2 - 1)
+    assert arg(e).args[0] == e
+
+    f = Function('f')
+    e = 2*x*(f(0) - 1) - 2*x*f(0)
+    assert arg(e) == arg(-2*x)
+    assert arg(f(0)).func == arg and arg(f(0)).args == (f(0),)
+
+
+def test_arg_rewrite():
+    assert arg(1 + I) == atan2(1, 1)
+
+    x = Symbol('x', real=True)
+    y = Symbol('y', real=True)
+    assert arg(x + I*y).rewrite(atan2) == atan2(y, x)
+
+
+def test_adjoint():
+    a = Symbol('a', antihermitian=True)
+    b = Symbol('b', hermitian=True)
+    assert adjoint(a) == -a
+    assert adjoint(I*a) == I*a
+    assert adjoint(b) == b
+    assert adjoint(I*b) == -I*b
+    assert adjoint(a*b) == -b*a
+    assert adjoint(I*a*b) == I*b*a
+
+    x, y = symbols('x y')
+    assert adjoint(adjoint(x)) == x
+    assert adjoint(x + y) == adjoint(x) + adjoint(y)
+    assert adjoint(x - y) == adjoint(x) - adjoint(y)
+    assert adjoint(x * y) == adjoint(x) * adjoint(y)
+    assert adjoint(x / y) == adjoint(x) / adjoint(y)
+    assert adjoint(-x) == -adjoint(x)
+
+    x, y = symbols('x y', commutative=False)
+    assert adjoint(adjoint(x)) == x
+    assert adjoint(x + y) == adjoint(x) + adjoint(y)
+    assert adjoint(x - y) == adjoint(x) - adjoint(y)
+    assert adjoint(x * y) == adjoint(y) * adjoint(x)
+    assert adjoint(x / y) == 1 / adjoint(y) * adjoint(x)
+    assert adjoint(-x) == -adjoint(x)
+
+
+def test_conjugate():
+    a = Symbol('a', real=True)
+    b = Symbol('b', imaginary=True)
+    assert conjugate(a) == a
+    assert conjugate(I*a) == -I*a
+    assert conjugate(b) == -b
+    assert conjugate(I*b) == I*b
+    assert conjugate(a*b) == -a*b
+    assert conjugate(I*a*b) == I*a*b
+
+    x, y = symbols('x y')
+    assert conjugate(conjugate(x)) == x
+    assert conjugate(x).inverse() == conjugate
+    assert conjugate(x + y) == conjugate(x) + conjugate(y)
+    assert conjugate(x - y) == conjugate(x) - conjugate(y)
+    assert conjugate(x * y) == conjugate(x) * conjugate(y)
+    assert conjugate(x / y) == conjugate(x) / conjugate(y)
+    assert conjugate(-x) == -conjugate(x)
+
+    a = Symbol('a', algebraic=True)
+    t = Symbol('t', transcendental=True)
+    assert re(a).is_algebraic
+    assert re(x).is_algebraic is None
+    assert re(t).is_algebraic is False
+
+
+def test_conjugate_transpose():
+    x = Symbol('x')
+    assert conjugate(transpose(x)) == adjoint(x)
+    assert transpose(conjugate(x)) == adjoint(x)
+    assert adjoint(transpose(x)) == conjugate(x)
+    assert transpose(adjoint(x)) == conjugate(x)
+    assert adjoint(conjugate(x)) == transpose(x)
+    assert conjugate(adjoint(x)) == transpose(x)
+
+    class Symmetric(Expr):
+        def _eval_adjoint(self):
+            return None
+
+        def _eval_conjugate(self):
+            return None
+
+        def _eval_transpose(self):
+            return self
+    x = Symmetric()
+    assert conjugate(x) == adjoint(x)
+    assert transpose(x) == x
+
+
+def test_transpose():
+    a = Symbol('a', complex=True)
+    assert transpose(a) == a
+    assert transpose(I*a) == I*a
+
+    x, y = symbols('x y')
+    assert transpose(transpose(x)) == x
+    assert transpose(x + y) == transpose(x) + transpose(y)
+    assert transpose(x - y) == transpose(x) - transpose(y)
+    assert transpose(x * y) == transpose(x) * transpose(y)
+    assert transpose(x / y) == transpose(x) / transpose(y)
+    assert transpose(-x) == -transpose(x)
+
+    x, y = symbols('x y', commutative=False)
+    assert transpose(transpose(x)) == x
+    assert transpose(x + y) == transpose(x) + transpose(y)
+    assert transpose(x - y) == transpose(x) - transpose(y)
+    assert transpose(x * y) == transpose(y) * transpose(x)
+    assert transpose(x / y) == 1 / transpose(y) * transpose(x)
+    assert transpose(-x) == -transpose(x)
+
+
+@_both_exp_pow
+def test_polarify():
+    from sympy.functions.elementary.complexes import (polar_lift, polarify)
+    x = Symbol('x')
+    z = Symbol('z', polar=True)
+    f = Function('f')
+    ES = {}
+
+    assert polarify(-1) == (polar_lift(-1), ES)
+    assert polarify(1 + I) == (polar_lift(1 + I), ES)
+
+    assert polarify(exp(x), subs=False) == exp(x)
+    assert polarify(1 + x, subs=False) == 1 + x
+    assert polarify(f(I) + x, subs=False) == f(polar_lift(I)) + x
+
+    assert polarify(x, lift=True) == polar_lift(x)
+    assert polarify(z, lift=True) == z
+    assert polarify(f(x), lift=True) == f(polar_lift(x))
+    assert polarify(1 + x, lift=True) == polar_lift(1 + x)
+    assert polarify(1 + f(x), lift=True) == polar_lift(1 + f(polar_lift(x)))
+
+    newex, subs = polarify(f(x) + z)
+    assert newex.subs(subs) == f(x) + z
+
+    mu = Symbol("mu")
+    sigma = Symbol("sigma", positive=True)
+
+    # Make sure polarify(lift=True) doesn't try to lift the integration
+    # variable
+    assert polarify(
+        Integral(sqrt(2)*x*exp(-(-mu + x)**2/(2*sigma**2))/(2*sqrt(pi)*sigma),
+        (x, -oo, oo)), lift=True) == Integral(sqrt(2)*(sigma*exp_polar(0))**exp_polar(I*pi)*
+        exp((sigma*exp_polar(0))**(2*exp_polar(I*pi))*exp_polar(I*pi)*polar_lift(-mu + x)**
+        (2*exp_polar(0))/2)*exp_polar(0)*polar_lift(x)/(2*sqrt(pi)), (x, -oo, oo))
+
+
+def test_unpolarify():
+    from sympy.functions.elementary.complexes import (polar_lift, principal_branch, unpolarify)
+    from sympy.core.relational import Ne
+    from sympy.functions.elementary.hyperbolic import tanh
+    from sympy.functions.special.error_functions import erf
+    from sympy.functions.special.gamma_functions import (gamma, uppergamma)
+    from sympy.abc import x
+    p = exp_polar(7*I) + 1
+    u = exp(7*I) + 1
+
+    assert unpolarify(1) == 1
+    assert unpolarify(p) == u
+    assert unpolarify(p**2) == u**2
+    assert unpolarify(p**x) == p**x
+    assert unpolarify(p*x) == u*x
+    assert unpolarify(p + x) == u + x
+    assert unpolarify(sqrt(sin(p))) == sqrt(sin(u))
+
+    # Test reduction to principal branch 2*pi.
+    t = principal_branch(x, 2*pi)
+    assert unpolarify(t) == x
+    assert unpolarify(sqrt(t)) == sqrt(t)
+
+    # Test exponents_only.
+    assert unpolarify(p**p, exponents_only=True) == p**u
+    assert unpolarify(uppergamma(x, p**p)) == uppergamma(x, p**u)
+
+    # Test functions.
+    assert unpolarify(sin(p)) == sin(u)
+    assert unpolarify(tanh(p)) == tanh(u)
+    assert unpolarify(gamma(p)) == gamma(u)
+    assert unpolarify(erf(p)) == erf(u)
+    assert unpolarify(uppergamma(x, p)) == uppergamma(x, p)
+
+    assert unpolarify(uppergamma(sin(p), sin(p + exp_polar(0)))) == \
+        uppergamma(sin(u), sin(u + 1))
+    assert unpolarify(uppergamma(polar_lift(0), 2*exp_polar(0))) == \
+        uppergamma(0, 2)
+
+    assert unpolarify(Eq(p, 0)) == Eq(u, 0)
+    assert unpolarify(Ne(p, 0)) == Ne(u, 0)
+    assert unpolarify(polar_lift(x) > 0) == (x > 0)
+
+    # Test bools
+    assert unpolarify(True) is True
+
+
+def test_issue_4035():
+    x = Symbol('x')
+    assert Abs(x).expand(trig=True) == Abs(x)
+    assert sign(x).expand(trig=True) == sign(x)
+    assert arg(x).expand(trig=True) == arg(x)
+
+
+def test_issue_3206():
+    x = Symbol('x')
+    assert Abs(Abs(x)) == Abs(x)
+
+
+def test_issue_4754_derivative_conjugate():
+    x = Symbol('x', real=True)
+    y = Symbol('y', imaginary=True)
+    f = Function('f')
+    assert (f(x).conjugate()).diff(x) == (f(x).diff(x)).conjugate()
+    assert (f(y).conjugate()).diff(y) == -(f(y).diff(y)).conjugate()
+
+
+def test_derivatives_issue_4757():
+    x = Symbol('x', real=True)
+    y = Symbol('y', imaginary=True)
+    f = Function('f')
+    assert re(f(x)).diff(x) == re(f(x).diff(x))
+    assert im(f(x)).diff(x) == im(f(x).diff(x))
+    assert re(f(y)).diff(y) == -I*im(f(y).diff(y))
+    assert im(f(y)).diff(y) == -I*re(f(y).diff(y))
+    assert Abs(f(x)).diff(x).subs(f(x), 1 + I*x).doit() == x/sqrt(1 + x**2)
+    assert arg(f(x)).diff(x).subs(f(x), 1 + I*x**2).doit() == 2*x/(1 + x**4)
+    assert Abs(f(y)).diff(y).subs(f(y), 1 + y).doit() == -y/sqrt(1 - y**2)
+    assert arg(f(y)).diff(y).subs(f(y), I + y**2).doit() == 2*y/(1 + y**4)
+
+
+def test_issue_11413():
+    from sympy.simplify.simplify import simplify
+    v0 = Symbol('v0')
+    v1 = Symbol('v1')
+    v2 = Symbol('v2')
+    V = Matrix([[v0],[v1],[v2]])
+    U = V.normalized()
+    assert U == Matrix([
+    [v0/sqrt(Abs(v0)**2 + Abs(v1)**2 + Abs(v2)**2)],
+    [v1/sqrt(Abs(v0)**2 + Abs(v1)**2 + Abs(v2)**2)],
+    [v2/sqrt(Abs(v0)**2 + Abs(v1)**2 + Abs(v2)**2)]])
+    U.norm = sqrt(v0**2/(v0**2 + v1**2 + v2**2) + v1**2/(v0**2 + v1**2 + v2**2) + v2**2/(v0**2 + v1**2 + v2**2))
+    assert simplify(U.norm) == 1
+
+
+def test_periodic_argument():
+    from sympy.functions.elementary.complexes import (periodic_argument, polar_lift, principal_branch, unbranched_argument)
+    x = Symbol('x')
+    p = Symbol('p', positive=True)
+
+    assert unbranched_argument(2 + I) == periodic_argument(2 + I, oo)
+    assert unbranched_argument(1 + x) == periodic_argument(1 + x, oo)
+    assert N_equals(unbranched_argument((1 + I)**2), pi/2)
+    assert N_equals(unbranched_argument((1 - I)**2), -pi/2)
+    assert N_equals(periodic_argument((1 + I)**2, 3*pi), pi/2)
+    assert N_equals(periodic_argument((1 - I)**2, 3*pi), -pi/2)
+
+    assert unbranched_argument(principal_branch(x, pi)) == \
+        periodic_argument(x, pi)
+
+    assert unbranched_argument(polar_lift(2 + I)) == unbranched_argument(2 + I)
+    assert periodic_argument(polar_lift(2 + I), 2*pi) == \
+        periodic_argument(2 + I, 2*pi)
+    assert periodic_argument(polar_lift(2 + I), 3*pi) == \
+        periodic_argument(2 + I, 3*pi)
+    assert periodic_argument(polar_lift(2 + I), pi) == \
+        periodic_argument(polar_lift(2 + I), pi)
+
+    assert unbranched_argument(polar_lift(1 + I)) == pi/4
+    assert periodic_argument(2*p, p) == periodic_argument(p, p)
+    assert periodic_argument(pi*p, p) == periodic_argument(p, p)
+
+    assert Abs(polar_lift(1 + I)) == Abs(1 + I)
+
+
+@XFAIL
+def test_principal_branch_fail():
+    # TODO XXX why does abs(x)._eval_evalf() not fall back to global evalf?
+    from sympy.functions.elementary.complexes import principal_branch
+    assert N_equals(principal_branch((1 + I)**2, pi/2), 0)
+
+
+def test_principal_branch():
+    from sympy.functions.elementary.complexes import (polar_lift, principal_branch)
+    p = Symbol('p', positive=True)
+    x = Symbol('x')
+    neg = Symbol('x', negative=True)
+
+    assert principal_branch(polar_lift(x), p) == principal_branch(x, p)
+    assert principal_branch(polar_lift(2 + I), p) == principal_branch(2 + I, p)
+    assert principal_branch(2*x, p) == 2*principal_branch(x, p)
+    assert principal_branch(1, pi) == exp_polar(0)
+    assert principal_branch(-1, 2*pi) == exp_polar(I*pi)
+    assert principal_branch(-1, pi) == exp_polar(0)
+    assert principal_branch(exp_polar(3*pi*I)*x, 2*pi) == \
+        principal_branch(exp_polar(I*pi)*x, 2*pi)
+    assert principal_branch(neg*exp_polar(pi*I), 2*pi) == neg*exp_polar(-I*pi)
+    # related to issue #14692
+    assert principal_branch(exp_polar(-I*pi/2)/polar_lift(neg), 2*pi) == \
+        exp_polar(-I*pi/2)/neg
+
+    assert N_equals(principal_branch((1 + I)**2, 2*pi), 2*I)
+    assert N_equals(principal_branch((1 + I)**2, 3*pi), 2*I)
+    assert N_equals(principal_branch((1 + I)**2, 1*pi), 2*I)
+
+    # test argument sanitization
+    assert principal_branch(x, I).func is principal_branch
+    assert principal_branch(x, -4).func is principal_branch
+    assert principal_branch(x, -oo).func is principal_branch
+    assert principal_branch(x, zoo).func is principal_branch
+
+
+@XFAIL
+def test_issue_6167_6151():
+    n = pi**1000
+    i = int(n)
+    assert sign(n - i) == 1
+    assert abs(n - i) == n - i
+    x = Symbol('x')
+    eps = pi**-1500
+    big = pi**1000
+    one = cos(x)**2 + sin(x)**2
+    e = big*one - big + eps
+    from sympy.simplify.simplify import simplify
+    assert sign(simplify(e)) == 1
+    for xi in (111, 11, 1, Rational(1, 10)):
+        assert sign(e.subs(x, xi)) == 1
+
+
+def test_issue_14216():
+    from sympy.functions.elementary.complexes import unpolarify
+    A = MatrixSymbol("A", 2, 2)
+    assert unpolarify(A[0, 0]) == A[0, 0]
+    assert unpolarify(A[0, 0]*A[1, 0]) == A[0, 0]*A[1, 0]
+
+
+def test_issue_14238():
+    # doesn't cause recursion error
+    r = Symbol('r', real=True)
+    assert Abs(r + Piecewise((0, r > 0), (1 - r, True)))
+
+
+def test_issue_22189():
+    x = Symbol('x')
+    for a in (sqrt(7 - 2*x) - 2, 1 - x):
+        assert Abs(a) - Abs(-a) == 0, a
+
+
+def test_zero_assumptions():
+    nr = Symbol('nonreal', real=False, finite=True)
+    ni = Symbol('nonimaginary', imaginary=False)
+    # imaginary implies not zero
+    nzni = Symbol('nonzerononimaginary', zero=False, imaginary=False)
+
+    assert re(nr).is_zero is None
+    assert im(nr).is_zero is False
+
+    assert re(ni).is_zero is None
+    assert im(ni).is_zero is None
+
+    assert re(nzni).is_zero is False
+    assert im(nzni).is_zero is None
+
+
+@_both_exp_pow
+def test_issue_15893():
+    f = Function('f', real=True)
+    x = Symbol('x', real=True)
+    eq = Derivative(Abs(f(x)), f(x))
+    assert eq.doit() == sign(f(x))

llmeval-env/lib/python3.10/site-packages/sympy/functions/elementary/tests/test_exponential.py

@@ -0,0 +1,806 @@
+from sympy.assumptions.refine import refine
+from sympy.calculus.accumulationbounds import AccumBounds
+from sympy.concrete.products import Product
+from sympy.concrete.summations import Sum
+from sympy.core.function import expand_log
+from sympy.core.numbers import (E, Float, I, Rational, nan, oo, pi, zoo)
+from sympy.core.power import Pow
+from sympy.core.singleton import S
+from sympy.core.symbol import (Symbol, symbols)
+from sympy.functions.elementary.complexes import (adjoint, conjugate, re, sign, transpose)
+from sympy.functions.elementary.exponential import (LambertW, exp, exp_polar, log)
+from sympy.functions.elementary.hyperbolic import (cosh, sinh, tanh)
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.trigonometric import (cos, sin, tan)
+from sympy.matrices.expressions.matexpr import MatrixSymbol
+from sympy.polys.polytools import gcd
+from sympy.series.order import O
+from sympy.simplify.simplify import simplify
+from sympy.core.parameters import global_parameters
+from sympy.functions.elementary.exponential import match_real_imag
+from sympy.abc import x, y, z
+from sympy.core.expr import unchanged
+from sympy.core.function import ArgumentIndexError
+from sympy.testing.pytest import raises, XFAIL, _both_exp_pow
+
+
+@_both_exp_pow
+def test_exp_values():
+    if global_parameters.exp_is_pow:
+        assert type(exp(x)) is Pow
+    else:
+        assert type(exp(x)) is exp
+
+    k = Symbol('k', integer=True)
+
+    assert exp(nan) is nan
+
+    assert exp(oo) is oo
+    assert exp(-oo) == 0
+
+    assert exp(0) == 1
+    assert exp(1) == E
+    assert exp(-1 + x).as_base_exp() == (S.Exp1, x - 1)
+    assert exp(1 + x).as_base_exp() == (S.Exp1, x + 1)
+
+    assert exp(pi*I/2) == I
+    assert exp(pi*I) == -1
+    assert exp(pi*I*Rational(3, 2)) == -I
+    assert exp(2*pi*I) == 1
+
+    assert refine(exp(pi*I*2*k)) == 1
+    assert refine(exp(pi*I*2*(k + S.Half))) == -1
+    assert refine(exp(pi*I*2*(k + Rational(1, 4)))) == I
+    assert refine(exp(pi*I*2*(k + Rational(3, 4)))) == -I
+
+    assert exp(log(x)) == x
+    assert exp(2*log(x)) == x**2
+    assert exp(pi*log(x)) == x**pi
+
+    assert exp(17*log(x) + E*log(y)) == x**17 * y**E
+
+    assert exp(x*log(x)) != x**x
+    assert exp(sin(x)*log(x)) != x
+
+    assert exp(3*log(x) + oo*x) == exp(oo*x) * x**3
+    assert exp(4*log(x)*log(y) + 3*log(x)) == x**3 * exp(4*log(x)*log(y))
+
+    assert exp(-oo, evaluate=False).is_finite is True
+    assert exp(oo, evaluate=False).is_finite is False
+
+
+@_both_exp_pow
+def test_exp_period():
+    assert exp(I*pi*Rational(9, 4)) == exp(I*pi/4)
+    assert exp(I*pi*Rational(46, 18)) == exp(I*pi*Rational(5, 9))
+    assert exp(I*pi*Rational(25, 7)) == exp(I*pi*Rational(-3, 7))
+    assert exp(I*pi*Rational(-19, 3)) == exp(-I*pi/3)
+    assert exp(I*pi*Rational(37, 8)) - exp(I*pi*Rational(-11, 8)) == 0
+    assert exp(I*pi*Rational(-5, 3)) / exp(I*pi*Rational(11, 5)) * exp(I*pi*Rational(148, 15)) == 1
+
+    assert exp(2 - I*pi*Rational(17, 5)) == exp(2 + I*pi*Rational(3, 5))
+    assert exp(log(3) + I*pi*Rational(29, 9)) == 3 * exp(I*pi*Rational(-7, 9))
+
+    n = Symbol('n', integer=True)
+    e = Symbol('e', even=True)
+    assert exp(e*I*pi) == 1
+    assert exp((e + 1)*I*pi) == -1
+    assert exp((1 + 4*n)*I*pi/2) == I
+    assert exp((-1 + 4*n)*I*pi/2) == -I
+
+
+@_both_exp_pow
+def test_exp_log():
+    x = Symbol("x", real=True)
+    assert log(exp(x)) == x
+    assert exp(log(x)) == x
+
+    if not global_parameters.exp_is_pow:
+        assert log(x).inverse() == exp
+        assert exp(x).inverse() == log
+
+    y = Symbol("y", polar=True)
+    assert log(exp_polar(z)) == z
+    assert exp(log(y)) == y
+
+
+@_both_exp_pow
+def test_exp_expand():
+    e = exp(log(Rational(2))*(1 + x) - log(Rational(2))*x)
+    assert e.expand() == 2
+    assert exp(x + y) != exp(x)*exp(y)
+    assert exp(x + y).expand() == exp(x)*exp(y)
+
+
+@_both_exp_pow
+def test_exp__as_base_exp():
+    assert exp(x).as_base_exp() == (E, x)
+    assert exp(2*x).as_base_exp() == (E, 2*x)
+    assert exp(x*y).as_base_exp() == (E, x*y)
+    assert exp(-x).as_base_exp() == (E, -x)
+
+    # Pow( *expr.as_base_exp() ) == expr    invariant should hold
+    assert E**x == exp(x)
+    assert E**(2*x) == exp(2*x)
+    assert E**(x*y) == exp(x*y)
+
+    assert exp(x).base is S.Exp1
+    assert exp(x).exp == x
+
+
+@_both_exp_pow
+def test_exp_infinity():
+    assert exp(I*y) != nan
+    assert refine(exp(I*oo)) is nan
+    assert refine(exp(-I*oo)) is nan
+    assert exp(y*I*oo) != nan
+    assert exp(zoo) is nan
+    x = Symbol('x', extended_real=True, finite=False)
+    assert exp(x).is_complex is None
+
+
+@_both_exp_pow
+def test_exp_subs():
+    x = Symbol('x')
+    e = (exp(3*log(x), evaluate=False))  # evaluates to x**3
+    assert e.subs(x**3, y**3) == e
+    assert e.subs(x**2, 5) == e
+    assert (x**3).subs(x**2, y) != y**Rational(3, 2)
+    assert exp(exp(x) + exp(x**2)).subs(exp(exp(x)), y) == y * exp(exp(x**2))
+    assert exp(x).subs(E, y) == y**x
+    x = symbols('x', real=True)
+    assert exp(5*x).subs(exp(7*x), y) == y**Rational(5, 7)
+    assert exp(2*x + 7).subs(exp(3*x), y) == y**Rational(2, 3) * exp(7)
+    x = symbols('x', positive=True)
+    assert exp(3*log(x)).subs(x**2, y) == y**Rational(3, 2)
+    # differentiate between E and exp
+    assert exp(exp(x + E)).subs(exp, 3) == 3**(3**(x + E))
+    assert exp(exp(x + E)).subs(exp, sin) == sin(sin(x + E))
+    assert exp(exp(x + E)).subs(E, 3) == 3**(3**(x + 3))
+    assert exp(3).subs(E, sin) == sin(3)
+
+
+def test_exp_adjoint():
+    assert adjoint(exp(x)) == exp(adjoint(x))
+
+
+def test_exp_conjugate():
+    assert conjugate(exp(x)) == exp(conjugate(x))
+
+
+@_both_exp_pow
+def test_exp_transpose():
+    assert transpose(exp(x)) == exp(transpose(x))
+
+
+@_both_exp_pow
+def test_exp_rewrite():
+    assert exp(x).rewrite(sin) == sinh(x) + cosh(x)
+    assert exp(x*I).rewrite(cos) == cos(x) + I*sin(x)
+    assert exp(1).rewrite(cos) == sinh(1) + cosh(1)
+    assert exp(1).rewrite(sin) == sinh(1) + cosh(1)
+    assert exp(1).rewrite(sin) == sinh(1) + cosh(1)
+    assert exp(x).rewrite(tanh) == (1 + tanh(x/2))/(1 - tanh(x/2))
+    assert exp(pi*I/4).rewrite(sqrt) == sqrt(2)/2 + sqrt(2)*I/2
+    assert exp(pi*I/3).rewrite(sqrt) == S.Half + sqrt(3)*I/2
+    if not global_parameters.exp_is_pow:
+        assert exp(x*log(y)).rewrite(Pow) == y**x
+        assert exp(log(x)*log(y)).rewrite(Pow) in [x**log(y), y**log(x)]
+        assert exp(log(log(x))*y).rewrite(Pow) == log(x)**y
+
+    n = Symbol('n', integer=True)
+
+    assert Sum((exp(pi*I/2)/2)**n, (n, 0, oo)).rewrite(sqrt).doit() == Rational(4, 5) + I*2/5
+    assert Sum((exp(pi*I/4)/2)**n, (n, 0, oo)).rewrite(sqrt).doit() == 1/(1 - sqrt(2)*(1 + I)/4)
+    assert (Sum((exp(pi*I/3)/2)**n, (n, 0, oo)).rewrite(sqrt).doit().cancel()
+            == 4*I/(sqrt(3) + 3*I))
+
+
+@_both_exp_pow
+def test_exp_leading_term():
+    assert exp(x).as_leading_term(x) == 1
+    assert exp(2 + x).as_leading_term(x) == exp(2)
+    assert exp((2*x + 3) / (x+1)).as_leading_term(x) == exp(3)
+
+    # The following tests are commented, since now SymPy returns the
+    # original function when the leading term in the series expansion does
+    # not exist.
+    # raises(NotImplementedError, lambda: exp(1/x).as_leading_term(x))
+    # raises(NotImplementedError, lambda: exp((x + 1) / x**2).as_leading_term(x))
+    # raises(NotImplementedError, lambda: exp(x + 1/x).as_leading_term(x))
+
+
+@_both_exp_pow
+def test_exp_taylor_term():
+    x = symbols('x')
+    assert exp(x).taylor_term(1, x) == x
+    assert exp(x).taylor_term(3, x) == x**3/6
+    assert exp(x).taylor_term(4, x) == x**4/24
+    assert exp(x).taylor_term(-1, x) is S.Zero
+
+
+def test_exp_MatrixSymbol():
+    A = MatrixSymbol("A", 2, 2)
+    assert exp(A).has(exp)
+
+
+def test_exp_fdiff():
+    x = Symbol('x')
+    raises(ArgumentIndexError, lambda: exp(x).fdiff(2))
+
+
+def test_log_values():
+    assert log(nan) is nan
+
+    assert log(oo) is oo
+    assert log(-oo) is oo
+
+    assert log(zoo) is zoo
+    assert log(-zoo) is zoo
+
+    assert log(0) is zoo
+
+    assert log(1) == 0
+    assert log(-1) == I*pi
+
+    assert log(E) == 1
+    assert log(-E).expand() == 1 + I*pi
+
+    assert unchanged(log, pi)
+    assert log(-pi).expand() == log(pi) + I*pi
+
+    assert unchanged(log, 17)
+    assert log(-17) == log(17) + I*pi
+
+    assert log(I) == I*pi/2
+    assert log(-I) == -I*pi/2
+
+    assert log(17*I) == I*pi/2 + log(17)
+    assert log(-17*I).expand() == -I*pi/2 + log(17)
+
+    assert log(oo*I) is oo
+    assert log(-oo*I) is oo
+    assert log(0, 2) is zoo
+    assert log(0, 5) is zoo
+
+    assert exp(-log(3))**(-1) == 3
+
+    assert log(S.Half) == -log(2)
+    assert log(2*3).func is log
+    assert log(2*3**2).func is log
+
+
+def test_match_real_imag():
+    x, y = symbols('x,y', real=True)
+    i = Symbol('i', imaginary=True)
+    assert match_real_imag(S.One) == (1, 0)
+    assert match_real_imag(I) == (0, 1)
+    assert match_real_imag(3 - 5*I) == (3, -5)
+    assert match_real_imag(-sqrt(3) + S.Half*I) == (-sqrt(3), S.Half)
+    assert match_real_imag(x + y*I) == (x, y)
+    assert match_real_imag(x*I + y*I) == (0, x + y)
+    assert match_real_imag((x + y)*I) == (0, x + y)
+    assert match_real_imag(Rational(-2, 3)*i*I) == (None, None)
+    assert match_real_imag(1 - 2*i) == (None, None)
+    assert match_real_imag(sqrt(2)*(3 - 5*I)) == (None, None)
+
+
+def test_log_exact():
+    # check for pi/2, pi/3, pi/4, pi/6, pi/8, pi/12; pi/5, pi/10:
+    for n in range(-23, 24):
+        if gcd(n, 24) != 1:
+            assert log(exp(n*I*pi/24).rewrite(sqrt)) == n*I*pi/24
+        for n in range(-9, 10):
+            assert log(exp(n*I*pi/10).rewrite(sqrt)) == n*I*pi/10
+
+    assert log(S.Half - I*sqrt(3)/2) == -I*pi/3
+    assert log(Rational(-1, 2) + I*sqrt(3)/2) == I*pi*Rational(2, 3)
+    assert log(-sqrt(2)/2 - I*sqrt(2)/2) == -I*pi*Rational(3, 4)
+    assert log(-sqrt(3)/2 - I*S.Half) == -I*pi*Rational(5, 6)
+
+    assert log(Rational(-1, 4) + sqrt(5)/4 - I*sqrt(sqrt(5)/8 + Rational(5, 8))) == -I*pi*Rational(2, 5)
+    assert log(sqrt(Rational(5, 8) - sqrt(5)/8) + I*(Rational(1, 4) + sqrt(5)/4)) == I*pi*Rational(3, 10)
+    assert log(-sqrt(sqrt(2)/4 + S.Half) + I*sqrt(S.Half - sqrt(2)/4)) == I*pi*Rational(7, 8)
+    assert log(-sqrt(6)/4 - sqrt(2)/4 + I*(-sqrt(6)/4 + sqrt(2)/4)) == -I*pi*Rational(11, 12)
+
+    assert log(-1 + I*sqrt(3)) == log(2) + I*pi*Rational(2, 3)
+    assert log(5 + 5*I) == log(5*sqrt(2)) + I*pi/4
+    assert log(sqrt(-12)) == log(2*sqrt(3)) + I*pi/2
+    assert log(-sqrt(6) + sqrt(2) - I*sqrt(6) - I*sqrt(2)) == log(4) - I*pi*Rational(7, 12)
+    assert log(-sqrt(6-3*sqrt(2)) - I*sqrt(6+3*sqrt(2))) == log(2*sqrt(3)) - I*pi*Rational(5, 8)
+    assert log(1 + I*sqrt(2-sqrt(2))/sqrt(2+sqrt(2))) == log(2/sqrt(sqrt(2) + 2)) + I*pi/8
+    assert log(cos(pi*Rational(7, 12)) + I*sin(pi*Rational(7, 12))) == I*pi*Rational(7, 12)
+    assert log(cos(pi*Rational(6, 5)) + I*sin(pi*Rational(6, 5))) == I*pi*Rational(-4, 5)
+
+    assert log(5*(1 + I)/sqrt(2)) == log(5) + I*pi/4
+    assert log(sqrt(2)*(-sqrt(3) + 1 - sqrt(3)*I - I)) == log(4) - I*pi*Rational(7, 12)
+    assert log(-sqrt(2)*(1 - I*sqrt(3))) == log(2*sqrt(2)) + I*pi*Rational(2, 3)
+    assert log(sqrt(3)*I*(-sqrt(6 - 3*sqrt(2)) - I*sqrt(3*sqrt(2) + 6))) == log(6) - I*pi/8
+
+    zero = (1 + sqrt(2))**2 - 3 - 2*sqrt(2)
+    assert log(zero - I*sqrt(3)) == log(sqrt(3)) - I*pi/2
+    assert unchanged(log, zero + I*zero) or log(zero + zero*I) is zoo
+
+    # bail quickly if no obvious simplification is possible:
+    assert unchanged(log, (sqrt(2)-1/sqrt(sqrt(3)+I))**1000)
+    # beware of non-real coefficients
+    assert unchanged(log, sqrt(2-sqrt(5))*(1 + I))
+
+
+def test_log_base():
+    assert log(1, 2) == 0
+    assert log(2, 2) == 1
+    assert log(3, 2) == log(3)/log(2)
+    assert log(6, 2) == 1 + log(3)/log(2)
+    assert log(6, 3) == 1 + log(2)/log(3)
+    assert log(2**3, 2) == 3
+    assert log(3**3, 3) == 3
+    assert log(5, 1) is zoo
+    assert log(1, 1) is nan
+    assert log(Rational(2, 3), 10) == log(Rational(2, 3))/log(10)
+    assert log(Rational(2, 3), Rational(1, 3)) == -log(2)/log(3) + 1
+    assert log(Rational(2, 3), Rational(2, 5)) == \
+        log(Rational(2, 3))/log(Rational(2, 5))
+    # issue 17148
+    assert log(Rational(8, 3), 2) == -log(3)/log(2) + 3
+
+
+def test_log_symbolic():
+    assert log(x, exp(1)) == log(x)
+    assert log(exp(x)) != x
+
+    assert log(x, exp(1)) == log(x)
+    assert log(x*y) != log(x) + log(y)
+    assert log(x/y).expand() != log(x) - log(y)
+    assert log(x/y).expand(force=True) == log(x) - log(y)
+    assert log(x**y).expand() != y*log(x)
+    assert log(x**y).expand(force=True) == y*log(x)
+
+    assert log(x, 2) == log(x)/log(2)
+    assert log(E, 2) == 1/log(2)
+
+    p, q = symbols('p,q', positive=True)
+    r = Symbol('r', real=True)
+
+    assert log(p**2) != 2*log(p)
+    assert log(p**2).expand() == 2*log(p)
+    assert log(x**2).expand() != 2*log(x)
+    assert log(p**q) != q*log(p)
+    assert log(exp(p)) == p
+    assert log(p*q) != log(p) + log(q)
+    assert log(p*q).expand() == log(p) + log(q)
+
+    assert log(-sqrt(3)) == log(sqrt(3)) + I*pi
+    assert log(-exp(p)) != p + I*pi
+    assert log(-exp(x)).expand() != x + I*pi
+    assert log(-exp(r)).expand() == r + I*pi
+
+    assert log(x**y) != y*log(x)
+
+    assert (log(x**-5)**-1).expand() != -1/log(x)/5
+    assert (log(p**-5)**-1).expand() == -1/log(p)/5
+    assert log(-x).func is log and log(-x).args[0] == -x
+    assert log(-p).func is log and log(-p).args[0] == -p
+
+
+def test_log_exp():
+    assert log(exp(4*I*pi)) == 0     # exp evaluates
+    assert log(exp(-5*I*pi)) == I*pi # exp evaluates
+    assert log(exp(I*pi*Rational(19, 4))) == I*pi*Rational(3, 4)
+    assert log(exp(I*pi*Rational(25, 7))) == I*pi*Rational(-3, 7)
+    assert log(exp(-5*I)) == -5*I + 2*I*pi
+
+
+@_both_exp_pow
+def test_exp_assumptions():
+    r = Symbol('r', real=True)
+    i = Symbol('i', imaginary=True)
+    for e in exp, exp_polar:
+        assert e(x).is_real is None
+        assert e(x).is_imaginary is None
+        assert e(i).is_real is None
+        assert e(i).is_imaginary is None
+        assert e(r).is_real is True
+        assert e(r).is_imaginary is False
+        assert e(re(x)).is_extended_real is True
+        assert e(re(x)).is_imaginary is False
+
+    assert Pow(E, I*pi, evaluate=False).is_imaginary == False
+    assert Pow(E, 2*I*pi, evaluate=False).is_imaginary == False
+    assert Pow(E, I*pi/2, evaluate=False).is_imaginary == True
+    assert Pow(E, I*pi/3, evaluate=False).is_imaginary is None
+
+    assert exp(0, evaluate=False).is_algebraic
+
+    a = Symbol('a', algebraic=True)
+    an = Symbol('an', algebraic=True, nonzero=True)
+    r = Symbol('r', rational=True)
+    rn = Symbol('rn', rational=True, nonzero=True)
+    assert exp(a).is_algebraic is None
+    assert exp(an).is_algebraic is False
+    assert exp(pi*r).is_algebraic is None
+    assert exp(pi*rn).is_algebraic is False
+
+    assert exp(0, evaluate=False).is_algebraic is True
+    assert exp(I*pi/3, evaluate=False).is_algebraic is True
+    assert exp(I*pi*r, evaluate=False).is_algebraic is True
+
+
+@_both_exp_pow
+def test_exp_AccumBounds():
+    assert exp(AccumBounds(1, 2)) == AccumBounds(E, E**2)
+
+
+def test_log_assumptions():
+    p = symbols('p', positive=True)
+    n = symbols('n', negative=True)
+    z = symbols('z', zero=True)
+    x = symbols('x', infinite=True, extended_positive=True)
+
+    assert log(z).is_positive is False
+    assert log(x).is_extended_positive is True
+    assert log(2) > 0
+    assert log(1, evaluate=False).is_zero
+    assert log(1 + z).is_zero
+    assert log(p).is_zero is None
+    assert log(n).is_zero is False
+    assert log(0.5).is_negative is True
+    assert log(exp(p) + 1).is_positive
+
+    assert log(1, evaluate=False).is_algebraic
+    assert log(42, evaluate=False).is_algebraic is False
+
+    assert log(1 + z).is_rational
+
+
+def test_log_hashing():
+    assert x != log(log(x))
+    assert hash(x) != hash(log(log(x)))
+    assert log(x) != log(log(log(x)))
+
+    e = 1/log(log(x) + log(log(x)))
+    assert e.base.func is log
+    e = 1/log(log(x) + log(log(log(x))))
+    assert e.base.func is log
+
+    e = log(log(x))
+    assert e.func is log
+    assert x.func is not log
+    assert hash(log(log(x))) != hash(x)
+    assert e != x
+
+
+def test_log_sign():
+    assert sign(log(2)) == 1
+
+
+def test_log_expand_complex():
+    assert log(1 + I).expand(complex=True) == log(2)/2 + I*pi/4
+    assert log(1 - sqrt(2)).expand(complex=True) == log(sqrt(2) - 1) + I*pi
+
+
+def test_log_apply_evalf():
+    value = (log(3)/log(2) - 1).evalf()
+    assert value.epsilon_eq(Float("0.58496250072115618145373"))
+
+
+def test_log_leading_term():
+    p = Symbol('p')
+
+    # Test for STEP 3
+    assert log(1 + x + x**2).as_leading_term(x, cdir=1) == x
+    # Test for STEP 4
+    assert log(2*x).as_leading_term(x, cdir=1) == log(x) + log(2)
+    assert log(2*x).as_leading_term(x, cdir=-1) == log(x) + log(2)
+    assert log(-2*x).as_leading_term(x, cdir=1, logx=p) == p + log(2) + I*pi
+    assert log(-2*x).as_leading_term(x, cdir=-1, logx=p) == p + log(2) - I*pi
+    # Test for STEP 5
+    assert log(-2*x + (3 - I)*x**2).as_leading_term(x, cdir=1) == log(x) + log(2) - I*pi
+    assert log(-2*x + (3 - I)*x**2).as_leading_term(x, cdir=-1) == log(x) + log(2) - I*pi
+    assert log(2*x + (3 - I)*x**2).as_leading_term(x, cdir=1) == log(x) + log(2)
+    assert log(2*x + (3 - I)*x**2).as_leading_term(x, cdir=-1) == log(x) + log(2) - 2*I*pi
+    assert log(-1 + x - I*x**2 + I*x**3).as_leading_term(x, cdir=1) == -I*pi
+    assert log(-1 + x - I*x**2 + I*x**3).as_leading_term(x, cdir=-1) == -I*pi
+    assert log(-1/(1 - x)).as_leading_term(x, cdir=1) == I*pi
+    assert log(-1/(1 - x)).as_leading_term(x, cdir=-1) == I*pi
+
+
+def test_log_nseries():
+    p = Symbol('p')
+    assert log(1/x)._eval_nseries(x, 4, logx=-p, cdir=1) == p
+    assert log(1/x)._eval_nseries(x, 4, logx=-p, cdir=-1) == p + 2*I*pi
+    assert log(x - 1)._eval_nseries(x, 4, None, I) == I*pi - x - x**2/2 - x**3/3 + O(x**4)
+    assert log(x - 1)._eval_nseries(x, 4, None, -I) == -I*pi - x - x**2/2 - x**3/3 + O(x**4)
+    assert log(I*x + I*x**3 - 1)._eval_nseries(x, 3, None, 1) == I*pi - I*x + x**2/2 + O(x**3)
+    assert log(I*x + I*x**3 - 1)._eval_nseries(x, 3, None, -1) == -I*pi - I*x + x**2/2 + O(x**3)
+    assert log(I*x**2 + I*x**3 - 1)._eval_nseries(x, 3, None, 1) == I*pi - I*x**2 + O(x**3)
+    assert log(I*x**2 + I*x**3 - 1)._eval_nseries(x, 3, None, -1) == I*pi - I*x**2 + O(x**3)
+    assert log(2*x + (3 - I)*x**2)._eval_nseries(x, 3, None, 1) == log(2) + log(x) + \
+    x*(S(3)/2 - I/2) + x**2*(-1 + 3*I/4) + O(x**3)
+    assert log(2*x + (3 - I)*x**2)._eval_nseries(x, 3, None, -1) == -2*I*pi + log(2) + \
+    log(x) - x*(-S(3)/2 + I/2) + x**2*(-1 + 3*I/4) + O(x**3)
+    assert log(-2*x + (3 - I)*x**2)._eval_nseries(x, 3, None, 1) == -I*pi + log(2) + log(x) + \
+    x*(-S(3)/2 + I/2) + x**2*(-1 + 3*I/4) + O(x**3)
+    assert log(-2*x + (3 - I)*x**2)._eval_nseries(x, 3, None, -1) == -I*pi + log(2) + log(x) - \
+    x*(S(3)/2 - I/2) + x**2*(-1 + 3*I/4) + O(x**3)
+    assert log(sqrt(-I*x**2 - 3)*sqrt(-I*x**2 - 1) - 2)._eval_nseries(x, 3, None, 1) == -I*pi + \
+    log(sqrt(3) + 2) + I*x**2*(-2 + 4*sqrt(3)/3) + O(x**3)
+    assert log(-1/(1 - x))._eval_nseries(x, 3, None, 1) == I*pi + x + x**2/2 + O(x**3)
+    assert log(-1/(1 - x))._eval_nseries(x, 3, None, -1) == I*pi + x + x**2/2 + O(x**3)
+
+
+def test_log_series():
+    # Note Series at infinities other than oo/-oo were introduced as a part of
+    # pull request 23798. Refer https://github.com/sympy/sympy/pull/23798 for
+    # more information.
+    expr1 = log(1 + x)
+    expr2 = log(x + sqrt(x**2 + 1))
+
+    assert expr1.series(x, x0=I*oo, n=4) == 1/(3*x**3) - 1/(2*x**2) + 1/x + \
+    I*pi/2 - log(I/x) + O(x**(-4), (x, oo*I))
+    assert expr1.series(x, x0=-I*oo, n=4) == 1/(3*x**3) - 1/(2*x**2) + 1/x - \
+    I*pi/2 - log(-I/x) + O(x**(-4), (x, -oo*I))
+    assert expr2.series(x, x0=I*oo, n=4) == 1/(4*x**2) + I*pi/2 + log(2) - \
+    log(I/x) + O(x**(-4), (x, oo*I))
+    assert expr2.series(x, x0=-I*oo, n=4) == -1/(4*x**2) - I*pi/2 - log(2) + \
+    log(-I/x) + O(x**(-4), (x, -oo*I))
+
+
+def test_log_expand():
+    w = Symbol("w", positive=True)
+    e = log(w**(log(5)/log(3)))
+    assert e.expand() == log(5)/log(3) * log(w)
+    x, y, z = symbols('x,y,z', positive=True)
+    assert log(x*(y + z)).expand(mul=False) == log(x) + log(y + z)
+    assert log(log(x**2)*log(y*z)).expand() in [log(2*log(x)*log(y) +
+        2*log(x)*log(z)), log(log(x)*log(z) + log(y)*log(x)) + log(2),
+        log((log(y) + log(z))*log(x)) + log(2)]
+    assert log(x**log(x**2)).expand(deep=False) == log(x)*log(x**2)
+    assert log(x**log(x**2)).expand() == 2*log(x)**2
+    x, y = symbols('x,y')
+    assert log(x*y).expand(force=True) == log(x) + log(y)
+    assert log(x**y).expand(force=True) == y*log(x)
+    assert log(exp(x)).expand(force=True) == x
+
+    # there's generally no need to expand out logs since this requires
+    # factoring and if simplification is sought, it's cheaper to put
+    # logs together than it is to take them apart.
+    assert log(2*3**2).expand() != 2*log(3) + log(2)
+
+
+@XFAIL
+def test_log_expand_fail():
+    x, y, z = symbols('x,y,z', positive=True)
+    assert (log(x*(y + z))*(x + y)).expand(mul=True, log=True) == y*log(
+        x) + y*log(y + z) + z*log(x) + z*log(y + z)
+
+
+def test_log_simplify():
+    x = Symbol("x", positive=True)
+    assert log(x**2).expand() == 2*log(x)
+    assert expand_log(log(x**(2 + log(2)))) == (2 + log(2))*log(x)
+
+    z = Symbol('z')
+    assert log(sqrt(z)).expand() == log(z)/2
+    assert expand_log(log(z**(log(2) - 1))) == (log(2) - 1)*log(z)
+    assert log(z**(-1)).expand() != -log(z)
+    assert log(z**(x/(x+1))).expand() == x*log(z)/(x + 1)
+
+
+def test_log_AccumBounds():
+    assert log(AccumBounds(1, E)) == AccumBounds(0, 1)
+    assert log(AccumBounds(0, E)) == AccumBounds(-oo, 1)
+    assert log(AccumBounds(-1, E)) == S.NaN
+    assert log(AccumBounds(0, oo)) == AccumBounds(-oo, oo)
+    assert log(AccumBounds(-oo, 0)) == S.NaN
+    assert log(AccumBounds(-oo, oo)) == S.NaN
+
+
+@_both_exp_pow
+def test_lambertw():
+    k = Symbol('k')
+
+    assert LambertW(x, 0) == LambertW(x)
+    assert LambertW(x, 0, evaluate=False) != LambertW(x)
+    assert LambertW(0) == 0
+    assert LambertW(E) == 1
+    assert LambertW(-1/E) == -1
+    assert LambertW(-log(2)/2) == -log(2)
+    assert LambertW(oo) is oo
+    assert LambertW(0, 1) is -oo
+    assert LambertW(0, 42) is -oo
+    assert LambertW(-pi/2, -1) == -I*pi/2
+    assert LambertW(-1/E, -1) == -1
+    assert LambertW(-2*exp(-2), -1) == -2
+    assert LambertW(2*log(2)) == log(2)
+    assert LambertW(-pi/2) == I*pi/2
+    assert LambertW(exp(1 + E)) == E
+
+    assert LambertW(x**2).diff(x) == 2*LambertW(x**2)/x/(1 + LambertW(x**2))
+    assert LambertW(x, k).diff(x) == LambertW(x, k)/x/(1 + LambertW(x, k))
+
+    assert LambertW(sqrt(2)).evalf(30).epsilon_eq(
+        Float("0.701338383413663009202120278965", 30), 1e-29)
+    assert re(LambertW(2, -1)).evalf().epsilon_eq(Float("-0.834310366631110"))
+
+    assert LambertW(-1).is_real is False  # issue 5215
+    assert LambertW(2, evaluate=False).is_real
+    p = Symbol('p', positive=True)
+    assert LambertW(p, evaluate=False).is_real
+    assert LambertW(p - 1, evaluate=False).is_real is None
+    assert LambertW(-p - 2/S.Exp1, evaluate=False).is_real is False
+    assert LambertW(S.Half, -1, evaluate=False).is_real is False
+    assert LambertW(Rational(-1, 10), -1, evaluate=False).is_real
+    assert LambertW(-10, -1, evaluate=False).is_real is False
+    assert LambertW(-2, 2, evaluate=False).is_real is False
+
+    assert LambertW(0, evaluate=False).is_algebraic
+    na = Symbol('na', nonzero=True, algebraic=True)
+    assert LambertW(na).is_algebraic is False
+    assert LambertW(p).is_zero is False
+    n = Symbol('n', negative=True)
+    assert LambertW(n).is_zero is False
+
+
+def test_issue_5673():
+    e = LambertW(-1)
+    assert e.is_comparable is False
+    assert e.is_positive is not True
+    e2 = 1 - 1/(1 - exp(-1000))
+    assert e2.is_positive is not True
+    e3 = -2 + exp(exp(LambertW(log(2)))*LambertW(log(2)))
+    assert e3.is_nonzero is not True
+
+
+def test_log_fdiff():
+    x = Symbol('x')
+    raises(ArgumentIndexError, lambda: log(x).fdiff(2))
+
+
+def test_log_taylor_term():
+    x = symbols('x')
+    assert log(x).taylor_term(0, x) == x
+    assert log(x).taylor_term(1, x) == -x**2/2
+    assert log(x).taylor_term(4, x) == x**5/5
+    assert log(x).taylor_term(-1, x) is S.Zero
+
+
+def test_exp_expand_NC():
+    A, B, C = symbols('A,B,C', commutative=False)
+
+    assert exp(A + B).expand() == exp(A + B)
+    assert exp(A + B + C).expand() == exp(A + B + C)
+    assert exp(x + y).expand() == exp(x)*exp(y)
+    assert exp(x + y + z).expand() == exp(x)*exp(y)*exp(z)
+
+
+@_both_exp_pow
+def test_as_numer_denom():
+    n = symbols('n', negative=True)
+    assert exp(x).as_numer_denom() == (exp(x), 1)
+    assert exp(-x).as_numer_denom() == (1, exp(x))
+    assert exp(-2*x).as_numer_denom() == (1, exp(2*x))
+    assert exp(-2).as_numer_denom() == (1, exp(2))
+    assert exp(n).as_numer_denom() == (1, exp(-n))
+    assert exp(-n).as_numer_denom() == (exp(-n), 1)
+    assert exp(-I*x).as_numer_denom() == (1, exp(I*x))
+    assert exp(-I*n).as_numer_denom() == (1, exp(I*n))
+    assert exp(-n).as_numer_denom() == (exp(-n), 1)
+
+
+@_both_exp_pow
+def test_polar():
+    x, y = symbols('x y', polar=True)
+
+    assert abs(exp_polar(I*4)) == 1
+    assert abs(exp_polar(0)) == 1
+    assert abs(exp_polar(2 + 3*I)) == exp(2)
+    assert exp_polar(I*10).n() == exp_polar(I*10)
+
+    assert log(exp_polar(z)) == z
+    assert log(x*y).expand() == log(x) + log(y)
+    assert log(x**z).expand() == z*log(x)
+
+    assert exp_polar(3).exp == 3
+
+    # Compare exp(1.0*pi*I).
+    assert (exp_polar(1.0*pi*I).n(n=5)).as_real_imag()[1] >= 0
+
+    assert exp_polar(0).is_rational is True  # issue 8008
+
+
+def test_exp_summation():
+    w = symbols("w")
+    m, n, i, j = symbols("m n i j")
+    expr = exp(Sum(w*i, (i, 0, n), (j, 0, m)))
+    assert expr.expand() == Product(exp(w*i), (i, 0, n), (j, 0, m))
+
+
+def test_log_product():
+    from sympy.abc import n, m
+
+    i, j = symbols('i,j', positive=True, integer=True)
+    x, y = symbols('x,y', positive=True)
+    z = symbols('z', real=True)
+    w = symbols('w')
+
+    expr = log(Product(x**i, (i, 1, n)))
+    assert simplify(expr) == expr
+    assert expr.expand() == Sum(i*log(x), (i, 1, n))
+    expr = log(Product(x**i*y**j, (i, 1, n), (j, 1, m)))
+    assert simplify(expr) == expr
+    assert expr.expand() == Sum(i*log(x) + j*log(y), (i, 1, n), (j, 1, m))
+
+    expr = log(Product(-2, (n, 0, 4)))
+    assert simplify(expr) == expr
+    assert expr.expand() == expr
+    assert expr.expand(force=True) == Sum(log(-2), (n, 0, 4))
+
+    expr = log(Product(exp(z*i), (i, 0, n)))
+    assert expr.expand() == Sum(z*i, (i, 0, n))
+
+    expr = log(Product(exp(w*i), (i, 0, n)))
+    assert expr.expand() == expr
+    assert expr.expand(force=True) == Sum(w*i, (i, 0, n))
+
+    expr = log(Product(i**2*abs(j), (i, 1, n), (j, 1, m)))
+    assert expr.expand() == Sum(2*log(i) + log(j), (i, 1, n), (j, 1, m))
+
+
+@XFAIL
+def test_log_product_simplify_to_sum():
+    from sympy.abc import n, m
+    i, j = symbols('i,j', positive=True, integer=True)
+    x, y = symbols('x,y', positive=True)
+    assert simplify(log(Product(x**i, (i, 1, n)))) == Sum(i*log(x), (i, 1, n))
+    assert simplify(log(Product(x**i*y**j, (i, 1, n), (j, 1, m)))) == \
+            Sum(i*log(x) + j*log(y), (i, 1, n), (j, 1, m))
+
+
+def test_issue_8866():
+    assert simplify(log(x, 10, evaluate=False)) == simplify(log(x, 10))
+    assert expand_log(log(x, 10, evaluate=False)) == expand_log(log(x, 10))
+
+    y = Symbol('y', positive=True)
+    l1 = log(exp(y), exp(10))
+    b1 = log(exp(y), exp(5))
+    l2 = log(exp(y), exp(10), evaluate=False)
+    b2 = log(exp(y), exp(5), evaluate=False)
+    assert simplify(log(l1, b1)) == simplify(log(l2, b2))
+    assert expand_log(log(l1, b1)) == expand_log(log(l2, b2))
+
+
+def test_log_expand_factor():
+    assert (log(18)/log(3) - 2).expand(factor=True) == log(2)/log(3)
+    assert (log(12)/log(2)).expand(factor=True) == log(3)/log(2) + 2
+    assert (log(15)/log(3)).expand(factor=True) == 1 + log(5)/log(3)
+    assert (log(2)/(-log(12) + log(24))).expand(factor=True) == 1
+
+    assert expand_log(log(12), factor=True) == log(3) + 2*log(2)
+    assert expand_log(log(21)/log(7), factor=False) == log(3)/log(7) + 1
+    assert expand_log(log(45)/log(5) + log(20), factor=False) == \
+        1 + 2*log(3)/log(5) + log(20)
+    assert expand_log(log(45)/log(5) + log(26), factor=True) == \
+        log(2) + log(13) + (log(5) + 2*log(3))/log(5)
+
+
+def test_issue_9116():
+    n = Symbol('n', positive=True, integer=True)
+    assert log(n).is_nonnegative is True
+
+
+def test_issue_18473():
+    assert exp(x*log(cos(1/x))).as_leading_term(x) == S.NaN
+    assert exp(x*log(tan(1/x))).as_leading_term(x) == S.NaN
+    assert log(cos(1/x)).as_leading_term(x) == S.NaN
+    assert log(tan(1/x)).as_leading_term(x) == S.NaN
+    assert log(cos(1/x) + 2).as_leading_term(x) == AccumBounds(0, log(3))
+    assert exp(x*log(cos(1/x) + 2)).as_leading_term(x) == 1
+    assert log(cos(1/x) - 2).as_leading_term(x) == S.NaN
+    assert exp(x*log(cos(1/x) - 2)).as_leading_term(x) == S.NaN
+    assert log(cos(1/x) + 1).as_leading_term(x) == AccumBounds(-oo, log(2))
+    assert exp(x*log(cos(1/x) + 1)).as_leading_term(x) == AccumBounds(0, 1)
+    assert log(sin(1/x)**2).as_leading_term(x) == AccumBounds(-oo, 0)
+    assert exp(x*log(sin(1/x)**2)).as_leading_term(x) == AccumBounds(0, 1)
+    assert log(tan(1/x)**2).as_leading_term(x) == AccumBounds(-oo, oo)
+    assert exp(2*x*(log(tan(1/x)**2))).as_leading_term(x) == AccumBounds(0, oo)

llmeval-env/lib/python3.10/site-packages/sympy/functions/elementary/tests/test_hyperbolic.py

@@ -0,0 +1,1460 @@
+from sympy.calculus.accumulationbounds import AccumBounds
+from sympy.core.function import (expand_mul, expand_trig)
+from sympy.core.numbers import (E, I, Integer, Rational, nan, oo, pi, zoo)
+from sympy.core.singleton import S
+from sympy.core.symbol import (Symbol, symbols)
+from sympy.functions.elementary.complexes import (im, re)
+from sympy.functions.elementary.exponential import (exp, log)
+from sympy.functions.elementary.hyperbolic import (acosh, acoth, acsch, asech, asinh, atanh, cosh, coth, csch, sech, sinh, tanh)
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.trigonometric import (acos, asin, cos, cot, sec, sin, tan)
+from sympy.series.order import O
+
+from sympy.core.expr import unchanged
+from sympy.core.function import ArgumentIndexError
+from sympy.testing.pytest import raises
+
+
+def test_sinh():
+    x, y = symbols('x,y')
+
+    k = Symbol('k', integer=True)
+
+    assert sinh(nan) is nan
+    assert sinh(zoo) is nan
+
+    assert sinh(oo) is oo
+    assert sinh(-oo) is -oo
+
+    assert sinh(0) == 0
+
+    assert unchanged(sinh, 1)
+    assert sinh(-1) == -sinh(1)
+
+    assert unchanged(sinh, x)
+    assert sinh(-x) == -sinh(x)
+
+    assert unchanged(sinh, pi)
+    assert sinh(-pi) == -sinh(pi)
+
+    assert unchanged(sinh, 2**1024 * E)
+    assert sinh(-2**1024 * E) == -sinh(2**1024 * E)
+
+    assert sinh(pi*I) == 0
+    assert sinh(-pi*I) == 0
+    assert sinh(2*pi*I) == 0
+    assert sinh(-2*pi*I) == 0
+    assert sinh(-3*10**73*pi*I) == 0
+    assert sinh(7*10**103*pi*I) == 0
+
+    assert sinh(pi*I/2) == I
+    assert sinh(-pi*I/2) == -I
+    assert sinh(pi*I*Rational(5, 2)) == I
+    assert sinh(pi*I*Rational(7, 2)) == -I
+
+    assert sinh(pi*I/3) == S.Half*sqrt(3)*I
+    assert sinh(pi*I*Rational(-2, 3)) == Rational(-1, 2)*sqrt(3)*I
+
+    assert sinh(pi*I/4) == S.Half*sqrt(2)*I
+    assert sinh(-pi*I/4) == Rational(-1, 2)*sqrt(2)*I
+    assert sinh(pi*I*Rational(17, 4)) == S.Half*sqrt(2)*I
+    assert sinh(pi*I*Rational(-3, 4)) == Rational(-1, 2)*sqrt(2)*I
+
+    assert sinh(pi*I/6) == S.Half*I
+    assert sinh(-pi*I/6) == Rational(-1, 2)*I
+    assert sinh(pi*I*Rational(7, 6)) == Rational(-1, 2)*I
+    assert sinh(pi*I*Rational(-5, 6)) == Rational(-1, 2)*I
+
+    assert sinh(pi*I/105) == sin(pi/105)*I
+    assert sinh(-pi*I/105) == -sin(pi/105)*I
+
+    assert unchanged(sinh, 2 + 3*I)
+
+    assert sinh(x*I) == sin(x)*I
+
+    assert sinh(k*pi*I) == 0
+    assert sinh(17*k*pi*I) == 0
+
+    assert sinh(k*pi*I/2) == sin(k*pi/2)*I
+
+    assert sinh(x).as_real_imag(deep=False) == (cos(im(x))*sinh(re(x)),
+                sin(im(x))*cosh(re(x)))
+    x = Symbol('x', extended_real=True)
+    assert sinh(x).as_real_imag(deep=False) == (sinh(x), 0)
+
+    x = Symbol('x', real=True)
+    assert sinh(I*x).is_finite is True
+    assert sinh(x).is_real is True
+    assert sinh(I).is_real is False
+    p = Symbol('p', positive=True)
+    assert sinh(p).is_zero is False
+    assert sinh(0, evaluate=False).is_zero is True
+    assert sinh(2*pi*I, evaluate=False).is_zero is True
+
+
+def test_sinh_series():
+    x = Symbol('x')
+    assert sinh(x).series(x, 0, 10) == \
+        x + x**3/6 + x**5/120 + x**7/5040 + x**9/362880 + O(x**10)
+
+
+def test_sinh_fdiff():
+    x = Symbol('x')
+    raises(ArgumentIndexError, lambda: sinh(x).fdiff(2))
+
+
+def test_cosh():
+    x, y = symbols('x,y')
+
+    k = Symbol('k', integer=True)
+
+    assert cosh(nan) is nan
+    assert cosh(zoo) is nan
+
+    assert cosh(oo) is oo
+    assert cosh(-oo) is oo
+
+    assert cosh(0) == 1
+
+    assert unchanged(cosh, 1)
+    assert cosh(-1) == cosh(1)
+
+    assert unchanged(cosh, x)
+    assert cosh(-x) == cosh(x)
+
+    assert cosh(pi*I) == cos(pi)
+    assert cosh(-pi*I) == cos(pi)
+
+    assert unchanged(cosh, 2**1024 * E)
+    assert cosh(-2**1024 * E) == cosh(2**1024 * E)
+
+    assert cosh(pi*I/2) == 0
+    assert cosh(-pi*I/2) == 0
+    assert cosh((-3*10**73 + 1)*pi*I/2) == 0
+    assert cosh((7*10**103 + 1)*pi*I/2) == 0
+
+    assert cosh(pi*I) == -1
+    assert cosh(-pi*I) == -1
+    assert cosh(5*pi*I) == -1
+    assert cosh(8*pi*I) == 1
+
+    assert cosh(pi*I/3) == S.Half
+    assert cosh(pi*I*Rational(-2, 3)) == Rational(-1, 2)
+
+    assert cosh(pi*I/4) == S.Half*sqrt(2)
+    assert cosh(-pi*I/4) == S.Half*sqrt(2)
+    assert cosh(pi*I*Rational(11, 4)) == Rational(-1, 2)*sqrt(2)
+    assert cosh(pi*I*Rational(-3, 4)) == Rational(-1, 2)*sqrt(2)
+
+    assert cosh(pi*I/6) == S.Half*sqrt(3)
+    assert cosh(-pi*I/6) == S.Half*sqrt(3)
+    assert cosh(pi*I*Rational(7, 6)) == Rational(-1, 2)*sqrt(3)
+    assert cosh(pi*I*Rational(-5, 6)) == Rational(-1, 2)*sqrt(3)
+
+    assert cosh(pi*I/105) == cos(pi/105)
+    assert cosh(-pi*I/105) == cos(pi/105)
+
+    assert unchanged(cosh, 2 + 3*I)
+
+    assert cosh(x*I) == cos(x)
+
+    assert cosh(k*pi*I) == cos(k*pi)
+    assert cosh(17*k*pi*I) == cos(17*k*pi)
+
+    assert unchanged(cosh, k*pi)
+
+    assert cosh(x).as_real_imag(deep=False) == (cos(im(x))*cosh(re(x)),
+                sin(im(x))*sinh(re(x)))
+    x = Symbol('x', extended_real=True)
+    assert cosh(x).as_real_imag(deep=False) == (cosh(x), 0)
+
+    x = Symbol('x', real=True)
+    assert cosh(I*x).is_finite is True
+    assert cosh(I*x).is_real is True
+    assert cosh(I*2 + 1).is_real is False
+    assert cosh(5*I*S.Pi/2, evaluate=False).is_zero is True
+    assert cosh(x).is_zero is False
+
+
+def test_cosh_series():
+    x = Symbol('x')
+    assert cosh(x).series(x, 0, 10) == \
+        1 + x**2/2 + x**4/24 + x**6/720 + x**8/40320 + O(x**10)
+
+
+def test_cosh_fdiff():
+    x = Symbol('x')
+    raises(ArgumentIndexError, lambda: cosh(x).fdiff(2))
+
+
+def test_tanh():
+    x, y = symbols('x,y')
+
+    k = Symbol('k', integer=True)
+
+    assert tanh(nan) is nan
+    assert tanh(zoo) is nan
+
+    assert tanh(oo) == 1
+    assert tanh(-oo) == -1
+
+    assert tanh(0) == 0
+
+    assert unchanged(tanh, 1)
+    assert tanh(-1) == -tanh(1)
+
+    assert unchanged(tanh, x)
+    assert tanh(-x) == -tanh(x)
+
+    assert unchanged(tanh, pi)
+    assert tanh(-pi) == -tanh(pi)
+
+    assert unchanged(tanh, 2**1024 * E)
+    assert tanh(-2**1024 * E) == -tanh(2**1024 * E)
+
+    assert tanh(pi*I) == 0
+    assert tanh(-pi*I) == 0
+    assert tanh(2*pi*I) == 0
+    assert tanh(-2*pi*I) == 0
+    assert tanh(-3*10**73*pi*I) == 0
+    assert tanh(7*10**103*pi*I) == 0
+
+    assert tanh(pi*I/2) is zoo
+    assert tanh(-pi*I/2) is zoo
+    assert tanh(pi*I*Rational(5, 2)) is zoo
+    assert tanh(pi*I*Rational(7, 2)) is zoo
+
+    assert tanh(pi*I/3) == sqrt(3)*I
+    assert tanh(pi*I*Rational(-2, 3)) == sqrt(3)*I
+
+    assert tanh(pi*I/4) == I
+    assert tanh(-pi*I/4) == -I
+    assert tanh(pi*I*Rational(17, 4)) == I
+    assert tanh(pi*I*Rational(-3, 4)) == I
+
+    assert tanh(pi*I/6) == I/sqrt(3)
+    assert tanh(-pi*I/6) == -I/sqrt(3)
+    assert tanh(pi*I*Rational(7, 6)) == I/sqrt(3)
+    assert tanh(pi*I*Rational(-5, 6)) == I/sqrt(3)
+
+    assert tanh(pi*I/105) == tan(pi/105)*I
+    assert tanh(-pi*I/105) == -tan(pi/105)*I
+
+    assert unchanged(tanh, 2 + 3*I)
+
+    assert tanh(x*I) == tan(x)*I
+
+    assert tanh(k*pi*I) == 0
+    assert tanh(17*k*pi*I) == 0
+
+    assert tanh(k*pi*I/2) == tan(k*pi/2)*I
+
+    assert tanh(x).as_real_imag(deep=False) == (sinh(re(x))*cosh(re(x))/(cos(im(x))**2
+                                + sinh(re(x))**2),
+                                sin(im(x))*cos(im(x))/(cos(im(x))**2 + sinh(re(x))**2))
+    x = Symbol('x', extended_real=True)
+    assert tanh(x).as_real_imag(deep=False) == (tanh(x), 0)
+    assert tanh(I*pi/3 + 1).is_real is False
+    assert tanh(x).is_real is True
+    assert tanh(I*pi*x/2).is_real is None
+
+
+def test_tanh_series():
+    x = Symbol('x')
+    assert tanh(x).series(x, 0, 10) == \
+        x - x**3/3 + 2*x**5/15 - 17*x**7/315 + 62*x**9/2835 + O(x**10)
+
+
+def test_tanh_fdiff():
+    x = Symbol('x')
+    raises(ArgumentIndexError, lambda: tanh(x).fdiff(2))
+
+
+def test_coth():
+    x, y = symbols('x,y')
+
+    k = Symbol('k', integer=True)
+
+    assert coth(nan) is nan
+    assert coth(zoo) is nan
+
+    assert coth(oo) == 1
+    assert coth(-oo) == -1
+
+    assert coth(0) is zoo
+    assert unchanged(coth, 1)
+    assert coth(-1) == -coth(1)
+
+    assert unchanged(coth, x)
+    assert coth(-x) == -coth(x)
+
+    assert coth(pi*I) == -I*cot(pi)
+    assert coth(-pi*I) == cot(pi)*I
+
+    assert unchanged(coth, 2**1024 * E)
+    assert coth(-2**1024 * E) == -coth(2**1024 * E)
+
+    assert coth(pi*I) == -I*cot(pi)
+    assert coth(-pi*I) == I*cot(pi)
+    assert coth(2*pi*I) == -I*cot(2*pi)
+    assert coth(-2*pi*I) == I*cot(2*pi)
+    assert coth(-3*10**73*pi*I) == I*cot(3*10**73*pi)
+    assert coth(7*10**103*pi*I) == -I*cot(7*10**103*pi)
+
+    assert coth(pi*I/2) == 0
+    assert coth(-pi*I/2) == 0
+    assert coth(pi*I*Rational(5, 2)) == 0
+    assert coth(pi*I*Rational(7, 2)) == 0
+
+    assert coth(pi*I/3) == -I/sqrt(3)
+    assert coth(pi*I*Rational(-2, 3)) == -I/sqrt(3)
+
+    assert coth(pi*I/4) == -I
+    assert coth(-pi*I/4) == I
+    assert coth(pi*I*Rational(17, 4)) == -I
+    assert coth(pi*I*Rational(-3, 4)) == -I
+
+    assert coth(pi*I/6) == -sqrt(3)*I
+    assert coth(-pi*I/6) == sqrt(3)*I
+    assert coth(pi*I*Rational(7, 6)) == -sqrt(3)*I
+    assert coth(pi*I*Rational(-5, 6)) == -sqrt(3)*I
+
+    assert coth(pi*I/105) == -cot(pi/105)*I
+    assert coth(-pi*I/105) == cot(pi/105)*I
+
+    assert unchanged(coth, 2 + 3*I)
+
+    assert coth(x*I) == -cot(x)*I
+
+    assert coth(k*pi*I) == -cot(k*pi)*I
+    assert coth(17*k*pi*I) == -cot(17*k*pi)*I
+
+    assert coth(k*pi*I) == -cot(k*pi)*I
+
+    assert coth(log(tan(2))) == coth(log(-tan(2)))
+    assert coth(1 + I*pi/2) == tanh(1)
+
+    assert coth(x).as_real_imag(deep=False) == (sinh(re(x))*cosh(re(x))/(sin(im(x))**2
+                                + sinh(re(x))**2),
+                                -sin(im(x))*cos(im(x))/(sin(im(x))**2 + sinh(re(x))**2))
+    x = Symbol('x', extended_real=True)
+    assert coth(x).as_real_imag(deep=False) == (coth(x), 0)
+
+    assert expand_trig(coth(2*x)) == (coth(x)**2 + 1)/(2*coth(x))
+    assert expand_trig(coth(3*x)) == (coth(x)**3 + 3*coth(x))/(1 + 3*coth(x)**2)
+
+    assert expand_trig(coth(x + y)) == (1 + coth(x)*coth(y))/(coth(x) + coth(y))
+
+
+def test_coth_series():
+    x = Symbol('x')
+    assert coth(x).series(x, 0, 8) == \
+        1/x + x/3 - x**3/45 + 2*x**5/945 - x**7/4725 + O(x**8)
+
+
+def test_coth_fdiff():
+    x = Symbol('x')
+    raises(ArgumentIndexError, lambda: coth(x).fdiff(2))
+
+
+def test_csch():
+    x, y = symbols('x,y')
+
+    k = Symbol('k', integer=True)
+    n = Symbol('n', positive=True)
+
+    assert csch(nan) is nan
+    assert csch(zoo) is nan
+
+    assert csch(oo) == 0
+    assert csch(-oo) == 0
+
+    assert csch(0) is zoo
+
+    assert csch(-1) == -csch(1)
+
+    assert csch(-x) == -csch(x)
+    assert csch(-pi) == -csch(pi)
+    assert csch(-2**1024 * E) == -csch(2**1024 * E)
+
+    assert csch(pi*I) is zoo
+    assert csch(-pi*I) is zoo
+    assert csch(2*pi*I) is zoo
+    assert csch(-2*pi*I) is zoo
+    assert csch(-3*10**73*pi*I) is zoo
+    assert csch(7*10**103*pi*I) is zoo
+
+    assert csch(pi*I/2) == -I
+    assert csch(-pi*I/2) == I
+    assert csch(pi*I*Rational(5, 2)) == -I
+    assert csch(pi*I*Rational(7, 2)) == I
+
+    assert csch(pi*I/3) == -2/sqrt(3)*I
+    assert csch(pi*I*Rational(-2, 3)) == 2/sqrt(3)*I
+
+    assert csch(pi*I/4) == -sqrt(2)*I
+    assert csch(-pi*I/4) == sqrt(2)*I
+    assert csch(pi*I*Rational(7, 4)) == sqrt(2)*I
+    assert csch(pi*I*Rational(-3, 4)) == sqrt(2)*I
+
+    assert csch(pi*I/6) == -2*I
+    assert csch(-pi*I/6) == 2*I
+    assert csch(pi*I*Rational(7, 6)) == 2*I
+    assert csch(pi*I*Rational(-7, 6)) == -2*I
+    assert csch(pi*I*Rational(-5, 6)) == 2*I
+
+    assert csch(pi*I/105) == -1/sin(pi/105)*I
+    assert csch(-pi*I/105) == 1/sin(pi/105)*I
+
+    assert csch(x*I) == -1/sin(x)*I
+
+    assert csch(k*pi*I) is zoo
+    assert csch(17*k*pi*I) is zoo
+
+    assert csch(k*pi*I/2) == -1/sin(k*pi/2)*I
+
+    assert csch(n).is_real is True
+
+    assert expand_trig(csch(x + y)) == 1/(sinh(x)*cosh(y) + cosh(x)*sinh(y))
+
+
+def test_csch_series():
+    x = Symbol('x')
+    assert csch(x).series(x, 0, 10) == \
+       1/ x - x/6 + 7*x**3/360 - 31*x**5/15120 + 127*x**7/604800 \
+          - 73*x**9/3421440 + O(x**10)
+
+
+def test_csch_fdiff():
+    x = Symbol('x')
+    raises(ArgumentIndexError, lambda: csch(x).fdiff(2))
+
+
+def test_sech():
+    x, y = symbols('x, y')
+
+    k = Symbol('k', integer=True)
+    n = Symbol('n', positive=True)
+
+    assert sech(nan) is nan
+    assert sech(zoo) is nan
+
+    assert sech(oo) == 0
+    assert sech(-oo) == 0
+
+    assert sech(0) == 1
+
+    assert sech(-1) == sech(1)
+    assert sech(-x) == sech(x)
+
+    assert sech(pi*I) == sec(pi)
+
+    assert sech(-pi*I) == sec(pi)
+    assert sech(-2**1024 * E) == sech(2**1024 * E)
+
+    assert sech(pi*I/2) is zoo
+    assert sech(-pi*I/2) is zoo
+    assert sech((-3*10**73 + 1)*pi*I/2) is zoo
+    assert sech((7*10**103 + 1)*pi*I/2) is zoo
+
+    assert sech(pi*I) == -1
+    assert sech(-pi*I) == -1
+    assert sech(5*pi*I) == -1
+    assert sech(8*pi*I) == 1
+
+    assert sech(pi*I/3) == 2
+    assert sech(pi*I*Rational(-2, 3)) == -2
+
+    assert sech(pi*I/4) == sqrt(2)
+    assert sech(-pi*I/4) == sqrt(2)
+    assert sech(pi*I*Rational(5, 4)) == -sqrt(2)
+    assert sech(pi*I*Rational(-5, 4)) == -sqrt(2)
+
+    assert sech(pi*I/6) == 2/sqrt(3)
+    assert sech(-pi*I/6) == 2/sqrt(3)
+    assert sech(pi*I*Rational(7, 6)) == -2/sqrt(3)
+    assert sech(pi*I*Rational(-5, 6)) == -2/sqrt(3)
+
+    assert sech(pi*I/105) == 1/cos(pi/105)
+    assert sech(-pi*I/105) == 1/cos(pi/105)
+
+    assert sech(x*I) == 1/cos(x)
+
+    assert sech(k*pi*I) == 1/cos(k*pi)
+    assert sech(17*k*pi*I) == 1/cos(17*k*pi)
+
+    assert sech(n).is_real is True
+
+    assert expand_trig(sech(x + y)) == 1/(cosh(x)*cosh(y) + sinh(x)*sinh(y))
+
+
+def test_sech_series():
+    x = Symbol('x')
+    assert sech(x).series(x, 0, 10) == \
+        1 - x**2/2 + 5*x**4/24 - 61*x**6/720 + 277*x**8/8064 + O(x**10)
+
+
+def test_sech_fdiff():
+    x = Symbol('x')
+    raises(ArgumentIndexError, lambda: sech(x).fdiff(2))
+
+
+def test_asinh():
+    x, y = symbols('x,y')
+    assert unchanged(asinh, x)
+    assert asinh(-x) == -asinh(x)
+
+    #at specific points
+    assert asinh(nan) is nan
+    assert asinh( 0) == 0
+    assert asinh(+1) == log(sqrt(2) + 1)
+
+    assert asinh(-1) == log(sqrt(2) - 1)
+    assert asinh(I) == pi*I/2
+    assert asinh(-I) == -pi*I/2
+    assert asinh(I/2) == pi*I/6
+    assert asinh(-I/2) == -pi*I/6
+
+    # at infinites
+    assert asinh(oo) is oo
+    assert asinh(-oo) is -oo
+
+    assert asinh(I*oo) is oo
+    assert asinh(-I *oo) is -oo
+
+    assert asinh(zoo) is zoo
+
+    #properties
+    assert asinh(I *(sqrt(3) - 1)/(2**Rational(3, 2))) == pi*I/12
+    assert asinh(-I *(sqrt(3) - 1)/(2**Rational(3, 2))) == -pi*I/12
+
+    assert asinh(I*(sqrt(5) - 1)/4) == pi*I/10
+    assert asinh(-I*(sqrt(5) - 1)/4) == -pi*I/10
+
+    assert asinh(I*(sqrt(5) + 1)/4) == pi*I*Rational(3, 10)
+    assert asinh(-I*(sqrt(5) + 1)/4) == pi*I*Rational(-3, 10)
+
+    # Symmetry
+    assert asinh(Rational(-1, 2)) == -asinh(S.Half)
+
+    # inverse composition
+    assert unchanged(asinh, sinh(Symbol('v1')))
+
+    assert asinh(sinh(0, evaluate=False)) == 0
+    assert asinh(sinh(-3, evaluate=False)) == -3
+    assert asinh(sinh(2, evaluate=False)) == 2
+    assert asinh(sinh(I, evaluate=False)) == I
+    assert asinh(sinh(-I, evaluate=False)) == -I
+    assert asinh(sinh(5*I, evaluate=False)) == -2*I*pi + 5*I
+    assert asinh(sinh(15 + 11*I)) == 15 - 4*I*pi + 11*I
+    assert asinh(sinh(-73 + 97*I)) == 73 - 97*I + 31*I*pi
+    assert asinh(sinh(-7 - 23*I)) == 7 - 7*I*pi + 23*I
+    assert asinh(sinh(13 - 3*I)) == -13 - I*pi + 3*I
+    p = Symbol('p', positive=True)
+    assert asinh(p).is_zero is False
+    assert asinh(sinh(0, evaluate=False), evaluate=False).is_zero is True
+
+
+def test_asinh_rewrite():
+    x = Symbol('x')
+    assert asinh(x).rewrite(log) == log(x + sqrt(x**2 + 1))
+    assert asinh(x).rewrite(atanh) == atanh(x/sqrt(1 + x**2))
+    assert asinh(x).rewrite(asin) == asinh(x)
+    assert asinh(x*(1 + I)).rewrite(asin) == -I*asin(I*x*(1+I))
+    assert asinh(x).rewrite(acos) == I*(-I*asinh(x) + pi/2) - I*pi/2
+
+
+def test_asinh_leading_term():
+    x = Symbol('x')
+    assert asinh(x).as_leading_term(x, cdir=1) == x
+    # Tests concerning branch points
+    assert asinh(x + I).as_leading_term(x, cdir=1) == I*pi/2
+    assert asinh(x - I).as_leading_term(x, cdir=1) == -I*pi/2
+    assert asinh(1/x).as_leading_term(x, cdir=1) == -log(x) + log(2)
+    assert asinh(1/x).as_leading_term(x, cdir=-1) == log(x) - log(2) - I*pi
+    # Tests concerning points lying on branch cuts
+    assert asinh(x + 2*I).as_leading_term(x, cdir=1) == I*asin(2)
+    assert asinh(x + 2*I).as_leading_term(x, cdir=-1) == -I*asin(2) + I*pi
+    assert asinh(x - 2*I).as_leading_term(x, cdir=1) == -I*pi + I*asin(2)
+    assert asinh(x - 2*I).as_leading_term(x, cdir=-1) == -I*asin(2)
+    # Tests concerning re(ndir) == 0
+    assert asinh(2*I + I*x - x**2).as_leading_term(x, cdir=1) == log(2 - sqrt(3)) + I*pi/2
+    assert asinh(2*I + I*x - x**2).as_leading_term(x, cdir=-1) == log(2 - sqrt(3)) + I*pi/2
+
+
+def test_asinh_series():
+    x = Symbol('x')
+    assert asinh(x).series(x, 0, 8) == \
+        x - x**3/6 + 3*x**5/40 - 5*x**7/112 + O(x**8)
+    t5 = asinh(x).taylor_term(5, x)
+    assert t5 == 3*x**5/40
+    assert asinh(x).taylor_term(7, x, t5, 0) == -5*x**7/112
+
+
+def test_asinh_nseries():
+    x = Symbol('x')
+    # Tests concerning branch points
+    assert asinh(x + I)._eval_nseries(x, 4, None) == I*pi/2 + \
+    sqrt(x)*(1 - I) + x**(S(3)/2)*(S(1)/12 + I/12) + x**(S(5)/2)*(-S(3)/160 + 3*I/160) + \
+    x**(S(7)/2)*(-S(5)/896 - 5*I/896) + O(x**4)
+    assert asinh(x - I)._eval_nseries(x, 4, None) == -I*pi/2 + \
+    sqrt(x)*(1 + I) + x**(S(3)/2)*(S(1)/12 - I/12) + x**(S(5)/2)*(-S(3)/160 - 3*I/160) + \
+    x**(S(7)/2)*(-S(5)/896 + 5*I/896) + O(x**4)
+    # Tests concerning points lying on branch cuts
+    assert asinh(x + 2*I)._eval_nseries(x, 4, None, cdir=1) == I*asin(2) - \
+    sqrt(3)*I*x/3 + sqrt(3)*x**2/9 + sqrt(3)*I*x**3/18 + O(x**4)
+    assert asinh(x + 2*I)._eval_nseries(x, 4, None, cdir=-1) == I*pi - I*asin(2) + \
+    sqrt(3)*I*x/3 - sqrt(3)*x**2/9 - sqrt(3)*I*x**3/18 + O(x**4)
+    assert asinh(x - 2*I)._eval_nseries(x, 4, None, cdir=1) == I*asin(2) - I*pi + \
+    sqrt(3)*I*x/3 + sqrt(3)*x**2/9 - sqrt(3)*I*x**3/18 + O(x**4)
+    assert asinh(x - 2*I)._eval_nseries(x, 4, None, cdir=-1) == -I*asin(2) - \
+    sqrt(3)*I*x/3 - sqrt(3)*x**2/9 + sqrt(3)*I*x**3/18 + O(x**4)
+    # Tests concerning re(ndir) == 0
+    assert asinh(2*I + I*x - x**2)._eval_nseries(x, 4, None) == I*pi/2 + log(2 - sqrt(3)) - \
+    sqrt(3)*x/3 + x**2*(sqrt(3)/9 - sqrt(3)*I/3) + x**3*(-sqrt(3)/18 + 2*sqrt(3)*I/9) + O(x**4)
+
+
+def test_asinh_fdiff():
+    x = Symbol('x')
+    raises(ArgumentIndexError, lambda: asinh(x).fdiff(2))
+
+
+def test_acosh():
+    x = Symbol('x')
+
+    assert unchanged(acosh, -x)
+
+    #at specific points
+    assert acosh(1) == 0
+    assert acosh(-1) == pi*I
+    assert acosh(0) == I*pi/2
+    assert acosh(S.Half) == I*pi/3
+    assert acosh(Rational(-1, 2)) == pi*I*Rational(2, 3)
+    assert acosh(nan) is nan
+
+    # at infinites
+    assert acosh(oo) is oo
+    assert acosh(-oo) is oo
+
+    assert acosh(I*oo) == oo + I*pi/2
+    assert acosh(-I*oo) == oo - I*pi/2
+
+    assert acosh(zoo) is zoo
+
+    assert acosh(I) == log(I*(1 + sqrt(2)))
+    assert acosh(-I) == log(-I*(1 + sqrt(2)))
+    assert acosh((sqrt(3) - 1)/(2*sqrt(2))) == pi*I*Rational(5, 12)
+    assert acosh(-(sqrt(3) - 1)/(2*sqrt(2))) == pi*I*Rational(7, 12)
+    assert acosh(sqrt(2)/2) == I*pi/4
+    assert acosh(-sqrt(2)/2) == I*pi*Rational(3, 4)
+    assert acosh(sqrt(3)/2) == I*pi/6
+    assert acosh(-sqrt(3)/2) == I*pi*Rational(5, 6)
+    assert acosh(sqrt(2 + sqrt(2))/2) == I*pi/8
+    assert acosh(-sqrt(2 + sqrt(2))/2) == I*pi*Rational(7, 8)
+    assert acosh(sqrt(2 - sqrt(2))/2) == I*pi*Rational(3, 8)
+    assert acosh(-sqrt(2 - sqrt(2))/2) == I*pi*Rational(5, 8)
+    assert acosh((1 + sqrt(3))/(2*sqrt(2))) == I*pi/12
+    assert acosh(-(1 + sqrt(3))/(2*sqrt(2))) == I*pi*Rational(11, 12)
+    assert acosh((sqrt(5) + 1)/4) == I*pi/5
+    assert acosh(-(sqrt(5) + 1)/4) == I*pi*Rational(4, 5)
+
+    assert str(acosh(5*I).n(6)) == '2.31244 + 1.5708*I'
+    assert str(acosh(-5*I).n(6)) == '2.31244 - 1.5708*I'
+
+    # inverse composition
+    assert unchanged(acosh, Symbol('v1'))
+
+    assert acosh(cosh(-3, evaluate=False)) == 3
+    assert acosh(cosh(3, evaluate=False)) == 3
+    assert acosh(cosh(0, evaluate=False)) == 0
+    assert acosh(cosh(I, evaluate=False)) == I
+    assert acosh(cosh(-I, evaluate=False)) == I
+    assert acosh(cosh(7*I, evaluate=False)) == -2*I*pi + 7*I
+    assert acosh(cosh(1 + I)) == 1 + I
+    assert acosh(cosh(3 - 3*I)) == 3 - 3*I
+    assert acosh(cosh(-3 + 2*I)) == 3 - 2*I
+    assert acosh(cosh(-5 - 17*I)) == 5 - 6*I*pi + 17*I
+    assert acosh(cosh(-21 + 11*I)) == 21 - 11*I + 4*I*pi
+    assert acosh(cosh(cosh(1) + I)) == cosh(1) + I
+    assert acosh(1, evaluate=False).is_zero is True
+
+
+def test_acosh_rewrite():
+    x = Symbol('x')
+    assert acosh(x).rewrite(log) == log(x + sqrt(x - 1)*sqrt(x + 1))
+    assert acosh(x).rewrite(asin) == sqrt(x - 1)*(-asin(x) + pi/2)/sqrt(1 - x)
+    assert acosh(x).rewrite(asinh) == sqrt(x - 1)*(-asin(x) + pi/2)/sqrt(1 - x)
+    assert acosh(x).rewrite(atanh) == \
+        (sqrt(x - 1)*sqrt(x + 1)*atanh(sqrt(x**2 - 1)/x)/sqrt(x**2 - 1) +
+         pi*sqrt(x - 1)*(-x*sqrt(x**(-2)) + 1)/(2*sqrt(1 - x)))
+    x = Symbol('x', positive=True)
+    assert acosh(x).rewrite(atanh) == \
+        sqrt(x - 1)*sqrt(x + 1)*atanh(sqrt(x**2 - 1)/x)/sqrt(x**2 - 1)
+
+
+def test_acosh_leading_term():
+    x = Symbol('x')
+    # Tests concerning branch points
+    assert acosh(x).as_leading_term(x) == I*pi/2
+    assert acosh(x + 1).as_leading_term(x) == sqrt(2)*sqrt(x)
+    assert acosh(x - 1).as_leading_term(x) == I*pi
+    assert acosh(1/x).as_leading_term(x, cdir=1) == -log(x) + log(2)
+    assert acosh(1/x).as_leading_term(x, cdir=-1) == -log(x) + log(2) + 2*I*pi
+    # Tests concerning points lying on branch cuts
+    assert acosh(I*x - 2).as_leading_term(x, cdir=1) == acosh(-2)
+    assert acosh(-I*x - 2).as_leading_term(x, cdir=1) == -2*I*pi + acosh(-2)
+    assert acosh(x**2 - I*x + S(1)/3).as_leading_term(x, cdir=1) == -acosh(S(1)/3)
+    assert acosh(x**2 - I*x + S(1)/3).as_leading_term(x, cdir=-1) == acosh(S(1)/3)
+    assert acosh(1/(I*x - 3)).as_leading_term(x, cdir=1) == -acosh(-S(1)/3)
+    assert acosh(1/(I*x - 3)).as_leading_term(x, cdir=-1) == acosh(-S(1)/3)
+    # Tests concerning im(ndir) == 0
+    assert acosh(-I*x**2 + x - 2).as_leading_term(x, cdir=1) == log(sqrt(3) + 2) - I*pi
+    assert acosh(-I*x**2 + x - 2).as_leading_term(x, cdir=-1) == log(sqrt(3) + 2) - I*pi
+
+
+def test_acosh_series():
+    x = Symbol('x')
+    assert acosh(x).series(x, 0, 8) == \
+        -I*x + pi*I/2 - I*x**3/6 - 3*I*x**5/40 - 5*I*x**7/112 + O(x**8)
+    t5 = acosh(x).taylor_term(5, x)
+    assert t5 == - 3*I*x**5/40
+    assert acosh(x).taylor_term(7, x, t5, 0) == - 5*I*x**7/112
+
+
+def test_acosh_nseries():
+    x = Symbol('x')
+    # Tests concerning branch points
+    assert acosh(x + 1)._eval_nseries(x, 4, None) == sqrt(2)*sqrt(x) - \
+    sqrt(2)*x**(S(3)/2)/12 + 3*sqrt(2)*x**(S(5)/2)/160 - 5*sqrt(2)*x**(S(7)/2)/896 + O(x**4)
+    # Tests concerning points lying on branch cuts
+    assert acosh(x - 1)._eval_nseries(x, 4, None) == I*pi - \
+    sqrt(2)*I*sqrt(x) - sqrt(2)*I*x**(S(3)/2)/12 - 3*sqrt(2)*I*x**(S(5)/2)/160 - \
+    5*sqrt(2)*I*x**(S(7)/2)/896 + O(x**4)
+    assert acosh(I*x - 2)._eval_nseries(x, 4, None, cdir=1) == acosh(-2) - \
+    sqrt(3)*I*x/3 + sqrt(3)*x**2/9 + sqrt(3)*I*x**3/18 + O(x**4)
+    assert acosh(-I*x - 2)._eval_nseries(x, 4, None, cdir=1) == acosh(-2) - \
+    2*I*pi + sqrt(3)*I*x/3 + sqrt(3)*x**2/9 - sqrt(3)*I*x**3/18 + O(x**4)
+    assert acosh(1/(I*x - 3))._eval_nseries(x, 4, None, cdir=1) == -acosh(-S(1)/3) + \
+    sqrt(2)*x/12 + 17*sqrt(2)*I*x**2/576 - 443*sqrt(2)*x**3/41472 + O(x**4)
+    assert acosh(1/(I*x - 3))._eval_nseries(x, 4, None, cdir=-1) == acosh(-S(1)/3) - \
+    sqrt(2)*x/12 - 17*sqrt(2)*I*x**2/576 + 443*sqrt(2)*x**3/41472 + O(x**4)
+    # Tests concerning im(ndir) == 0
+    assert acosh(-I*x**2 + x - 2)._eval_nseries(x, 4, None) == -I*pi + log(sqrt(3) + 2) - \
+    sqrt(3)*x/3 + x**2*(-sqrt(3)/9 + sqrt(3)*I/3) + x**3*(-sqrt(3)/18 + 2*sqrt(3)*I/9) + O(x**4)
+
+
+def test_acosh_fdiff():
+    x = Symbol('x')
+    raises(ArgumentIndexError, lambda: acosh(x).fdiff(2))
+
+
+def test_asech():
+    x = Symbol('x')
+
+    assert unchanged(asech, -x)
+
+    # values at fixed points
+    assert asech(1) == 0
+    assert asech(-1) == pi*I
+    assert asech(0) is oo
+    assert asech(2) == I*pi/3
+    assert asech(-2) == 2*I*pi / 3
+    assert asech(nan) is nan
+
+    # at infinites
+    assert asech(oo) == I*pi/2
+    assert asech(-oo) == I*pi/2
+    assert asech(zoo) == I*AccumBounds(-pi/2, pi/2)
+
+    assert asech(I) == log(1 + sqrt(2)) - I*pi/2
+    assert asech(-I) == log(1 + sqrt(2)) + I*pi/2
+    assert asech(sqrt(2) - sqrt(6)) == 11*I*pi / 12
+    assert asech(sqrt(2 - 2/sqrt(5))) == I*pi / 10
+    assert asech(-sqrt(2 - 2/sqrt(5))) == 9*I*pi / 10
+    assert asech(2 / sqrt(2 + sqrt(2))) == I*pi / 8
+    assert asech(-2 / sqrt(2 + sqrt(2))) == 7*I*pi / 8
+    assert asech(sqrt(5) - 1) == I*pi / 5
+    assert asech(1 - sqrt(5)) == 4*I*pi / 5
+    assert asech(-sqrt(2*(2 + sqrt(2)))) == 5*I*pi / 8
+
+    # properties
+    # asech(x) == acosh(1/x)
+    assert asech(sqrt(2)) == acosh(1/sqrt(2))
+    assert asech(2/sqrt(3)) == acosh(sqrt(3)/2)
+    assert asech(2/sqrt(2 + sqrt(2))) == acosh(sqrt(2 + sqrt(2))/2)
+    assert asech(2) == acosh(S.Half)
+
+    # asech(x) == I*acos(1/x)
+    # (Note: the exact formula is asech(x) == +/- I*acos(1/x))
+    assert asech(-sqrt(2)) == I*acos(-1/sqrt(2))
+    assert asech(-2/sqrt(3)) == I*acos(-sqrt(3)/2)
+    assert asech(-S(2)) == I*acos(Rational(-1, 2))
+    assert asech(-2/sqrt(2)) == I*acos(-sqrt(2)/2)
+
+    # sech(asech(x)) / x == 1
+    assert expand_mul(sech(asech(sqrt(6) - sqrt(2))) / (sqrt(6) - sqrt(2))) == 1
+    assert expand_mul(sech(asech(sqrt(6) + sqrt(2))) / (sqrt(6) + sqrt(2))) == 1
+    assert (sech(asech(sqrt(2 + 2/sqrt(5)))) / (sqrt(2 + 2/sqrt(5)))).simplify() == 1
+    assert (sech(asech(-sqrt(2 + 2/sqrt(5)))) / (-sqrt(2 + 2/sqrt(5)))).simplify() == 1
+    assert (sech(asech(sqrt(2*(2 + sqrt(2))))) / (sqrt(2*(2 + sqrt(2))))).simplify() == 1
+    assert expand_mul(sech(asech(1 + sqrt(5))) / (1 + sqrt(5))) == 1
+    assert expand_mul(sech(asech(-1 - sqrt(5))) / (-1 - sqrt(5))) == 1
+    assert expand_mul(sech(asech(-sqrt(6) - sqrt(2))) / (-sqrt(6) - sqrt(2))) == 1
+
+    # numerical evaluation
+    assert str(asech(5*I).n(6)) == '0.19869 - 1.5708*I'
+    assert str(asech(-5*I).n(6)) == '0.19869 + 1.5708*I'
+
+
+def test_asech_leading_term():
+    x = Symbol('x')
+    # Tests concerning branch points
+    assert asech(x).as_leading_term(x, cdir=1) == -log(x) + log(2)
+    assert asech(x).as_leading_term(x, cdir=-1) == -log(x) + log(2) + 2*I*pi
+    assert asech(x + 1).as_leading_term(x, cdir=1) == sqrt(2)*I*sqrt(x)
+    assert asech(1/x).as_leading_term(x, cdir=1) == I*pi/2
+    # Tests concerning points lying on branch cuts
+    assert asech(x - 1).as_leading_term(x, cdir=1) == I*pi
+    assert asech(I*x + 3).as_leading_term(x, cdir=1) == -asech(3)
+    assert asech(-I*x + 3).as_leading_term(x, cdir=1) == asech(3)
+    assert asech(I*x - 3).as_leading_term(x, cdir=1) == -asech(-3)
+    assert asech(-I*x - 3).as_leading_term(x, cdir=1) == asech(-3)
+    assert asech(I*x - S(1)/3).as_leading_term(x, cdir=1) == -2*I*pi + asech(-S(1)/3)
+    assert asech(I*x - S(1)/3).as_leading_term(x, cdir=-1) == asech(-S(1)/3)
+    # Tests concerning im(ndir) == 0
+    assert asech(-I*x**2 + x - 3).as_leading_term(x, cdir=1) == log(-S(1)/3 + 2*sqrt(2)*I/3)
+    assert asech(-I*x**2 + x - 3).as_leading_term(x, cdir=-1) == log(-S(1)/3 + 2*sqrt(2)*I/3)
+
+
+def test_asech_series():
+    x = Symbol('x')
+    assert asech(x).series(x, 0, 9, cdir=1) == log(2) - log(x) - x**2/4 - 3*x**4/32 \
+    - 5*x**6/96 - 35*x**8/1024 + O(x**9)
+    assert asech(x).series(x, 0, 9, cdir=-1) == I*pi + log(2) - log(-x) - x**2/4 - \
+    3*x**4/32 - 5*x**6/96 - 35*x**8/1024 + O(x**9)
+    t6 = asech(x).taylor_term(6, x)
+    assert t6 == -5*x**6/96
+    assert asech(x).taylor_term(8, x, t6, 0) == -35*x**8/1024
+
+
+def test_asech_nseries():
+    x = Symbol('x')
+    # Tests concerning branch points
+    assert asech(x + 1)._eval_nseries(x, 4, None) == sqrt(2)*sqrt(-x) + 5*sqrt(2)*(-x)**(S(3)/2)/12 + \
+    43*sqrt(2)*(-x)**(S(5)/2)/160 + 177*sqrt(2)*(-x)**(S(7)/2)/896 + O(x**4)
+    # Tests concerning points lying on branch cuts
+    assert asech(x - 1)._eval_nseries(x, 4, None) == I*pi + sqrt(2)*sqrt(x) + \
+    5*sqrt(2)*x**(S(3)/2)/12 + 43*sqrt(2)*x**(S(5)/2)/160 + 177*sqrt(2)*x**(S(7)/2)/896 + O(x**4)
+    assert asech(I*x + 3)._eval_nseries(x, 4, None) == -asech(3) + sqrt(2)*x/12 - \
+    17*sqrt(2)*I*x**2/576 - 443*sqrt(2)*x**3/41472 + O(x**4)
+    assert asech(-I*x + 3)._eval_nseries(x, 4, None) == asech(3) + sqrt(2)*x/12 + \
+    17*sqrt(2)*I*x**2/576 - 443*sqrt(2)*x**3/41472 + O(x**4)
+    assert asech(I*x - 3)._eval_nseries(x, 4, None) == -asech(-3) - sqrt(2)*x/12 - \
+    17*sqrt(2)*I*x**2/576 + 443*sqrt(2)*x**3/41472 + O(x**4)
+    assert asech(-I*x - 3)._eval_nseries(x, 4, None) == asech(-3) - sqrt(2)*x/12 + \
+    17*sqrt(2)*I*x**2/576 + 443*sqrt(2)*x**3/41472 + O(x**4)
+    # Tests concerning im(ndir) == 0
+    assert asech(-I*x**2 + x - 2)._eval_nseries(x, 3, None) == 2*I*pi/3 + sqrt(3)*I*x/6 + \
+    x**2*(sqrt(3)/6 + 7*sqrt(3)*I/72) + O(x**3)
+
+
+def test_asech_rewrite():
+    x = Symbol('x')
+    assert asech(x).rewrite(log) == log(1/x + sqrt(1/x - 1) * sqrt(1/x + 1))
+    assert asech(x).rewrite(acosh) == acosh(1/x)
+    assert asech(x).rewrite(asinh) == sqrt(-1 + 1/x)*(-asin(1/x) + pi/2)/sqrt(1 - 1/x)
+    assert asech(x).rewrite(atanh) == \
+        sqrt(x + 1)*sqrt(1/(x + 1))*atanh(sqrt(1 - x**2)) + I*pi*(-sqrt(x)*sqrt(1/x) + 1 - I*sqrt(x**2)/(2*sqrt(-x**2)) - I*sqrt(-x)/(2*sqrt(x)))
+
+
+def test_asech_fdiff():
+    x = Symbol('x')
+    raises(ArgumentIndexError, lambda: asech(x).fdiff(2))
+
+
+def test_acsch():
+    x = Symbol('x')
+
+    assert unchanged(acsch, x)
+    assert acsch(-x) == -acsch(x)
+
+    # values at fixed points
+    assert acsch(1) == log(1 + sqrt(2))
+    assert acsch(-1) == - log(1 + sqrt(2))
+    assert acsch(0) is zoo
+    assert acsch(2) == log((1+sqrt(5))/2)
+    assert acsch(-2) == - log((1+sqrt(5))/2)
+
+    assert acsch(I) == - I*pi/2
+    assert acsch(-I) == I*pi/2
+    assert acsch(-I*(sqrt(6) + sqrt(2))) == I*pi / 12
+    assert acsch(I*(sqrt(2) + sqrt(6))) == -I*pi / 12
+    assert acsch(-I*(1 + sqrt(5))) == I*pi / 10
+    assert acsch(I*(1 + sqrt(5))) == -I*pi / 10
+    assert acsch(-I*2 / sqrt(2 - sqrt(2))) == I*pi / 8
+    assert acsch(I*2 / sqrt(2 - sqrt(2))) == -I*pi / 8
+    assert acsch(-I*2) == I*pi / 6
+    assert acsch(I*2) == -I*pi / 6
+    assert acsch(-I*sqrt(2 + 2/sqrt(5))) == I*pi / 5
+    assert acsch(I*sqrt(2 + 2/sqrt(5))) == -I*pi / 5
+    assert acsch(-I*sqrt(2)) == I*pi / 4
+    assert acsch(I*sqrt(2)) == -I*pi / 4
+    assert acsch(-I*(sqrt(5)-1)) == 3*I*pi / 10
+    assert acsch(I*(sqrt(5)-1)) == -3*I*pi / 10
+    assert acsch(-I*2 / sqrt(3)) == I*pi / 3
+    assert acsch(I*2 / sqrt(3)) == -I*pi / 3
+    assert acsch(-I*2 / sqrt(2 + sqrt(2))) == 3*I*pi / 8
+    assert acsch(I*2 / sqrt(2 + sqrt(2))) == -3*I*pi / 8
+    assert acsch(-I*sqrt(2 - 2/sqrt(5))) == 2*I*pi / 5
+    assert acsch(I*sqrt(2 - 2/sqrt(5))) == -2*I*pi / 5
+    assert acsch(-I*(sqrt(6) - sqrt(2))) == 5*I*pi / 12
+    assert acsch(I*(sqrt(6) - sqrt(2))) == -5*I*pi / 12
+    assert acsch(nan) is nan
+
+    # properties
+    # acsch(x) == asinh(1/x)
+    assert acsch(-I*sqrt(2)) == asinh(I/sqrt(2))
+    assert acsch(-I*2 / sqrt(3)) == asinh(I*sqrt(3) / 2)
+
+    # acsch(x) == -I*asin(I/x)
+    assert acsch(-I*sqrt(2)) == -I*asin(-1/sqrt(2))
+    assert acsch(-I*2 / sqrt(3)) == -I*asin(-sqrt(3)/2)
+
+    # csch(acsch(x)) / x == 1
+    assert expand_mul(csch(acsch(-I*(sqrt(6) + sqrt(2)))) / (-I*(sqrt(6) + sqrt(2)))) == 1
+    assert expand_mul(csch(acsch(I*(1 + sqrt(5)))) / (I*(1 + sqrt(5)))) == 1
+    assert (csch(acsch(I*sqrt(2 - 2/sqrt(5)))) / (I*sqrt(2 - 2/sqrt(5)))).simplify() == 1
+    assert (csch(acsch(-I*sqrt(2 - 2/sqrt(5)))) / (-I*sqrt(2 - 2/sqrt(5)))).simplify() == 1
+
+    # numerical evaluation
+    assert str(acsch(5*I+1).n(6)) == '0.0391819 - 0.193363*I'
+    assert str(acsch(-5*I+1).n(6)) == '0.0391819 + 0.193363*I'
+
+
+def test_acsch_infinities():
+    assert acsch(oo) == 0
+    assert acsch(-oo) == 0
+    assert acsch(zoo) == 0
+
+
+def test_acsch_leading_term():
+    x = Symbol('x')
+    assert acsch(1/x).as_leading_term(x) == x
+    # Tests concerning branch points
+    assert acsch(x + I).as_leading_term(x) == -I*pi/2
+    assert acsch(x - I).as_leading_term(x) == I*pi/2
+    # Tests concerning points lying on branch cuts
+    assert acsch(x).as_leading_term(x, cdir=1) == -log(x) + log(2)
+    assert acsch(x).as_leading_term(x, cdir=-1) == log(x) - log(2) - I*pi
+    assert acsch(x + I/2).as_leading_term(x, cdir=1) == -I*pi - acsch(I/2)
+    assert acsch(x + I/2).as_leading_term(x, cdir=-1) == acsch(I/2)
+    assert acsch(x - I/2).as_leading_term(x, cdir=1) == -acsch(I/2)
+    assert acsch(x - I/2).as_leading_term(x, cdir=-1) == acsch(I/2) + I*pi
+    # Tests concerning re(ndir) == 0
+    assert acsch(I/2 + I*x - x**2).as_leading_term(x, cdir=1) == log(2 - sqrt(3)) - I*pi/2
+    assert acsch(I/2 + I*x - x**2).as_leading_term(x, cdir=-1) == log(2 - sqrt(3)) - I*pi/2
+
+
+def test_acsch_series():
+    x = Symbol('x')
+    assert acsch(x).series(x, 0, 9) == log(2) - log(x) + x**2/4 - 3*x**4/32 \
+    + 5*x**6/96 - 35*x**8/1024 + O(x**9)
+    t4 = acsch(x).taylor_term(4, x)
+    assert t4 == -3*x**4/32
+    assert acsch(x).taylor_term(6, x, t4, 0) == 5*x**6/96
+
+
+def test_acsch_nseries():
+    x = Symbol('x')
+    # Tests concerning branch points
+    assert acsch(x + I)._eval_nseries(x, 4, None) == -I*pi/2 + I*sqrt(x) + \
+    sqrt(x) + 5*I*x**(S(3)/2)/12 - 5*x**(S(3)/2)/12 - 43*I*x**(S(5)/2)/160 - \
+    43*x**(S(5)/2)/160 - 177*I*x**(S(7)/2)/896 + 177*x**(S(7)/2)/896 + O(x**4)
+    assert acsch(x - I)._eval_nseries(x, 4, None) == I*pi/2 - I*sqrt(x) + \
+    sqrt(x) - 5*I*x**(S(3)/2)/12 - 5*x**(S(3)/2)/12 + 43*I*x**(S(5)/2)/160 - \
+    43*x**(S(5)/2)/160 + 177*I*x**(S(7)/2)/896 + 177*x**(S(7)/2)/896 + O(x**4)
+    # Tests concerning points lying on branch cuts
+    assert acsch(x + I/2)._eval_nseries(x, 4, None, cdir=1) == -acsch(I/2) - \
+    I*pi + 4*sqrt(3)*I*x/3 - 8*sqrt(3)*x**2/9 - 16*sqrt(3)*I*x**3/9 + O(x**4)
+    assert acsch(x + I/2)._eval_nseries(x, 4, None, cdir=-1) == acsch(I/2) - \
+    4*sqrt(3)*I*x/3 + 8*sqrt(3)*x**2/9 + 16*sqrt(3)*I*x**3/9 + O(x**4)
+    assert acsch(x - I/2)._eval_nseries(x, 4, None, cdir=1) == -acsch(I/2) - \
+    4*sqrt(3)*I*x/3 - 8*sqrt(3)*x**2/9 + 16*sqrt(3)*I*x**3/9 + O(x**4)
+    assert acsch(x - I/2)._eval_nseries(x, 4, None, cdir=-1) == I*pi + \
+    acsch(I/2) + 4*sqrt(3)*I*x/3 + 8*sqrt(3)*x**2/9 - 16*sqrt(3)*I*x**3/9 + O(x**4)
+    # TODO: Tests concerning re(ndir) == 0
+    assert acsch(I/2 + I*x - x**2)._eval_nseries(x, 4, None) == -I*pi/2 + \
+    log(2 - sqrt(3)) + 4*sqrt(3)*x/3 + x**2*(-8*sqrt(3)/9 + 4*sqrt(3)*I/3) + \
+    x**3*(16*sqrt(3)/9 - 16*sqrt(3)*I/9) + O(x**4)
+
+
+def test_acsch_rewrite():
+    x = Symbol('x')
+    assert acsch(x).rewrite(log) == log(1/x + sqrt(1/x**2 + 1))
+    assert acsch(x).rewrite(asinh) == asinh(1/x)
+    assert acsch(x).rewrite(atanh) == (sqrt(-x**2)*(-sqrt(-(x**2 + 1)**2)
+                                                    *atanh(sqrt(x**2 + 1))/(x**2 + 1)
+                                                    + pi/2)/x)
+
+
+def test_acsch_fdiff():
+    x = Symbol('x')
+    raises(ArgumentIndexError, lambda: acsch(x).fdiff(2))
+
+
+def test_atanh():
+    x = Symbol('x')
+
+    #at specific points
+    assert atanh(0) == 0
+    assert atanh(I) == I*pi/4
+    assert atanh(-I) == -I*pi/4
+    assert atanh(1) is oo
+    assert atanh(-1) is -oo
+    assert atanh(nan) is nan
+
+    # at infinites
+    assert atanh(oo) == -I*pi/2
+    assert atanh(-oo) == I*pi/2
+
+    assert atanh(I*oo) == I*pi/2
+    assert atanh(-I*oo) == -I*pi/2
+
+    assert atanh(zoo) == I*AccumBounds(-pi/2, pi/2)
+
+    #properties
+    assert atanh(-x) == -atanh(x)
+
+    assert atanh(I/sqrt(3)) == I*pi/6
+    assert atanh(-I/sqrt(3)) == -I*pi/6
+    assert atanh(I*sqrt(3)) == I*pi/3
+    assert atanh(-I*sqrt(3)) == -I*pi/3
+    assert atanh(I*(1 + sqrt(2))) == pi*I*Rational(3, 8)
+    assert atanh(I*(sqrt(2) - 1)) == pi*I/8
+    assert atanh(I*(1 - sqrt(2))) == -pi*I/8
+    assert atanh(-I*(1 + sqrt(2))) == pi*I*Rational(-3, 8)
+    assert atanh(I*sqrt(5 + 2*sqrt(5))) == I*pi*Rational(2, 5)
+    assert atanh(-I*sqrt(5 + 2*sqrt(5))) == I*pi*Rational(-2, 5)
+    assert atanh(I*(2 - sqrt(3))) == pi*I/12
+    assert atanh(I*(sqrt(3) - 2)) == -pi*I/12
+    assert atanh(oo) == -I*pi/2
+
+    # Symmetry
+    assert atanh(Rational(-1, 2)) == -atanh(S.Half)
+
+    # inverse composition
+    assert unchanged(atanh, tanh(Symbol('v1')))
+
+    assert atanh(tanh(-5, evaluate=False)) == -5
+    assert atanh(tanh(0, evaluate=False)) == 0
+    assert atanh(tanh(7, evaluate=False)) == 7
+    assert atanh(tanh(I, evaluate=False)) == I
+    assert atanh(tanh(-I, evaluate=False)) == -I
+    assert atanh(tanh(-11*I, evaluate=False)) == -11*I + 4*I*pi
+    assert atanh(tanh(3 + I)) == 3 + I
+    assert atanh(tanh(4 + 5*I)) == 4 - 2*I*pi + 5*I
+    assert atanh(tanh(pi/2)) == pi/2
+    assert atanh(tanh(pi)) == pi
+    assert atanh(tanh(-3 + 7*I)) == -3 - 2*I*pi + 7*I
+    assert atanh(tanh(9 - I*2/3)) == 9 - I*2/3
+    assert atanh(tanh(-32 - 123*I)) == -32 - 123*I + 39*I*pi
+
+
+def test_atanh_rewrite():
+    x = Symbol('x')
+    assert atanh(x).rewrite(log) == (log(1 + x) - log(1 - x)) / 2
+    assert atanh(x).rewrite(asinh) == \
+        pi*x/(2*sqrt(-x**2)) - sqrt(-x)*sqrt(1 - x**2)*sqrt(1/(x**2 - 1))*asinh(sqrt(1/(x**2 - 1)))/sqrt(x)
+
+
+def test_atanh_leading_term():
+    x = Symbol('x')
+    assert atanh(x).as_leading_term(x) == x
+    # Tests concerning branch points
+    assert atanh(x + 1).as_leading_term(x, cdir=1) == -log(x)/2 + log(2)/2 - I*pi/2
+    assert atanh(x + 1).as_leading_term(x, cdir=-1) == -log(x)/2 + log(2)/2 + I*pi/2
+    assert atanh(x - 1).as_leading_term(x, cdir=1) == log(x)/2 - log(2)/2
+    assert atanh(x - 1).as_leading_term(x, cdir=-1) == log(x)/2 - log(2)/2
+    assert atanh(1/x).as_leading_term(x, cdir=1) == -I*pi/2
+    assert atanh(1/x).as_leading_term(x, cdir=-1) == I*pi/2
+    # Tests concerning points lying on branch cuts
+    assert atanh(I*x + 2).as_leading_term(x, cdir=1) == atanh(2) + I*pi
+    assert atanh(-I*x + 2).as_leading_term(x, cdir=1) == atanh(2)
+    assert atanh(I*x - 2).as_leading_term(x, cdir=1) == -atanh(2)
+    assert atanh(-I*x - 2).as_leading_term(x, cdir=1) == -I*pi - atanh(2)
+    # Tests concerning im(ndir) == 0
+    assert atanh(-I*x**2 + x - 2).as_leading_term(x, cdir=1) == -log(3)/2 - I*pi/2
+    assert atanh(-I*x**2 + x - 2).as_leading_term(x, cdir=-1) == -log(3)/2 - I*pi/2
+
+
+def test_atanh_series():
+    x = Symbol('x')
+    assert atanh(x).series(x, 0, 10) == \
+        x + x**3/3 + x**5/5 + x**7/7 + x**9/9 + O(x**10)
+
+
+def test_atanh_nseries():
+    x = Symbol('x')
+    # Tests concerning branch points
+    assert atanh(x + 1)._eval_nseries(x, 4, None, cdir=1) == -I*pi/2 + log(2)/2 - \
+    log(x)/2 + x/4 - x**2/16 + x**3/48 + O(x**4)
+    assert atanh(x + 1)._eval_nseries(x, 4, None, cdir=-1) == I*pi/2 + log(2)/2 - \
+    log(x)/2 + x/4 - x**2/16 + x**3/48 + O(x**4)
+    assert atanh(x - 1)._eval_nseries(x, 4, None, cdir=1) == -log(2)/2 + log(x)/2 + \
+    x/4 + x**2/16 + x**3/48 + O(x**4)
+    assert atanh(x - 1)._eval_nseries(x, 4, None, cdir=-1) == -log(2)/2 + log(x)/2 + \
+    x/4 + x**2/16 + x**3/48 + O(x**4)
+    # Tests concerning points lying on branch cuts
+    assert atanh(I*x + 2)._eval_nseries(x, 4, None, cdir=1) == I*pi + atanh(2) - \
+    I*x/3 - 2*x**2/9 + 13*I*x**3/81 + O(x**4)
+    assert atanh(I*x + 2)._eval_nseries(x, 4, None, cdir=-1) == atanh(2) - I*x/3 - \
+    2*x**2/9 + 13*I*x**3/81 + O(x**4)
+    assert atanh(I*x - 2)._eval_nseries(x, 4, None, cdir=1) == -atanh(2) - I*x/3 + \
+    2*x**2/9 + 13*I*x**3/81 + O(x**4)
+    assert atanh(I*x - 2)._eval_nseries(x, 4, None, cdir=-1) == -atanh(2) - I*pi - \
+    I*x/3 + 2*x**2/9 + 13*I*x**3/81 + O(x**4)
+    # Tests concerning im(ndir) == 0
+    assert atanh(-I*x**2 + x - 2)._eval_nseries(x, 4, None) == -I*pi/2 - log(3)/2 - x/3 + \
+    x**2*(-S(1)/4 + I/2) + x**2*(S(1)/36 - I/6) + x**3*(-S(1)/6 + I/2) + x**3*(S(1)/162 - I/18) + O(x**4)
+
+
+def test_atanh_fdiff():
+    x = Symbol('x')
+    raises(ArgumentIndexError, lambda: atanh(x).fdiff(2))
+
+
+def test_acoth():
+    x = Symbol('x')
+
+    #at specific points
+    assert acoth(0) == I*pi/2
+    assert acoth(I) == -I*pi/4
+    assert acoth(-I) == I*pi/4
+    assert acoth(1) is oo
+    assert acoth(-1) is -oo
+    assert acoth(nan) is nan
+
+    # at infinites
+    assert acoth(oo) == 0
+    assert acoth(-oo) == 0
+    assert acoth(I*oo) == 0
+    assert acoth(-I*oo) == 0
+    assert acoth(zoo) == 0
+
+    #properties
+    assert acoth(-x) == -acoth(x)
+
+    assert acoth(I/sqrt(3)) == -I*pi/3
+    assert acoth(-I/sqrt(3)) == I*pi/3
+    assert acoth(I*sqrt(3)) == -I*pi/6
+    assert acoth(-I*sqrt(3)) == I*pi/6
+    assert acoth(I*(1 + sqrt(2))) == -pi*I/8
+    assert acoth(-I*(sqrt(2) + 1)) == pi*I/8
+    assert acoth(I*(1 - sqrt(2))) == pi*I*Rational(3, 8)
+    assert acoth(I*(sqrt(2) - 1)) == pi*I*Rational(-3, 8)
+    assert acoth(I*sqrt(5 + 2*sqrt(5))) == -I*pi/10
+    assert acoth(-I*sqrt(5 + 2*sqrt(5))) == I*pi/10
+    assert acoth(I*(2 + sqrt(3))) == -pi*I/12
+    assert acoth(-I*(2 + sqrt(3))) == pi*I/12
+    assert acoth(I*(2 - sqrt(3))) == pi*I*Rational(-5, 12)
+    assert acoth(I*(sqrt(3) - 2)) == pi*I*Rational(5, 12)
+
+    # Symmetry
+    assert acoth(Rational(-1, 2)) == -acoth(S.Half)
+
+
+def test_acoth_rewrite():
+    x = Symbol('x')
+    assert acoth(x).rewrite(log) == (log(1 + 1/x) - log(1 - 1/x)) / 2
+    assert acoth(x).rewrite(atanh) == atanh(1/x)
+    assert acoth(x).rewrite(asinh) == \
+        x*sqrt(x**(-2))*asinh(sqrt(1/(x**2 - 1))) + I*pi*(sqrt((x - 1)/x)*sqrt(x/(x - 1)) - sqrt(x/(x + 1))*sqrt(1 + 1/x))/2
+
+
+def test_acoth_leading_term():
+    x = Symbol('x')
+    # Tests concerning branch points
+    assert acoth(x + 1).as_leading_term(x, cdir=1) == -log(x)/2 + log(2)/2
+    assert acoth(x + 1).as_leading_term(x, cdir=-1) == -log(x)/2 + log(2)/2
+    assert acoth(x - 1).as_leading_term(x, cdir=1) == log(x)/2 - log(2)/2 + I*pi/2
+    assert acoth(x - 1).as_leading_term(x, cdir=-1) == log(x)/2 - log(2)/2 - I*pi/2
+    # Tests concerning points lying on branch cuts
+    assert acoth(x).as_leading_term(x, cdir=-1) == I*pi/2
+    assert acoth(x).as_leading_term(x, cdir=1) == -I*pi/2
+    assert acoth(I*x + 1/2).as_leading_term(x, cdir=1) == acoth(1/2)
+    assert acoth(-I*x + 1/2).as_leading_term(x, cdir=1) == acoth(1/2) + I*pi
+    assert acoth(I*x - 1/2).as_leading_term(x, cdir=1) == -I*pi - acoth(1/2)
+    assert acoth(-I*x - 1/2).as_leading_term(x, cdir=1) == -acoth(1/2)
+    # Tests concerning im(ndir) == 0
+    assert acoth(-I*x**2 - x - S(1)/2).as_leading_term(x, cdir=1) == -log(3)/2 + I*pi/2
+    assert acoth(-I*x**2 - x - S(1)/2).as_leading_term(x, cdir=-1) == -log(3)/2 + I*pi/2
+
+
+def test_acoth_series():
+    x = Symbol('x')
+    assert acoth(x).series(x, 0, 10) == \
+        -I*pi/2 + x + x**3/3 + x**5/5 + x**7/7 + x**9/9 + O(x**10)
+
+
+def test_acoth_nseries():
+    x = Symbol('x')
+    # Tests concerning branch points
+    assert acoth(x + 1)._eval_nseries(x, 4, None) == log(2)/2 - log(x)/2 + x/4 - \
+    x**2/16 + x**3/48 + O(x**4)
+    assert acoth(x - 1)._eval_nseries(x, 4, None, cdir=1) == I*pi/2 - log(2)/2 + \
+    log(x)/2 + x/4 + x**2/16 + x**3/48 + O(x**4)
+    assert acoth(x - 1)._eval_nseries(x, 4, None, cdir=-1) == -I*pi/2 - log(2)/2 + \
+    log(x)/2 + x/4 + x**2/16 + x**3/48 + O(x**4)
+    # Tests concerning points lying on branch cuts
+    assert acoth(I*x + S(1)/2)._eval_nseries(x, 4, None, cdir=1) == acoth(S(1)/2) + \
+    4*I*x/3 - 8*x**2/9 - 112*I*x**3/81 + O(x**4)
+    assert acoth(I*x + S(1)/2)._eval_nseries(x, 4, None, cdir=-1) == I*pi + \
+    acoth(S(1)/2) + 4*I*x/3 - 8*x**2/9 - 112*I*x**3/81 + O(x**4)
+    assert acoth(I*x - S(1)/2)._eval_nseries(x, 4, None, cdir=1) == -acoth(S(1)/2) - \
+    I*pi + 4*I*x/3 + 8*x**2/9 - 112*I*x**3/81 + O(x**4)
+    assert acoth(I*x - S(1)/2)._eval_nseries(x, 4, None, cdir=-1) == -acoth(S(1)/2) + \
+    4*I*x/3 + 8*x**2/9 - 112*I*x**3/81 + O(x**4)
+    # Tests concerning im(ndir) == 0
+    assert acoth(-I*x**2 - x - S(1)/2)._eval_nseries(x, 4, None) == I*pi/2 - log(3)/2 - \
+    4*x/3 + x**2*(-S(8)/9 + 2*I/3) - 2*I*x**2 + x**3*(S(104)/81 - 16*I/9) - 8*x**3/3 + O(x**4)
+
+
+def test_acoth_fdiff():
+    x = Symbol('x')
+    raises(ArgumentIndexError, lambda: acoth(x).fdiff(2))
+
+
+def test_inverses():
+    x = Symbol('x')
+    assert sinh(x).inverse() == asinh
+    raises(AttributeError, lambda: cosh(x).inverse())
+    assert tanh(x).inverse() == atanh
+    assert coth(x).inverse() == acoth
+    assert asinh(x).inverse() == sinh
+    assert acosh(x).inverse() == cosh
+    assert atanh(x).inverse() == tanh
+    assert acoth(x).inverse() == coth
+    assert asech(x).inverse() == sech
+    assert acsch(x).inverse() == csch
+
+
+def test_leading_term():
+    x = Symbol('x')
+    assert cosh(x).as_leading_term(x) == 1
+    assert coth(x).as_leading_term(x) == 1/x
+    for func in [sinh, tanh]:
+        assert func(x).as_leading_term(x) == x
+    for func in [sinh, cosh, tanh, coth]:
+        for ar in (1/x, S.Half):
+            eq = func(ar)
+            assert eq.as_leading_term(x) == eq
+    for func in [csch, sech]:
+        eq = func(S.Half)
+        assert eq.as_leading_term(x) == eq
+
+
+def test_complex():
+    a, b = symbols('a,b', real=True)
+    z = a + b*I
+    for func in [sinh, cosh, tanh, coth, sech, csch]:
+        assert func(z).conjugate() == func(a - b*I)
+    for deep in [True, False]:
+        assert sinh(z).expand(
+            complex=True, deep=deep) == sinh(a)*cos(b) + I*cosh(a)*sin(b)
+        assert cosh(z).expand(
+            complex=True, deep=deep) == cosh(a)*cos(b) + I*sinh(a)*sin(b)
+        assert tanh(z).expand(complex=True, deep=deep) == sinh(a)*cosh(
+            a)/(cos(b)**2 + sinh(a)**2) + I*sin(b)*cos(b)/(cos(b)**2 + sinh(a)**2)
+        assert coth(z).expand(complex=True, deep=deep) == sinh(a)*cosh(
+            a)/(sin(b)**2 + sinh(a)**2) - I*sin(b)*cos(b)/(sin(b)**2 + sinh(a)**2)
+        assert csch(z).expand(complex=True, deep=deep) == cos(b) * sinh(a) / (sin(b)**2\
+            *cosh(a)**2 + cos(b)**2 * sinh(a)**2) - I*sin(b) * cosh(a) / (sin(b)**2\
+            *cosh(a)**2 + cos(b)**2 * sinh(a)**2)
+        assert sech(z).expand(complex=True, deep=deep) == cos(b) * cosh(a) / (sin(b)**2\
+            *sinh(a)**2 + cos(b)**2 * cosh(a)**2) - I*sin(b) * sinh(a) / (sin(b)**2\
+            *sinh(a)**2 + cos(b)**2 * cosh(a)**2)
+
+
+def test_complex_2899():
+    a, b = symbols('a,b', real=True)
+    for deep in [True, False]:
+        for func in [sinh, cosh, tanh, coth]:
+            assert func(a).expand(complex=True, deep=deep) == func(a)
+
+
+def test_simplifications():
+    x = Symbol('x')
+    assert sinh(asinh(x)) == x
+    assert sinh(acosh(x)) == sqrt(x - 1) * sqrt(x + 1)
+    assert sinh(atanh(x)) == x/sqrt(1 - x**2)
+    assert sinh(acoth(x)) == 1/(sqrt(x - 1) * sqrt(x + 1))
+
+    assert cosh(asinh(x)) == sqrt(1 + x**2)
+    assert cosh(acosh(x)) == x
+    assert cosh(atanh(x)) == 1/sqrt(1 - x**2)
+    assert cosh(acoth(x)) == x/(sqrt(x - 1) * sqrt(x + 1))
+
+    assert tanh(asinh(x)) == x/sqrt(1 + x**2)
+    assert tanh(acosh(x)) == sqrt(x - 1) * sqrt(x + 1) / x
+    assert tanh(atanh(x)) == x
+    assert tanh(acoth(x)) == 1/x
+
+    assert coth(asinh(x)) == sqrt(1 + x**2)/x
+    assert coth(acosh(x)) == x/(sqrt(x - 1) * sqrt(x + 1))
+    assert coth(atanh(x)) == 1/x
+    assert coth(acoth(x)) == x
+
+    assert csch(asinh(x)) == 1/x
+    assert csch(acosh(x)) == 1/(sqrt(x - 1) * sqrt(x + 1))
+    assert csch(atanh(x)) == sqrt(1 - x**2)/x
+    assert csch(acoth(x)) == sqrt(x - 1) * sqrt(x + 1)
+
+    assert sech(asinh(x)) == 1/sqrt(1 + x**2)
+    assert sech(acosh(x)) == 1/x
+    assert sech(atanh(x)) == sqrt(1 - x**2)
+    assert sech(acoth(x)) == sqrt(x - 1) * sqrt(x + 1)/x
+
+
+def test_issue_4136():
+    assert cosh(asinh(Integer(3)/2)) == sqrt(Integer(13)/4)
+
+
+def test_sinh_rewrite():
+    x = Symbol('x')
+    assert sinh(x).rewrite(exp) == (exp(x) - exp(-x))/2 \
+        == sinh(x).rewrite('tractable')
+    assert sinh(x).rewrite(cosh) == -I*cosh(x + I*pi/2)
+    tanh_half = tanh(S.Half*x)
+    assert sinh(x).rewrite(tanh) == 2*tanh_half/(1 - tanh_half**2)
+    coth_half = coth(S.Half*x)
+    assert sinh(x).rewrite(coth) == 2*coth_half/(coth_half**2 - 1)
+
+
+def test_cosh_rewrite():
+    x = Symbol('x')
+    assert cosh(x).rewrite(exp) == (exp(x) + exp(-x))/2 \
+        == cosh(x).rewrite('tractable')
+    assert cosh(x).rewrite(sinh) == -I*sinh(x + I*pi/2)
+    tanh_half = tanh(S.Half*x)**2
+    assert cosh(x).rewrite(tanh) == (1 + tanh_half)/(1 - tanh_half)
+    coth_half = coth(S.Half*x)**2
+    assert cosh(x).rewrite(coth) == (coth_half + 1)/(coth_half - 1)
+
+
+def test_tanh_rewrite():
+    x = Symbol('x')
+    assert tanh(x).rewrite(exp) == (exp(x) - exp(-x))/(exp(x) + exp(-x)) \
+        == tanh(x).rewrite('tractable')
+    assert tanh(x).rewrite(sinh) == I*sinh(x)/sinh(I*pi/2 - x)
+    assert tanh(x).rewrite(cosh) == I*cosh(I*pi/2 - x)/cosh(x)
+    assert tanh(x).rewrite(coth) == 1/coth(x)
+
+
+def test_coth_rewrite():
+    x = Symbol('x')
+    assert coth(x).rewrite(exp) == (exp(x) + exp(-x))/(exp(x) - exp(-x)) \
+        == coth(x).rewrite('tractable')
+    assert coth(x).rewrite(sinh) == -I*sinh(I*pi/2 - x)/sinh(x)
+    assert coth(x).rewrite(cosh) == -I*cosh(x)/cosh(I*pi/2 - x)
+    assert coth(x).rewrite(tanh) == 1/tanh(x)
+
+
+def test_csch_rewrite():
+    x = Symbol('x')
+    assert csch(x).rewrite(exp) == 1 / (exp(x)/2 - exp(-x)/2) \
+        == csch(x).rewrite('tractable')
+    assert csch(x).rewrite(cosh) == I/cosh(x + I*pi/2)
+    tanh_half = tanh(S.Half*x)
+    assert csch(x).rewrite(tanh) == (1 - tanh_half**2)/(2*tanh_half)
+    coth_half = coth(S.Half*x)
+    assert csch(x).rewrite(coth) == (coth_half**2 - 1)/(2*coth_half)
+
+
+def test_sech_rewrite():
+    x = Symbol('x')
+    assert sech(x).rewrite(exp) == 1 / (exp(x)/2 + exp(-x)/2) \
+        == sech(x).rewrite('tractable')
+    assert sech(x).rewrite(sinh) == I/sinh(x + I*pi/2)
+    tanh_half = tanh(S.Half*x)**2
+    assert sech(x).rewrite(tanh) == (1 - tanh_half)/(1 + tanh_half)
+    coth_half = coth(S.Half*x)**2
+    assert sech(x).rewrite(coth) == (coth_half - 1)/(coth_half + 1)
+
+
+def test_derivs():
+    x = Symbol('x')
+    assert coth(x).diff(x) == -sinh(x)**(-2)
+    assert sinh(x).diff(x) == cosh(x)
+    assert cosh(x).diff(x) == sinh(x)
+    assert tanh(x).diff(x) == -tanh(x)**2 + 1
+    assert csch(x).diff(x) == -coth(x)*csch(x)
+    assert sech(x).diff(x) == -tanh(x)*sech(x)
+    assert acoth(x).diff(x) == 1/(-x**2 + 1)
+    assert asinh(x).diff(x) == 1/sqrt(x**2 + 1)
+    assert acosh(x).diff(x) == 1/(sqrt(x - 1)*sqrt(x + 1))
+    assert acosh(x).diff(x) == acosh(x).rewrite(log).diff(x).together()
+    assert atanh(x).diff(x) == 1/(-x**2 + 1)
+    assert asech(x).diff(x) == -1/(x*sqrt(1 - x**2))
+    assert acsch(x).diff(x) == -1/(x**2*sqrt(1 + x**(-2)))
+
+
+def test_sinh_expansion():
+    x, y = symbols('x,y')
+    assert sinh(x+y).expand(trig=True) == sinh(x)*cosh(y) + cosh(x)*sinh(y)
+    assert sinh(2*x).expand(trig=True) == 2*sinh(x)*cosh(x)
+    assert sinh(3*x).expand(trig=True).expand() == \
+        sinh(x)**3 + 3*sinh(x)*cosh(x)**2
+
+
+def test_cosh_expansion():
+    x, y = symbols('x,y')
+    assert cosh(x+y).expand(trig=True) == cosh(x)*cosh(y) + sinh(x)*sinh(y)
+    assert cosh(2*x).expand(trig=True) == cosh(x)**2 + sinh(x)**2
+    assert cosh(3*x).expand(trig=True).expand() == \
+        3*sinh(x)**2*cosh(x) + cosh(x)**3
+
+def test_cosh_positive():
+    # See issue 11721
+    # cosh(x) is positive for real values of x
+    k = symbols('k', real=True)
+    n = symbols('n', integer=True)
+
+    assert cosh(k, evaluate=False).is_positive is True
+    assert cosh(k + 2*n*pi*I, evaluate=False).is_positive is True
+    assert cosh(I*pi/4, evaluate=False).is_positive is True
+    assert cosh(3*I*pi/4, evaluate=False).is_positive is False
+
+def test_cosh_nonnegative():
+    k = symbols('k', real=True)
+    n = symbols('n', integer=True)
+
+    assert cosh(k, evaluate=False).is_nonnegative is True
+    assert cosh(k + 2*n*pi*I, evaluate=False).is_nonnegative is True
+    assert cosh(I*pi/4, evaluate=False).is_nonnegative is True
+    assert cosh(3*I*pi/4, evaluate=False).is_nonnegative is False
+    assert cosh(S.Zero, evaluate=False).is_nonnegative is True
+
+def test_real_assumptions():
+    z = Symbol('z', real=False)
+    assert sinh(z).is_real is None
+    assert cosh(z).is_real is None
+    assert tanh(z).is_real is None
+    assert sech(z).is_real is None
+    assert csch(z).is_real is None
+    assert coth(z).is_real is None
+
+def test_sign_assumptions():
+    p = Symbol('p', positive=True)
+    n = Symbol('n', negative=True)
+    assert sinh(n).is_negative is True
+    assert sinh(p).is_positive is True
+    assert cosh(n).is_positive is True
+    assert cosh(p).is_positive is True
+    assert tanh(n).is_negative is True
+    assert tanh(p).is_positive is True
+    assert csch(n).is_negative is True
+    assert csch(p).is_positive is True
+    assert sech(n).is_positive is True
+    assert sech(p).is_positive is True
+    assert coth(n).is_negative is True
+    assert coth(p).is_positive is True

llmeval-env/lib/python3.10/site-packages/sympy/functions/elementary/tests/test_integers.py

@@ -0,0 +1,632 @@
+from sympy.calculus.accumulationbounds import AccumBounds
+from sympy.core.numbers import (E, Float, I, Rational, nan, oo, pi, zoo)
+from sympy.core.relational import (Eq, Ge, Gt, Le, Lt, Ne)
+from sympy.core.singleton import S
+from sympy.core.symbol import (Symbol, symbols)
+from sympy.functions.combinatorial.factorials import factorial
+from sympy.functions.elementary.exponential import (exp, log)
+from sympy.functions.elementary.integers import (ceiling, floor, frac)
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.trigonometric import sin, cos, tan
+
+from sympy.core.expr import unchanged
+from sympy.testing.pytest import XFAIL
+
+x = Symbol('x')
+i = Symbol('i', imaginary=True)
+y = Symbol('y', real=True)
+k, n = symbols('k,n', integer=True)
+
+
+def test_floor():
+
+    assert floor(nan) is nan
+
+    assert floor(oo) is oo
+    assert floor(-oo) is -oo
+    assert floor(zoo) is zoo
+
+    assert floor(0) == 0
+
+    assert floor(1) == 1
+    assert floor(-1) == -1
+
+    assert floor(E) == 2
+    assert floor(-E) == -3
+
+    assert floor(2*E) == 5
+    assert floor(-2*E) == -6
+
+    assert floor(pi) == 3
+    assert floor(-pi) == -4
+
+    assert floor(S.Half) == 0
+    assert floor(Rational(-1, 2)) == -1
+
+    assert floor(Rational(7, 3)) == 2
+    assert floor(Rational(-7, 3)) == -3
+    assert floor(-Rational(7, 3)) == -3
+
+    assert floor(Float(17.0)) == 17
+    assert floor(-Float(17.0)) == -17
+
+    assert floor(Float(7.69)) == 7
+    assert floor(-Float(7.69)) == -8
+
+    assert floor(I) == I
+    assert floor(-I) == -I
+    e = floor(i)
+    assert e.func is floor and e.args[0] == i
+
+    assert floor(oo*I) == oo*I
+    assert floor(-oo*I) == -oo*I
+    assert floor(exp(I*pi/4)*oo) == exp(I*pi/4)*oo
+
+    assert floor(2*I) == 2*I
+    assert floor(-2*I) == -2*I
+
+    assert floor(I/2) == 0
+    assert floor(-I/2) == -I
+
+    assert floor(E + 17) == 19
+    assert floor(pi + 2) == 5
+
+    assert floor(E + pi) == 5
+    assert floor(I + pi) == 3 + I
+
+    assert floor(floor(pi)) == 3
+    assert floor(floor(y)) == floor(y)
+    assert floor(floor(x)) == floor(x)
+
+    assert unchanged(floor, x)
+    assert unchanged(floor, 2*x)
+    assert unchanged(floor, k*x)
+
+    assert floor(k) == k
+    assert floor(2*k) == 2*k
+    assert floor(k*n) == k*n
+
+    assert unchanged(floor, k/2)
+
+    assert unchanged(floor, x + y)
+
+    assert floor(x + 3) == floor(x) + 3
+    assert floor(x + k) == floor(x) + k
+
+    assert floor(y + 3) == floor(y) + 3
+    assert floor(y + k) == floor(y) + k
+
+    assert floor(3 + I*y + pi) == 6 + floor(y)*I
+
+    assert floor(k + n) == k + n
+
+    assert unchanged(floor, x*I)
+    assert floor(k*I) == k*I
+
+    assert floor(Rational(23, 10) - E*I) == 2 - 3*I
+
+    assert floor(sin(1)) == 0
+    assert floor(sin(-1)) == -1
+
+    assert floor(exp(2)) == 7
+
+    assert floor(log(8)/log(2)) != 2
+    assert int(floor(log(8)/log(2)).evalf(chop=True)) == 3
+
+    assert floor(factorial(50)/exp(1)) == \
+        11188719610782480504630258070757734324011354208865721592720336800
+
+    assert (floor(y) < y) == False
+    assert (floor(y) <= y) == True
+    assert (floor(y) > y) == False
+    assert (floor(y) >= y) == False
+    assert (floor(x) <= x).is_Relational  # x could be non-real
+    assert (floor(x) > x).is_Relational
+    assert (floor(x) <= y).is_Relational  # arg is not same as rhs
+    assert (floor(x) > y).is_Relational
+    assert (floor(y) <= oo) == True
+    assert (floor(y) < oo) == True
+    assert (floor(y) >= -oo) == True
+    assert (floor(y) > -oo) == True
+
+    assert floor(y).rewrite(frac) == y - frac(y)
+    assert floor(y).rewrite(ceiling) == -ceiling(-y)
+    assert floor(y).rewrite(frac).subs(y, -pi) == floor(-pi)
+    assert floor(y).rewrite(frac).subs(y, E) == floor(E)
+    assert floor(y).rewrite(ceiling).subs(y, E) == -ceiling(-E)
+    assert floor(y).rewrite(ceiling).subs(y, -pi) == -ceiling(pi)
+
+    assert Eq(floor(y), y - frac(y))
+    assert Eq(floor(y), -ceiling(-y))
+
+    neg = Symbol('neg', negative=True)
+    nn = Symbol('nn', nonnegative=True)
+    pos = Symbol('pos', positive=True)
+    np = Symbol('np', nonpositive=True)
+
+    assert (floor(neg) < 0) == True
+    assert (floor(neg) <= 0) == True
+    assert (floor(neg) > 0) == False
+    assert (floor(neg) >= 0) == False
+    assert (floor(neg) <= -1) == True
+    assert (floor(neg) >= -3) == (neg >= -3)
+    assert (floor(neg) < 5) == (neg < 5)
+
+    assert (floor(nn) < 0) == False
+    assert (floor(nn) >= 0) == True
+
+    assert (floor(pos) < 0) == False
+    assert (floor(pos) <= 0) == (pos < 1)
+    assert (floor(pos) > 0) == (pos >= 1)
+    assert (floor(pos) >= 0) == True
+    assert (floor(pos) >= 3) == (pos >= 3)
+
+    assert (floor(np) <= 0) == True
+    assert (floor(np) > 0) == False
+
+    assert floor(neg).is_negative == True
+    assert floor(neg).is_nonnegative == False
+    assert floor(nn).is_negative == False
+    assert floor(nn).is_nonnegative == True
+    assert floor(pos).is_negative == False
+    assert floor(pos).is_nonnegative == True
+    assert floor(np).is_negative is None
+    assert floor(np).is_nonnegative is None
+
+    assert (floor(7, evaluate=False) >= 7) == True
+    assert (floor(7, evaluate=False) > 7) == False
+    assert (floor(7, evaluate=False) <= 7) == True
+    assert (floor(7, evaluate=False) < 7) == False
+
+    assert (floor(7, evaluate=False) >= 6) == True
+    assert (floor(7, evaluate=False) > 6) == True
+    assert (floor(7, evaluate=False) <= 6) == False
+    assert (floor(7, evaluate=False) < 6) == False
+
+    assert (floor(7, evaluate=False) >= 8) == False
+    assert (floor(7, evaluate=False) > 8) == False
+    assert (floor(7, evaluate=False) <= 8) == True
+    assert (floor(7, evaluate=False) < 8) == True
+
+    assert (floor(x) <= 5.5) == Le(floor(x), 5.5, evaluate=False)
+    assert (floor(x) >= -3.2) == Ge(floor(x), -3.2, evaluate=False)
+    assert (floor(x) < 2.9) == Lt(floor(x), 2.9, evaluate=False)
+    assert (floor(x) > -1.7) == Gt(floor(x), -1.7, evaluate=False)
+
+    assert (floor(y) <= 5.5) == (y < 6)
+    assert (floor(y) >= -3.2) == (y >= -3)
+    assert (floor(y) < 2.9) == (y < 3)
+    assert (floor(y) > -1.7) == (y >= -1)
+
+    assert (floor(y) <= n) == (y < n + 1)
+    assert (floor(y) >= n) == (y >= n)
+    assert (floor(y) < n) == (y < n)
+    assert (floor(y) > n) == (y >= n + 1)
+
+
+def test_ceiling():
+
+    assert ceiling(nan) is nan
+
+    assert ceiling(oo) is oo
+    assert ceiling(-oo) is -oo
+    assert ceiling(zoo) is zoo
+
+    assert ceiling(0) == 0
+
+    assert ceiling(1) == 1
+    assert ceiling(-1) == -1
+
+    assert ceiling(E) == 3
+    assert ceiling(-E) == -2
+
+    assert ceiling(2*E) == 6
+    assert ceiling(-2*E) == -5
+
+    assert ceiling(pi) == 4
+    assert ceiling(-pi) == -3
+
+    assert ceiling(S.Half) == 1
+    assert ceiling(Rational(-1, 2)) == 0
+
+    assert ceiling(Rational(7, 3)) == 3
+    assert ceiling(-Rational(7, 3)) == -2
+
+    assert ceiling(Float(17.0)) == 17
+    assert ceiling(-Float(17.0)) == -17
+
+    assert ceiling(Float(7.69)) == 8
+    assert ceiling(-Float(7.69)) == -7
+
+    assert ceiling(I) == I
+    assert ceiling(-I) == -I
+    e = ceiling(i)
+    assert e.func is ceiling and e.args[0] == i
+
+    assert ceiling(oo*I) == oo*I
+    assert ceiling(-oo*I) == -oo*I
+    assert ceiling(exp(I*pi/4)*oo) == exp(I*pi/4)*oo
+
+    assert ceiling(2*I) == 2*I
+    assert ceiling(-2*I) == -2*I
+
+    assert ceiling(I/2) == I
+    assert ceiling(-I/2) == 0
+
+    assert ceiling(E + 17) == 20
+    assert ceiling(pi + 2) == 6
+
+    assert ceiling(E + pi) == 6
+    assert ceiling(I + pi) == I + 4
+
+    assert ceiling(ceiling(pi)) == 4
+    assert ceiling(ceiling(y)) == ceiling(y)
+    assert ceiling(ceiling(x)) == ceiling(x)
+
+    assert unchanged(ceiling, x)
+    assert unchanged(ceiling, 2*x)
+    assert unchanged(ceiling, k*x)
+
+    assert ceiling(k) == k
+    assert ceiling(2*k) == 2*k
+    assert ceiling(k*n) == k*n
+
+    assert unchanged(ceiling, k/2)
+
+    assert unchanged(ceiling, x + y)
+
+    assert ceiling(x + 3) == ceiling(x) + 3
+    assert ceiling(x + k) == ceiling(x) + k
+
+    assert ceiling(y + 3) == ceiling(y) + 3
+    assert ceiling(y + k) == ceiling(y) + k
+
+    assert ceiling(3 + pi + y*I) == 7 + ceiling(y)*I
+
+    assert ceiling(k + n) == k + n
+
+    assert unchanged(ceiling, x*I)
+    assert ceiling(k*I) == k*I
+
+    assert ceiling(Rational(23, 10) - E*I) == 3 - 2*I
+
+    assert ceiling(sin(1)) == 1
+    assert ceiling(sin(-1)) == 0
+
+    assert ceiling(exp(2)) == 8
+
+    assert ceiling(-log(8)/log(2)) != -2
+    assert int(ceiling(-log(8)/log(2)).evalf(chop=True)) == -3
+
+    assert ceiling(factorial(50)/exp(1)) == \
+        11188719610782480504630258070757734324011354208865721592720336801
+
+    assert (ceiling(y) >= y) == True
+    assert (ceiling(y) > y) == False
+    assert (ceiling(y) < y) == False
+    assert (ceiling(y) <= y) == False
+    assert (ceiling(x) >= x).is_Relational  # x could be non-real
+    assert (ceiling(x) < x).is_Relational
+    assert (ceiling(x) >= y).is_Relational  # arg is not same as rhs
+    assert (ceiling(x) < y).is_Relational
+    assert (ceiling(y) >= -oo) == True
+    assert (ceiling(y) > -oo) == True
+    assert (ceiling(y) <= oo) == True
+    assert (ceiling(y) < oo) == True
+
+    assert ceiling(y).rewrite(floor) == -floor(-y)
+    assert ceiling(y).rewrite(frac) == y + frac(-y)
+    assert ceiling(y).rewrite(floor).subs(y, -pi) == -floor(pi)
+    assert ceiling(y).rewrite(floor).subs(y, E) == -floor(-E)
+    assert ceiling(y).rewrite(frac).subs(y, pi) == ceiling(pi)
+    assert ceiling(y).rewrite(frac).subs(y, -E) == ceiling(-E)
+
+    assert Eq(ceiling(y), y + frac(-y))
+    assert Eq(ceiling(y), -floor(-y))
+
+    neg = Symbol('neg', negative=True)
+    nn = Symbol('nn', nonnegative=True)
+    pos = Symbol('pos', positive=True)
+    np = Symbol('np', nonpositive=True)
+
+    assert (ceiling(neg) <= 0) == True
+    assert (ceiling(neg) < 0) == (neg <= -1)
+    assert (ceiling(neg) > 0) == False
+    assert (ceiling(neg) >= 0) == (neg > -1)
+    assert (ceiling(neg) > -3) == (neg > -3)
+    assert (ceiling(neg) <= 10) == (neg <= 10)
+
+    assert (ceiling(nn) < 0) == False
+    assert (ceiling(nn) >= 0) == True
+
+    assert (ceiling(pos) < 0) == False
+    assert (ceiling(pos) <= 0) == False
+    assert (ceiling(pos) > 0) == True
+    assert (ceiling(pos) >= 0) == True
+    assert (ceiling(pos) >= 1) == True
+    assert (ceiling(pos) > 5) == (pos > 5)
+
+    assert (ceiling(np) <= 0) == True
+    assert (ceiling(np) > 0) == False
+
+    assert ceiling(neg).is_positive == False
+    assert ceiling(neg).is_nonpositive == True
+    assert ceiling(nn).is_positive is None
+    assert ceiling(nn).is_nonpositive is None
+    assert ceiling(pos).is_positive == True
+    assert ceiling(pos).is_nonpositive == False
+    assert ceiling(np).is_positive == False
+    assert ceiling(np).is_nonpositive == True
+
+    assert (ceiling(7, evaluate=False) >= 7) == True
+    assert (ceiling(7, evaluate=False) > 7) == False
+    assert (ceiling(7, evaluate=False) <= 7) == True
+    assert (ceiling(7, evaluate=False) < 7) == False
+
+    assert (ceiling(7, evaluate=False) >= 6) == True
+    assert (ceiling(7, evaluate=False) > 6) == True
+    assert (ceiling(7, evaluate=False) <= 6) == False
+    assert (ceiling(7, evaluate=False) < 6) == False
+
+    assert (ceiling(7, evaluate=False) >= 8) == False
+    assert (ceiling(7, evaluate=False) > 8) == False
+    assert (ceiling(7, evaluate=False) <= 8) == True
+    assert (ceiling(7, evaluate=False) < 8) == True
+
+    assert (ceiling(x) <= 5.5) == Le(ceiling(x), 5.5, evaluate=False)
+    assert (ceiling(x) >= -3.2) == Ge(ceiling(x), -3.2, evaluate=False)
+    assert (ceiling(x) < 2.9) == Lt(ceiling(x), 2.9, evaluate=False)
+    assert (ceiling(x) > -1.7) == Gt(ceiling(x), -1.7, evaluate=False)
+
+    assert (ceiling(y) <= 5.5) == (y <= 5)
+    assert (ceiling(y) >= -3.2) == (y > -4)
+    assert (ceiling(y) < 2.9) == (y <= 2)
+    assert (ceiling(y) > -1.7) == (y > -2)
+
+    assert (ceiling(y) <= n) == (y <= n)
+    assert (ceiling(y) >= n) == (y > n - 1)
+    assert (ceiling(y) < n) == (y <= n - 1)
+    assert (ceiling(y) > n) == (y > n)
+
+
+def test_frac():
+    assert isinstance(frac(x), frac)
+    assert frac(oo) == AccumBounds(0, 1)
+    assert frac(-oo) == AccumBounds(0, 1)
+    assert frac(zoo) is nan
+
+    assert frac(n) == 0
+    assert frac(nan) is nan
+    assert frac(Rational(4, 3)) == Rational(1, 3)
+    assert frac(-Rational(4, 3)) == Rational(2, 3)
+    assert frac(Rational(-4, 3)) == Rational(2, 3)
+
+    r = Symbol('r', real=True)
+    assert frac(I*r) == I*frac(r)
+    assert frac(1 + I*r) == I*frac(r)
+    assert frac(0.5 + I*r) == 0.5 + I*frac(r)
+    assert frac(n + I*r) == I*frac(r)
+    assert frac(n + I*k) == 0
+    assert unchanged(frac, x + I*x)
+    assert frac(x + I*n) == frac(x)
+
+    assert frac(x).rewrite(floor) == x - floor(x)
+    assert frac(x).rewrite(ceiling) == x + ceiling(-x)
+    assert frac(y).rewrite(floor).subs(y, pi) == frac(pi)
+    assert frac(y).rewrite(floor).subs(y, -E) == frac(-E)
+    assert frac(y).rewrite(ceiling).subs(y, -pi) == frac(-pi)
+    assert frac(y).rewrite(ceiling).subs(y, E) == frac(E)
+
+    assert Eq(frac(y), y - floor(y))
+    assert Eq(frac(y), y + ceiling(-y))
+
+    r = Symbol('r', real=True)
+    p_i = Symbol('p_i', integer=True, positive=True)
+    n_i = Symbol('p_i', integer=True, negative=True)
+    np_i = Symbol('np_i', integer=True, nonpositive=True)
+    nn_i = Symbol('nn_i', integer=True, nonnegative=True)
+    p_r = Symbol('p_r', positive=True)
+    n_r = Symbol('n_r', negative=True)
+    np_r = Symbol('np_r', real=True, nonpositive=True)
+    nn_r = Symbol('nn_r', real=True, nonnegative=True)
+
+    # Real frac argument, integer rhs
+    assert frac(r) <= p_i
+    assert not frac(r) <= n_i
+    assert (frac(r) <= np_i).has(Le)
+    assert (frac(r) <= nn_i).has(Le)
+    assert frac(r) < p_i
+    assert not frac(r) < n_i
+    assert not frac(r) < np_i
+    assert (frac(r) < nn_i).has(Lt)
+    assert not frac(r) >= p_i
+    assert frac(r) >= n_i
+    assert frac(r) >= np_i
+    assert (frac(r) >= nn_i).has(Ge)
+    assert not frac(r) > p_i
+    assert frac(r) > n_i
+    assert (frac(r) > np_i).has(Gt)
+    assert (frac(r) > nn_i).has(Gt)
+
+    assert not Eq(frac(r), p_i)
+    assert not Eq(frac(r), n_i)
+    assert Eq(frac(r), np_i).has(Eq)
+    assert Eq(frac(r), nn_i).has(Eq)
+
+    assert Ne(frac(r), p_i)
+    assert Ne(frac(r), n_i)
+    assert Ne(frac(r), np_i).has(Ne)
+    assert Ne(frac(r), nn_i).has(Ne)
+
+
+    # Real frac argument, real rhs
+    assert (frac(r) <= p_r).has(Le)
+    assert not frac(r) <= n_r
+    assert (frac(r) <= np_r).has(Le)
+    assert (frac(r) <= nn_r).has(Le)
+    assert (frac(r) < p_r).has(Lt)
+    assert not frac(r) < n_r
+    assert not frac(r) < np_r
+    assert (frac(r) < nn_r).has(Lt)
+    assert (frac(r) >= p_r).has(Ge)
+    assert frac(r) >= n_r
+    assert frac(r) >= np_r
+    assert (frac(r) >= nn_r).has(Ge)
+    assert (frac(r) > p_r).has(Gt)
+    assert frac(r) > n_r
+    assert (frac(r) > np_r).has(Gt)
+    assert (frac(r) > nn_r).has(Gt)
+
+    assert not Eq(frac(r), n_r)
+    assert Eq(frac(r), p_r).has(Eq)
+    assert Eq(frac(r), np_r).has(Eq)
+    assert Eq(frac(r), nn_r).has(Eq)
+
+    assert Ne(frac(r), p_r).has(Ne)
+    assert Ne(frac(r), n_r)
+    assert Ne(frac(r), np_r).has(Ne)
+    assert Ne(frac(r), nn_r).has(Ne)
+
+    # Real frac argument, +/- oo rhs
+    assert frac(r) < oo
+    assert frac(r) <= oo
+    assert not frac(r) > oo
+    assert not frac(r) >= oo
+
+    assert not frac(r) < -oo
+    assert not frac(r) <= -oo
+    assert frac(r) > -oo
+    assert frac(r) >= -oo
+
+    assert frac(r) < 1
+    assert frac(r) <= 1
+    assert not frac(r) > 1
+    assert not frac(r) >= 1
+
+    assert not frac(r) < 0
+    assert (frac(r) <= 0).has(Le)
+    assert (frac(r) > 0).has(Gt)
+    assert frac(r) >= 0
+
+    # Some test for numbers
+    assert frac(r) <= sqrt(2)
+    assert (frac(r) <= sqrt(3) - sqrt(2)).has(Le)
+    assert not frac(r) <= sqrt(2) - sqrt(3)
+    assert not frac(r) >= sqrt(2)
+    assert (frac(r) >= sqrt(3) - sqrt(2)).has(Ge)
+    assert frac(r) >= sqrt(2) - sqrt(3)
+
+    assert not Eq(frac(r), sqrt(2))
+    assert Eq(frac(r), sqrt(3) - sqrt(2)).has(Eq)
+    assert not Eq(frac(r), sqrt(2) - sqrt(3))
+    assert Ne(frac(r), sqrt(2))
+    assert Ne(frac(r), sqrt(3) - sqrt(2)).has(Ne)
+    assert Ne(frac(r), sqrt(2) - sqrt(3))
+
+    assert frac(p_i, evaluate=False).is_zero
+    assert frac(p_i, evaluate=False).is_finite
+    assert frac(p_i, evaluate=False).is_integer
+    assert frac(p_i, evaluate=False).is_real
+    assert frac(r).is_finite
+    assert frac(r).is_real
+    assert frac(r).is_zero is None
+    assert frac(r).is_integer is None
+
+    assert frac(oo).is_finite
+    assert frac(oo).is_real
+
+
+def test_series():
+    x, y = symbols('x,y')
+    assert floor(x).nseries(x, y, 100) == floor(y)
+    assert ceiling(x).nseries(x, y, 100) == ceiling(y)
+    assert floor(x).nseries(x, pi, 100) == 3
+    assert ceiling(x).nseries(x, pi, 100) == 4
+    assert floor(x).nseries(x, 0, 100) == 0
+    assert ceiling(x).nseries(x, 0, 100) == 1
+    assert floor(-x).nseries(x, 0, 100) == -1
+    assert ceiling(-x).nseries(x, 0, 100) == 0
+
+
+def test_issue_14355():
+    # This test checks the leading term and series for the floor and ceil
+    # function when arg0 evaluates to S.NaN.
+    assert floor((x**3 + x)/(x**2 - x)).as_leading_term(x, cdir = 1) == -2
+    assert floor((x**3 + x)/(x**2 - x)).as_leading_term(x, cdir = -1) == -1
+    assert floor((cos(x) - 1)/x).as_leading_term(x, cdir = 1) == -1
+    assert floor((cos(x) - 1)/x).as_leading_term(x, cdir = -1) == 0
+    assert floor(sin(x)/x).as_leading_term(x, cdir = 1) == 0
+    assert floor(sin(x)/x).as_leading_term(x, cdir = -1) == 0
+    assert floor(-tan(x)/x).as_leading_term(x, cdir = 1) == -2
+    assert floor(-tan(x)/x).as_leading_term(x, cdir = -1) == -2
+    assert floor(sin(x)/x/3).as_leading_term(x, cdir = 1) == 0
+    assert floor(sin(x)/x/3).as_leading_term(x, cdir = -1) == 0
+    assert ceiling((x**3 + x)/(x**2 - x)).as_leading_term(x, cdir = 1) == -1
+    assert ceiling((x**3 + x)/(x**2 - x)).as_leading_term(x, cdir = -1) == 0
+    assert ceiling((cos(x) - 1)/x).as_leading_term(x, cdir = 1) == 0
+    assert ceiling((cos(x) - 1)/x).as_leading_term(x, cdir = -1) == 1
+    assert ceiling(sin(x)/x).as_leading_term(x, cdir = 1) == 1
+    assert ceiling(sin(x)/x).as_leading_term(x, cdir = -1) == 1
+    assert ceiling(-tan(x)/x).as_leading_term(x, cdir = 1) == -1
+    assert ceiling(-tan(x)/x).as_leading_term(x, cdir = 1) == -1
+    assert ceiling(sin(x)/x/3).as_leading_term(x, cdir = 1) == 1
+    assert ceiling(sin(x)/x/3).as_leading_term(x, cdir = -1) == 1
+    # test for series
+    assert floor(sin(x)/x).series(x, 0, 100, cdir = 1) == 0
+    assert floor(sin(x)/x).series(x, 0, 100, cdir = 1) == 0
+    assert floor((x**3 + x)/(x**2 - x)).series(x, 0, 100, cdir = 1) == -2
+    assert floor((x**3 + x)/(x**2 - x)).series(x, 0, 100, cdir = -1) == -1
+    assert ceiling(sin(x)/x).series(x, 0, 100, cdir = 1) == 1
+    assert ceiling(sin(x)/x).series(x, 0, 100, cdir = -1) == 1
+    assert ceiling((x**3 + x)/(x**2 - x)).series(x, 0, 100, cdir = 1) == -1
+    assert ceiling((x**3 + x)/(x**2 - x)).series(x, 0, 100, cdir = -1) == 0
+
+
+def test_frac_leading_term():
+    assert frac(x).as_leading_term(x) == x
+    assert frac(x).as_leading_term(x, cdir = 1) == x
+    assert frac(x).as_leading_term(x, cdir = -1) == 1
+    assert frac(x + S.Half).as_leading_term(x, cdir = 1) == S.Half
+    assert frac(x + S.Half).as_leading_term(x, cdir = -1) == S.Half
+    assert frac(-2*x + 1).as_leading_term(x, cdir = 1) == S.One
+    assert frac(-2*x + 1).as_leading_term(x, cdir = -1) == -2*x
+    assert frac(sin(x) + 5).as_leading_term(x, cdir = 1) == x
+    assert frac(sin(x) + 5).as_leading_term(x, cdir = -1) == S.One
+    assert frac(sin(x**2) + 5).as_leading_term(x, cdir = 1) == x**2
+    assert frac(sin(x**2) + 5).as_leading_term(x, cdir = -1) == x**2
+
+
+@XFAIL
+def test_issue_4149():
+    assert floor(3 + pi*I + y*I) == 3 + floor(pi + y)*I
+    assert floor(3*I + pi*I + y*I) == floor(3 + pi + y)*I
+    assert floor(3 + E + pi*I + y*I) == 5 + floor(pi + y)*I
+
+
+def test_issue_21651():
+    k = Symbol('k', positive=True, integer=True)
+    exp = 2*2**(-k)
+    assert isinstance(floor(exp), floor)
+
+
+def test_issue_11207():
+    assert floor(floor(x)) == floor(x)
+    assert floor(ceiling(x)) == ceiling(x)
+    assert ceiling(floor(x)) == floor(x)
+    assert ceiling(ceiling(x)) == ceiling(x)
+
+
+def test_nested_floor_ceiling():
+    assert floor(-floor(ceiling(x**3)/y)) == -floor(ceiling(x**3)/y)
+    assert ceiling(-floor(ceiling(x**3)/y)) == -floor(ceiling(x**3)/y)
+    assert floor(ceiling(-floor(x**Rational(7, 2)/y))) == -floor(x**Rational(7, 2)/y)
+    assert -ceiling(-ceiling(floor(x)/y)) == ceiling(floor(x)/y)
+
+def test_issue_18689():
+    assert floor(floor(floor(x)) + 3) == floor(x) + 3
+    assert ceiling(ceiling(ceiling(x)) + 1) == ceiling(x) + 1
+    assert ceiling(ceiling(floor(x)) + 3) == floor(x) + 3
+
+def test_issue_18421():
+    assert floor(float(0)) is S.Zero
+    assert ceiling(float(0)) is S.Zero

llmeval-env/lib/python3.10/site-packages/sympy/functions/elementary/tests/test_interface.py

@@ -0,0 +1,72 @@
+# This test file tests the SymPy function interface, that people use to create
+# their own new functions. It should be as easy as possible.
+from sympy.core.function import Function
+from sympy.core.sympify import sympify
+from sympy.functions.elementary.hyperbolic import tanh
+from sympy.functions.elementary.trigonometric import (cos, sin)
+from sympy.series.limits import limit
+from sympy.abc import x
+
+
+def test_function_series1():
+    """Create our new "sin" function."""
+
+    class my_function(Function):
+
+        def fdiff(self, argindex=1):
+            return cos(self.args[0])
+
+        @classmethod
+        def eval(cls, arg):
+            arg = sympify(arg)
+            if arg == 0:
+                return sympify(0)
+
+    #Test that the taylor series is correct
+    assert my_function(x).series(x, 0, 10) == sin(x).series(x, 0, 10)
+    assert limit(my_function(x)/x, x, 0) == 1
+
+
+def test_function_series2():
+    """Create our new "cos" function."""
+
+    class my_function2(Function):
+
+        def fdiff(self, argindex=1):
+            return -sin(self.args[0])
+
+        @classmethod
+        def eval(cls, arg):
+            arg = sympify(arg)
+            if arg == 0:
+                return sympify(1)
+
+    #Test that the taylor series is correct
+    assert my_function2(x).series(x, 0, 10) == cos(x).series(x, 0, 10)
+
+
+def test_function_series3():
+    """
+    Test our easy "tanh" function.
+
+    This test tests two things:
+      * that the Function interface works as expected and it's easy to use
+      * that the general algorithm for the series expansion works even when the
+        derivative is defined recursively in terms of the original function,
+        since tanh(x).diff(x) == 1-tanh(x)**2
+    """
+
+    class mytanh(Function):
+
+        def fdiff(self, argindex=1):
+            return 1 - mytanh(self.args[0])**2
+
+        @classmethod
+        def eval(cls, arg):
+            arg = sympify(arg)
+            if arg == 0:
+                return sympify(0)
+
+    e = tanh(x)
+    f = mytanh(x)
+    assert e.series(x, 0, 6) == f.series(x, 0, 6)

llmeval-env/lib/python3.10/site-packages/sympy/functions/elementary/tests/test_miscellaneous.py

@@ -0,0 +1,504 @@
+import itertools as it
+
+from sympy.core.expr import unchanged
+from sympy.core.function import Function
+from sympy.core.numbers import I, oo, Rational
+from sympy.core.power import Pow
+from sympy.core.singleton import S
+from sympy.core.symbol import Symbol
+from sympy.external import import_module
+from sympy.functions.elementary.exponential import log
+from sympy.functions.elementary.integers import floor, ceiling
+from sympy.functions.elementary.miscellaneous import (sqrt, cbrt, root, Min,
+                                                      Max, real_root, Rem)
+from sympy.functions.elementary.trigonometric import cos, sin
+from sympy.functions.special.delta_functions import Heaviside
+
+from sympy.utilities.lambdify import lambdify
+from sympy.testing.pytest import raises, skip, ignore_warnings
+
+def test_Min():
+    from sympy.abc import x, y, z
+    n = Symbol('n', negative=True)
+    n_ = Symbol('n_', negative=True)
+    nn = Symbol('nn', nonnegative=True)
+    nn_ = Symbol('nn_', nonnegative=True)
+    p = Symbol('p', positive=True)
+    p_ = Symbol('p_', positive=True)
+    np = Symbol('np', nonpositive=True)
+    np_ = Symbol('np_', nonpositive=True)
+    r = Symbol('r', real=True)
+
+    assert Min(5, 4) == 4
+    assert Min(-oo, -oo) is -oo
+    assert Min(-oo, n) is -oo
+    assert Min(n, -oo) is -oo
+    assert Min(-oo, np) is -oo
+    assert Min(np, -oo) is -oo
+    assert Min(-oo, 0) is -oo
+    assert Min(0, -oo) is -oo
+    assert Min(-oo, nn) is -oo
+    assert Min(nn, -oo) is -oo
+    assert Min(-oo, p) is -oo
+    assert Min(p, -oo) is -oo
+    assert Min(-oo, oo) is -oo
+    assert Min(oo, -oo) is -oo
+    assert Min(n, n) == n
+    assert unchanged(Min, n, np)
+    assert Min(np, n) == Min(n, np)
+    assert Min(n, 0) == n
+    assert Min(0, n) == n
+    assert Min(n, nn) == n
+    assert Min(nn, n) == n
+    assert Min(n, p) == n
+    assert Min(p, n) == n
+    assert Min(n, oo) == n
+    assert Min(oo, n) == n
+    assert Min(np, np) == np
+    assert Min(np, 0) == np
+    assert Min(0, np) == np
+    assert Min(np, nn) == np
+    assert Min(nn, np) == np
+    assert Min(np, p) == np
+    assert Min(p, np) == np
+    assert Min(np, oo) == np
+    assert Min(oo, np) == np
+    assert Min(0, 0) == 0
+    assert Min(0, nn) == 0
+    assert Min(nn, 0) == 0
+    assert Min(0, p) == 0
+    assert Min(p, 0) == 0
+    assert Min(0, oo) == 0
+    assert Min(oo, 0) == 0
+    assert Min(nn, nn) == nn
+    assert unchanged(Min, nn, p)
+    assert Min(p, nn) == Min(nn, p)
+    assert Min(nn, oo) == nn
+    assert Min(oo, nn) == nn
+    assert Min(p, p) == p
+    assert Min(p, oo) == p
+    assert Min(oo, p) == p
+    assert Min(oo, oo) is oo
+
+    assert Min(n, n_).func is Min
+    assert Min(nn, nn_).func is Min
+    assert Min(np, np_).func is Min
+    assert Min(p, p_).func is Min
+
+    # lists
+    assert Min() is S.Infinity
+    assert Min(x) == x
+    assert Min(x, y) == Min(y, x)
+    assert Min(x, y, z) == Min(z, y, x)
+    assert Min(x, Min(y, z)) == Min(z, y, x)
+    assert Min(x, Max(y, -oo)) == Min(x, y)
+    assert Min(p, oo, n, p, p, p_) == n
+    assert Min(p_, n_, p) == n_
+    assert Min(n, oo, -7, p, p, 2) == Min(n, -7)
+    assert Min(2, x, p, n, oo, n_, p, 2, -2, -2) == Min(-2, x, n, n_)
+    assert Min(0, x, 1, y) == Min(0, x, y)
+    assert Min(1000, 100, -100, x, p, n) == Min(n, x, -100)
+    assert unchanged(Min, sin(x), cos(x))
+    assert Min(sin(x), cos(x)) == Min(cos(x), sin(x))
+    assert Min(cos(x), sin(x)).subs(x, 1) == cos(1)
+    assert Min(cos(x), sin(x)).subs(x, S.Half) == sin(S.Half)
+    raises(ValueError, lambda: Min(cos(x), sin(x)).subs(x, I))
+    raises(ValueError, lambda: Min(I))
+    raises(ValueError, lambda: Min(I, x))
+    raises(ValueError, lambda: Min(S.ComplexInfinity, x))
+
+    assert Min(1, x).diff(x) == Heaviside(1 - x)
+    assert Min(x, 1).diff(x) == Heaviside(1 - x)
+    assert Min(0, -x, 1 - 2*x).diff(x) == -Heaviside(x + Min(0, -2*x + 1)) \
+        - 2*Heaviside(2*x + Min(0, -x) - 1)
+
+    # issue 7619
+    f = Function('f')
+    assert Min(1, 2*Min(f(1), 2))  # doesn't fail
+
+    # issue 7233
+    e = Min(0, x)
+    assert e.n().args == (0, x)
+
+    # issue 8643
+    m = Min(n, p_, n_, r)
+    assert m.is_positive is False
+    assert m.is_nonnegative is False
+    assert m.is_negative is True
+
+    m = Min(p, p_)
+    assert m.is_positive is True
+    assert m.is_nonnegative is True
+    assert m.is_negative is False
+
+    m = Min(p, nn_, p_)
+    assert m.is_positive is None
+    assert m.is_nonnegative is True
+    assert m.is_negative is False
+
+    m = Min(nn, p, r)
+    assert m.is_positive is None
+    assert m.is_nonnegative is None
+    assert m.is_negative is None
+
+
+def test_Max():
+    from sympy.abc import x, y, z
+    n = Symbol('n', negative=True)
+    n_ = Symbol('n_', negative=True)
+    nn = Symbol('nn', nonnegative=True)
+    p = Symbol('p', positive=True)
+    p_ = Symbol('p_', positive=True)
+    r = Symbol('r', real=True)
+
+    assert Max(5, 4) == 5
+
+    # lists
+
+    assert Max() is S.NegativeInfinity
+    assert Max(x) == x
+    assert Max(x, y) == Max(y, x)
+    assert Max(x, y, z) == Max(z, y, x)
+    assert Max(x, Max(y, z)) == Max(z, y, x)
+    assert Max(x, Min(y, oo)) == Max(x, y)
+    assert Max(n, -oo, n_, p, 2) == Max(p, 2)
+    assert Max(n, -oo, n_, p) == p
+    assert Max(2, x, p, n, -oo, S.NegativeInfinity, n_, p, 2) == Max(2, x, p)
+    assert Max(0, x, 1, y) == Max(1, x, y)
+    assert Max(r, r + 1, r - 1) == 1 + r
+    assert Max(1000, 100, -100, x, p, n) == Max(p, x, 1000)
+    assert Max(cos(x), sin(x)) == Max(sin(x), cos(x))
+    assert Max(cos(x), sin(x)).subs(x, 1) == sin(1)
+    assert Max(cos(x), sin(x)).subs(x, S.Half) == cos(S.Half)
+    raises(ValueError, lambda: Max(cos(x), sin(x)).subs(x, I))
+    raises(ValueError, lambda: Max(I))
+    raises(ValueError, lambda: Max(I, x))
+    raises(ValueError, lambda: Max(S.ComplexInfinity, 1))
+    assert Max(n, -oo, n_,  p, 2) == Max(p, 2)
+    assert Max(n, -oo, n_,  p, 1000) == Max(p, 1000)
+
+    assert Max(1, x).diff(x) == Heaviside(x - 1)
+    assert Max(x, 1).diff(x) == Heaviside(x - 1)
+    assert Max(x**2, 1 + x, 1).diff(x) == \
+        2*x*Heaviside(x**2 - Max(1, x + 1)) \
+        + Heaviside(x - Max(1, x**2) + 1)
+
+    e = Max(0, x)
+    assert e.n().args == (0, x)
+
+    # issue 8643
+    m = Max(p, p_, n, r)
+    assert m.is_positive is True
+    assert m.is_nonnegative is True
+    assert m.is_negative is False
+
+    m = Max(n, n_)
+    assert m.is_positive is False
+    assert m.is_nonnegative is False
+    assert m.is_negative is True
+
+    m = Max(n, n_, r)
+    assert m.is_positive is None
+    assert m.is_nonnegative is None
+    assert m.is_negative is None
+
+    m = Max(n, nn, r)
+    assert m.is_positive is None
+    assert m.is_nonnegative is True
+    assert m.is_negative is False
+
+
+def test_minmax_assumptions():
+    r = Symbol('r', real=True)
+    a = Symbol('a', real=True, algebraic=True)
+    t = Symbol('t', real=True, transcendental=True)
+    q = Symbol('q', rational=True)
+    p = Symbol('p', irrational=True)
+    n = Symbol('n', rational=True, integer=False)
+    i = Symbol('i', integer=True)
+    o = Symbol('o', odd=True)
+    e = Symbol('e', even=True)
+    k = Symbol('k', prime=True)
+    reals = [r, a, t, q, p, n, i, o, e, k]
+
+    for ext in (Max, Min):
+        for x, y in it.product(reals, repeat=2):
+
+            # Must be real
+            assert ext(x, y).is_real
+
+            # Algebraic?
+            if x.is_algebraic and y.is_algebraic:
+                assert ext(x, y).is_algebraic
+            elif x.is_transcendental and y.is_transcendental:
+                assert ext(x, y).is_transcendental
+            else:
+                assert ext(x, y).is_algebraic is None
+
+            # Rational?
+            if x.is_rational and y.is_rational:
+                assert ext(x, y).is_rational
+            elif x.is_irrational and y.is_irrational:
+                assert ext(x, y).is_irrational
+            else:
+                assert ext(x, y).is_rational is None
+
+            # Integer?
+            if x.is_integer and y.is_integer:
+                assert ext(x, y).is_integer
+            elif x.is_noninteger and y.is_noninteger:
+                assert ext(x, y).is_noninteger
+            else:
+                assert ext(x, y).is_integer is None
+
+            # Odd?
+            if x.is_odd and y.is_odd:
+                assert ext(x, y).is_odd
+            elif x.is_odd is False and y.is_odd is False:
+                assert ext(x, y).is_odd is False
+            else:
+                assert ext(x, y).is_odd is None
+
+            # Even?
+            if x.is_even and y.is_even:
+                assert ext(x, y).is_even
+            elif x.is_even is False and y.is_even is False:
+                assert ext(x, y).is_even is False
+            else:
+                assert ext(x, y).is_even is None
+
+            # Prime?
+            if x.is_prime and y.is_prime:
+                assert ext(x, y).is_prime
+            elif x.is_prime is False and y.is_prime is False:
+                assert ext(x, y).is_prime is False
+            else:
+                assert ext(x, y).is_prime is None
+
+
+def test_issue_8413():
+    x = Symbol('x', real=True)
+    # we can't evaluate in general because non-reals are not
+    # comparable: Min(floor(3.2 + I), 3.2 + I) -> ValueError
+    assert Min(floor(x), x) == floor(x)
+    assert Min(ceiling(x), x) == x
+    assert Max(floor(x), x) == x
+    assert Max(ceiling(x), x) == ceiling(x)
+
+
+def test_root():
+    from sympy.abc import x
+    n = Symbol('n', integer=True)
+    k = Symbol('k', integer=True)
+
+    assert root(2, 2) == sqrt(2)
+    assert root(2, 1) == 2
+    assert root(2, 3) == 2**Rational(1, 3)
+    assert root(2, 3) == cbrt(2)
+    assert root(2, -5) == 2**Rational(4, 5)/2
+
+    assert root(-2, 1) == -2
+
+    assert root(-2, 2) == sqrt(2)*I
+    assert root(-2, 1) == -2
+
+    assert root(x, 2) == sqrt(x)
+    assert root(x, 1) == x
+    assert root(x, 3) == x**Rational(1, 3)
+    assert root(x, 3) == cbrt(x)
+    assert root(x, -5) == x**Rational(-1, 5)
+
+    assert root(x, n) == x**(1/n)
+    assert root(x, -n) == x**(-1/n)
+
+    assert root(x, n, k) == (-1)**(2*k/n)*x**(1/n)
+
+
+def test_real_root():
+    assert real_root(-8, 3) == -2
+    assert real_root(-16, 4) == root(-16, 4)
+    r = root(-7, 4)
+    assert real_root(r) == r
+    r1 = root(-1, 3)
+    r2 = r1**2
+    r3 = root(-1, 4)
+    assert real_root(r1 + r2 + r3) == -1 + r2 + r3
+    assert real_root(root(-2, 3)) == -root(2, 3)
+    assert real_root(-8., 3) == -2.0
+    x = Symbol('x')
+    n = Symbol('n')
+    g = real_root(x, n)
+    assert g.subs({"x": -8, "n": 3}) == -2
+    assert g.subs({"x": 8, "n": 3}) == 2
+    # give principle root if there is no real root -- if this is not desired
+    # then maybe a Root class is needed to raise an error instead
+    assert g.subs({"x": I, "n": 3}) == cbrt(I)
+    assert g.subs({"x": -8, "n": 2}) == sqrt(-8)
+    assert g.subs({"x": I, "n": 2}) == sqrt(I)
+
+
+def test_issue_11463():
+    numpy = import_module('numpy')
+    if not numpy:
+        skip("numpy not installed.")
+    x = Symbol('x')
+    f = lambdify(x, real_root((log(x/(x-2))), 3), 'numpy')
+    # numpy.select evaluates all options before considering conditions,
+    # so it raises a warning about root of negative number which does
+    # not affect the outcome. This warning is suppressed here
+    with ignore_warnings(RuntimeWarning):
+        assert f(numpy.array(-1)) < -1
+
+
+def test_rewrite_MaxMin_as_Heaviside():
+    from sympy.abc import x
+    assert Max(0, x).rewrite(Heaviside) == x*Heaviside(x)
+    assert Max(3, x).rewrite(Heaviside) == x*Heaviside(x - 3) + \
+        3*Heaviside(-x + 3)
+    assert Max(0, x+2, 2*x).rewrite(Heaviside) == \
+        2*x*Heaviside(2*x)*Heaviside(x - 2) + \
+        (x + 2)*Heaviside(-x + 2)*Heaviside(x + 2)
+
+    assert Min(0, x).rewrite(Heaviside) == x*Heaviside(-x)
+    assert Min(3, x).rewrite(Heaviside) == x*Heaviside(-x + 3) + \
+        3*Heaviside(x - 3)
+    assert Min(x, -x, -2).rewrite(Heaviside) == \
+        x*Heaviside(-2*x)*Heaviside(-x - 2) - \
+        x*Heaviside(2*x)*Heaviside(x - 2) \
+        - 2*Heaviside(-x + 2)*Heaviside(x + 2)
+
+
+def test_rewrite_MaxMin_as_Piecewise():
+    from sympy.core.symbol import symbols
+    from sympy.functions.elementary.piecewise import Piecewise
+    x, y, z, a, b = symbols('x y z a b', real=True)
+    vx, vy, va = symbols('vx vy va')
+    assert Max(a, b).rewrite(Piecewise) == Piecewise((a, a >= b), (b, True))
+    assert Max(x, y, z).rewrite(Piecewise) == Piecewise((x, (x >= y) & (x >= z)), (y, y >= z), (z, True))
+    assert Max(x, y, a, b).rewrite(Piecewise) == Piecewise((a, (a >= b) & (a >= x) & (a >= y)),
+        (b, (b >= x) & (b >= y)), (x, x >= y), (y, True))
+    assert Min(a, b).rewrite(Piecewise) == Piecewise((a, a <= b), (b, True))
+    assert Min(x, y, z).rewrite(Piecewise) == Piecewise((x, (x <= y) & (x <= z)), (y, y <= z), (z, True))
+    assert Min(x,  y, a, b).rewrite(Piecewise) ==  Piecewise((a, (a <= b) & (a <= x) & (a <= y)),
+        (b, (b <= x) & (b <= y)), (x, x <= y), (y, True))
+
+    # Piecewise rewriting of Min/Max does also takes place for not explicitly real arguments
+    assert Max(vx, vy).rewrite(Piecewise) == Piecewise((vx, vx >= vy), (vy, True))
+    assert Min(va, vx, vy).rewrite(Piecewise) == Piecewise((va, (va <= vx) & (va <= vy)), (vx, vx <= vy), (vy, True))
+
+
+def test_issue_11099():
+    from sympy.abc import x, y
+    # some fixed value tests
+    fixed_test_data = {x: -2, y: 3}
+    assert Min(x, y).evalf(subs=fixed_test_data) == \
+        Min(x, y).subs(fixed_test_data).evalf()
+    assert Max(x, y).evalf(subs=fixed_test_data) == \
+        Max(x, y).subs(fixed_test_data).evalf()
+    # randomly generate some test data
+    from sympy.core.random import randint
+    for i in range(20):
+        random_test_data = {x: randint(-100, 100), y: randint(-100, 100)}
+        assert Min(x, y).evalf(subs=random_test_data) == \
+            Min(x, y).subs(random_test_data).evalf()
+        assert Max(x, y).evalf(subs=random_test_data) == \
+            Max(x, y).subs(random_test_data).evalf()
+
+
+def test_issue_12638():
+    from sympy.abc import a, b, c
+    assert Min(a, b, c, Max(a, b)) == Min(a, b, c)
+    assert Min(a, b, Max(a, b, c)) == Min(a, b)
+    assert Min(a, b, Max(a, c)) == Min(a, b)
+
+def test_issue_21399():
+    from sympy.abc import a, b, c
+    assert Max(Min(a, b), Min(a, b, c)) == Min(a, b)
+
+
+def test_instantiation_evaluation():
+    from sympy.abc import v, w, x, y, z
+    assert Min(1, Max(2, x)) == 1
+    assert Max(3, Min(2, x)) == 3
+    assert Min(Max(x, y), Max(x, z)) == Max(x, Min(y, z))
+    assert set(Min(Max(w, x), Max(y, z)).args) == {
+        Max(w, x), Max(y, z)}
+    assert Min(Max(x, y), Max(x, z), w) == Min(
+        w, Max(x, Min(y, z)))
+    A, B = Min, Max
+    for i in range(2):
+        assert A(x, B(x, y)) == x
+        assert A(x, B(y, A(x, w, z))) == A(x, B(y, A(w, z)))
+        A, B = B, A
+    assert Min(w, Max(x, y), Max(v, x, z)) == Min(
+        w, Max(x, Min(y, Max(v, z))))
+
+def test_rewrite_as_Abs():
+    from itertools import permutations
+    from sympy.functions.elementary.complexes import Abs
+    from sympy.abc import x, y, z, w
+    def test(e):
+        free = e.free_symbols
+        a = e.rewrite(Abs)
+        assert not a.has(Min, Max)
+        for i in permutations(range(len(free))):
+            reps = dict(zip(free, i))
+            assert a.xreplace(reps) == e.xreplace(reps)
+    test(Min(x, y))
+    test(Max(x, y))
+    test(Min(x, y, z))
+    test(Min(Max(w, x), Max(y, z)))
+
+def test_issue_14000():
+    assert isinstance(sqrt(4, evaluate=False), Pow) == True
+    assert isinstance(cbrt(3.5, evaluate=False), Pow) == True
+    assert isinstance(root(16, 4, evaluate=False), Pow) == True
+
+    assert sqrt(4, evaluate=False) == Pow(4, S.Half, evaluate=False)
+    assert cbrt(3.5, evaluate=False) == Pow(3.5, Rational(1, 3), evaluate=False)
+    assert root(4, 2, evaluate=False) == Pow(4, S.Half, evaluate=False)
+
+    assert root(16, 4, 2, evaluate=False).has(Pow) == True
+    assert real_root(-8, 3, evaluate=False).has(Pow) == True
+
+def test_issue_6899():
+    from sympy.core.function import Lambda
+    x = Symbol('x')
+    eqn = Lambda(x, x)
+    assert eqn.func(*eqn.args) == eqn
+
+def test_Rem():
+    from sympy.abc import x, y
+    assert Rem(5, 3) == 2
+    assert Rem(-5, 3) == -2
+    assert Rem(5, -3) == 2
+    assert Rem(-5, -3) == -2
+    assert Rem(x**3, y) == Rem(x**3, y)
+    assert Rem(Rem(-5, 3) + 3, 3) == 1
+
+
+def test_minmax_no_evaluate():
+    from sympy import evaluate
+    p = Symbol('p', positive=True)
+
+    assert Max(1, 3) == 3
+    assert Max(1, 3).args == ()
+    assert Max(0, p) == p
+    assert Max(0, p).args == ()
+    assert Min(0, p) == 0
+    assert Min(0, p).args == ()
+
+    assert Max(1, 3, evaluate=False) != 3
+    assert Max(1, 3, evaluate=False).args == (1, 3)
+    assert Max(0, p, evaluate=False) != p
+    assert Max(0, p, evaluate=False).args == (0, p)
+    assert Min(0, p, evaluate=False) != 0
+    assert Min(0, p, evaluate=False).args == (0, p)
+
+    with evaluate(False):
+        assert Max(1, 3) != 3
+        assert Max(1, 3).args == (1, 3)
+        assert Max(0, p) != p
+        assert Max(0, p).args == (0, p)
+        assert Min(0, p) != 0
+        assert Min(0, p).args == (0, p)

llmeval-env/lib/python3.10/site-packages/sympy/functions/elementary/tests/test_piecewise.py

@@ -0,0 +1,1606 @@
+from sympy.concrete.summations import Sum
+from sympy.core.add import Add
+from sympy.core.basic import Basic
+from sympy.core.containers import Tuple
+from sympy.core.expr import unchanged
+from sympy.core.function import (Function, diff, expand)
+from sympy.core.mul import Mul
+from sympy.core.mod import Mod
+from sympy.core.numbers import (Float, I, Rational, oo, pi, zoo)
+from sympy.core.relational import (Eq, Ge, Gt, Ne)
+from sympy.core.singleton import S
+from sympy.core.symbol import (Symbol, symbols)
+from sympy.functions.combinatorial.factorials import factorial
+from sympy.functions.elementary.complexes import (Abs, adjoint, arg, conjugate, im, re, transpose)
+from sympy.functions.elementary.exponential import (exp, log)
+from sympy.functions.elementary.miscellaneous import (Max, Min, sqrt)
+from sympy.functions.elementary.piecewise import (Piecewise,
+    piecewise_fold, piecewise_exclusive, Undefined, ExprCondPair)
+from sympy.functions.elementary.trigonometric import (cos, sin)
+from sympy.functions.special.delta_functions import (DiracDelta, Heaviside)
+from sympy.functions.special.tensor_functions import KroneckerDelta
+from sympy.integrals.integrals import (Integral, integrate)
+from sympy.logic.boolalg import (And, ITE, Not, Or)
+from sympy.matrices.expressions.matexpr import MatrixSymbol
+from sympy.printing import srepr
+from sympy.sets.contains import Contains
+from sympy.sets.sets import Interval
+from sympy.solvers.solvers import solve
+from sympy.testing.pytest import raises, slow
+from sympy.utilities.lambdify import lambdify
+
+a, b, c, d, x, y = symbols('a:d, x, y')
+z = symbols('z', nonzero=True)
+
+
+def test_piecewise1():
+
+    # Test canonicalization
+    assert unchanged(Piecewise, ExprCondPair(x, x < 1), ExprCondPair(0, True))
+    assert Piecewise((x, x < 1), (0, True)) == Piecewise(ExprCondPair(x, x < 1),
+                                                         ExprCondPair(0, True))
+    assert Piecewise((x, x < 1), (0, True), (1, True)) == \
+        Piecewise((x, x < 1), (0, True))
+    assert Piecewise((x, x < 1), (0, False), (-1, 1 > 2)) == \
+        Piecewise((x, x < 1))
+    assert Piecewise((x, x < 1), (0, x < 1), (0, True)) == \
+        Piecewise((x, x < 1), (0, True))
+    assert Piecewise((x, x < 1), (0, x < 2), (0, True)) == \
+        Piecewise((x, x < 1), (0, True))
+    assert Piecewise((x, x < 1), (x, x < 2), (0, True)) == \
+        Piecewise((x, Or(x < 1, x < 2)), (0, True))
+    assert Piecewise((x, x < 1), (x, x < 2), (x, True)) == x
+    assert Piecewise((x, True)) == x
+    # Explicitly constructed empty Piecewise not accepted
+    raises(TypeError, lambda: Piecewise())
+    # False condition is never retained
+    assert Piecewise((2*x, x < 0), (x, False)) == \
+        Piecewise((2*x, x < 0), (x, False), evaluate=False) == \
+        Piecewise((2*x, x < 0))
+    assert Piecewise((x, False)) == Undefined
+    raises(TypeError, lambda: Piecewise(x))
+    assert Piecewise((x, 1)) == x  # 1 and 0 are accepted as True/False
+    raises(TypeError, lambda: Piecewise((x, 2)))
+    raises(TypeError, lambda: Piecewise((x, x**2)))
+    raises(TypeError, lambda: Piecewise(([1], True)))
+    assert Piecewise(((1, 2), True)) == Tuple(1, 2)
+    cond = (Piecewise((1, x < 0), (2, True)) < y)
+    assert Piecewise((1, cond)
+        ) == Piecewise((1, ITE(x < 0, y > 1, y > 2)))
+
+    assert Piecewise((1, x > 0), (2, And(x <= 0, x > -1))
+        ) == Piecewise((1, x > 0), (2, x > -1))
+    assert Piecewise((1, x <= 0), (2, (x < 0) & (x > -1))
+        ) == Piecewise((1, x <= 0))
+
+    # test for supporting Contains in Piecewise
+    pwise = Piecewise(
+        (1, And(x <= 6, x > 1, Contains(x, S.Integers))),
+        (0, True))
+    assert pwise.subs(x, pi) == 0
+    assert pwise.subs(x, 2) == 1
+    assert pwise.subs(x, 7) == 0
+
+    # Test subs
+    p = Piecewise((-1, x < -1), (x**2, x < 0), (log(x), x >= 0))
+    p_x2 = Piecewise((-1, x**2 < -1), (x**4, x**2 < 0), (log(x**2), x**2 >= 0))
+    assert p.subs(x, x**2) == p_x2
+    assert p.subs(x, -5) == -1
+    assert p.subs(x, -1) == 1
+    assert p.subs(x, 1) == log(1)
+
+    # More subs tests
+    p2 = Piecewise((1, x < pi), (-1, x < 2*pi), (0, x > 2*pi))
+    p3 = Piecewise((1, Eq(x, 0)), (1/x, True))
+    p4 = Piecewise((1, Eq(x, 0)), (2, 1/x>2))
+    assert p2.subs(x, 2) == 1
+    assert p2.subs(x, 4) == -1
+    assert p2.subs(x, 10) == 0
+    assert p3.subs(x, 0.0) == 1
+    assert p4.subs(x, 0.0) == 1
+
+
+    f, g, h = symbols('f,g,h', cls=Function)
+    pf = Piecewise((f(x), x < -1), (f(x) + h(x) + 2, x <= 1))
+    pg = Piecewise((g(x), x < -1), (g(x) + h(x) + 2, x <= 1))
+    assert pg.subs(g, f) == pf
+
+    assert Piecewise((1, Eq(x, 0)), (0, True)).subs(x, 0) == 1
+    assert Piecewise((1, Eq(x, 0)), (0, True)).subs(x, 1) == 0
+    assert Piecewise((1, Eq(x, y)), (0, True)).subs(x, y) == 1
+    assert Piecewise((1, Eq(x, z)), (0, True)).subs(x, z) == 1
+    assert Piecewise((1, Eq(exp(x), cos(z))), (0, True)).subs(x, z) == \
+        Piecewise((1, Eq(exp(z), cos(z))), (0, True))
+
+    p5 = Piecewise( (0, Eq(cos(x) + y, 0)), (1, True))
+    assert p5.subs(y, 0) == Piecewise( (0, Eq(cos(x), 0)), (1, True))
+
+    assert Piecewise((-1, y < 1), (0, x < 0), (1, Eq(x, 0)), (2, True)
+        ).subs(x, 1) == Piecewise((-1, y < 1), (2, True))
+    assert Piecewise((1, Eq(x**2, -1)), (2, x < 0)).subs(x, I) == 1
+
+    p6 = Piecewise((x, x > 0))
+    n = symbols('n', negative=True)
+    assert p6.subs(x, n) == Undefined
+
+    # Test evalf
+    assert p.evalf() == Piecewise((-1.0, x < -1), (x**2, x < 0), (log(x), True))
+    assert p.evalf(subs={x: -2}) == -1.0
+    assert p.evalf(subs={x: -1}) == 1.0
+    assert p.evalf(subs={x: 1}) == log(1)
+    assert p6.evalf(subs={x: -5}) == Undefined
+
+    # Test doit
+    f_int = Piecewise((Integral(x, (x, 0, 1)), x < 1))
+    assert f_int.doit() == Piecewise( (S.Half, x < 1) )
+
+    # Test differentiation
+    f = x
+    fp = x*p
+    dp = Piecewise((0, x < -1), (2*x, x < 0), (1/x, x >= 0))
+    fp_dx = x*dp + p
+    assert diff(p, x) == dp
+    assert diff(f*p, x) == fp_dx
+
+    # Test simple arithmetic
+    assert x*p == fp
+    assert x*p + p == p + x*p
+    assert p + f == f + p
+    assert p + dp == dp + p
+    assert p - dp == -(dp - p)
+
+    # Test power
+    dp2 = Piecewise((0, x < -1), (4*x**2, x < 0), (1/x**2, x >= 0))
+    assert dp**2 == dp2
+
+    # Test _eval_interval
+    f1 = x*y + 2
+    f2 = x*y**2 + 3
+    peval = Piecewise((f1, x < 0), (f2, x > 0))
+    peval_interval = f1.subs(
+        x, 0) - f1.subs(x, -1) + f2.subs(x, 1) - f2.subs(x, 0)
+    assert peval._eval_interval(x, 0, 0) == 0
+    assert peval._eval_interval(x, -1, 1) == peval_interval
+    peval2 = Piecewise((f1, x < 0), (f2, True))
+    assert peval2._eval_interval(x, 0, 0) == 0
+    assert peval2._eval_interval(x, 1, -1) == -peval_interval
+    assert peval2._eval_interval(x, -1, -2) == f1.subs(x, -2) - f1.subs(x, -1)
+    assert peval2._eval_interval(x, -1, 1) == peval_interval
+    assert peval2._eval_interval(x, None, 0) == peval2.subs(x, 0)
+    assert peval2._eval_interval(x, -1, None) == -peval2.subs(x, -1)
+
+    # Test integration
+    assert p.integrate() == Piecewise(
+        (-x, x < -1),
+        (x**3/3 + Rational(4, 3), x < 0),
+        (x*log(x) - x + Rational(4, 3), True))
+    p = Piecewise((x, x < 1), (x**2, -1 <= x), (x, 3 < x))
+    assert integrate(p, (x, -2, 2)) == Rational(5, 6)
+    assert integrate(p, (x, 2, -2)) == Rational(-5, 6)
+    p = Piecewise((0, x < 0), (1, x < 1), (0, x < 2), (1, x < 3), (0, True))
+    assert integrate(p, (x, -oo, oo)) == 2
+    p = Piecewise((x, x < -10), (x**2, x <= -1), (x, 1 < x))
+    assert integrate(p, (x, -2, 2)) == Undefined
+
+    # Test commutativity
+    assert isinstance(p, Piecewise) and p.is_commutative is True
+
+
+def test_piecewise_free_symbols():
+    f = Piecewise((x, a < 0), (y, True))
+    assert f.free_symbols == {x, y, a}
+
+
+def test_piecewise_integrate1():
+    x, y = symbols('x y', real=True)
+
+    f = Piecewise(((x - 2)**2, x >= 0), (1, True))
+    assert integrate(f, (x, -2, 2)) == Rational(14, 3)
+
+    g = Piecewise(((x - 5)**5, x >= 4), (f, True))
+    assert integrate(g, (x, -2, 2)) == Rational(14, 3)
+    assert integrate(g, (x, -2, 5)) == Rational(43, 6)
+
+    assert g == Piecewise(((x - 5)**5, x >= 4), (f, x < 4))
+
+    g = Piecewise(((x - 5)**5, 2 <= x), (f, x < 2))
+    assert integrate(g, (x, -2, 2)) == Rational(14, 3)
+    assert integrate(g, (x, -2, 5)) == Rational(-701, 6)
+
+    assert g == Piecewise(((x - 5)**5, 2 <= x), (f, True))
+
+    g = Piecewise(((x - 5)**5, 2 <= x), (2*f, True))
+    assert integrate(g, (x, -2, 2)) == Rational(28, 3)
+    assert integrate(g, (x, -2, 5)) == Rational(-673, 6)
+
+
+def test_piecewise_integrate1b():
+    g = Piecewise((1, x > 0), (0, Eq(x, 0)), (-1, x < 0))
+    assert integrate(g, (x, -1, 1)) == 0
+
+    g = Piecewise((1, x - y < 0), (0, True))
+    assert integrate(g, (y, -oo, 0)) == -Min(0, x)
+    assert g.subs(x, -3).integrate((y, -oo, 0)) == 3
+    assert integrate(g, (y, 0, -oo)) == Min(0, x)
+    assert integrate(g, (y, 0, oo)) == -Max(0, x) + oo
+    assert integrate(g, (y, -oo, 42)) == -Min(42, x) + 42
+    assert integrate(g, (y, -oo, oo)) == -x + oo
+
+    g = Piecewise((0, x < 0), (x, x <= 1), (1, True))
+    gy1 = g.integrate((x, y, 1))
+    g1y = g.integrate((x, 1, y))
+    for yy in (-1, S.Half, 2):
+        assert g.integrate((x, yy, 1)) == gy1.subs(y, yy)
+        assert g.integrate((x, 1, yy)) == g1y.subs(y, yy)
+    assert gy1 == Piecewise(
+        (-Min(1, Max(0, y))**2/2 + S.Half, y < 1),
+        (-y + 1, True))
+    assert g1y == Piecewise(
+        (Min(1, Max(0, y))**2/2 - S.Half, y < 1),
+        (y - 1, True))
+
+
+@slow
+def test_piecewise_integrate1ca():
+    y = symbols('y', real=True)
+    g = Piecewise(
+        (1 - x, Interval(0, 1).contains(x)),
+        (1 + x, Interval(-1, 0).contains(x)),
+        (0, True)
+        )
+    gy1 = g.integrate((x, y, 1))
+    g1y = g.integrate((x, 1, y))
+
+    assert g.integrate((x, -2, 1)) == gy1.subs(y, -2)
+    assert g.integrate((x, 1, -2)) == g1y.subs(y, -2)
+    assert g.integrate((x, 0, 1)) == gy1.subs(y, 0)
+    assert g.integrate((x, 1, 0)) == g1y.subs(y, 0)
+    assert g.integrate((x, 2, 1)) == gy1.subs(y, 2)
+    assert g.integrate((x, 1, 2)) == g1y.subs(y, 2)
+    assert piecewise_fold(gy1.rewrite(Piecewise)
+        ).simplify() == Piecewise(
+            (1, y <= -1),
+            (-y**2/2 - y + S.Half, y <= 0),
+            (y**2/2 - y + S.Half, y < 1),
+            (0, True))
+    assert piecewise_fold(g1y.rewrite(Piecewise)
+        ).simplify() == Piecewise(
+            (-1, y <= -1),
+            (y**2/2 + y - S.Half, y <= 0),
+            (-y**2/2 + y - S.Half, y < 1),
+            (0, True))
+    assert gy1 == Piecewise(
+        (
+            -Min(1, Max(-1, y))**2/2 - Min(1, Max(-1, y)) +
+            Min(1, Max(0, y))**2 + S.Half, y < 1),
+        (0, True)
+        )
+    assert g1y == Piecewise(
+        (
+            Min(1, Max(-1, y))**2/2 + Min(1, Max(-1, y)) -
+            Min(1, Max(0, y))**2 - S.Half, y < 1),
+        (0, True))
+
+
+@slow
+def test_piecewise_integrate1cb():
+    y = symbols('y', real=True)
+    g = Piecewise(
+        (0, Or(x <= -1, x >= 1)),
+        (1 - x, x > 0),
+        (1 + x, True)
+        )
+    gy1 = g.integrate((x, y, 1))
+    g1y = g.integrate((x, 1, y))
+
+    assert g.integrate((x, -2, 1)) == gy1.subs(y, -2)
+    assert g.integrate((x, 1, -2)) == g1y.subs(y, -2)
+    assert g.integrate((x, 0, 1)) == gy1.subs(y, 0)
+    assert g.integrate((x, 1, 0)) == g1y.subs(y, 0)
+    assert g.integrate((x, 2, 1)) == gy1.subs(y, 2)
+    assert g.integrate((x, 1, 2)) == g1y.subs(y, 2)
+
+    assert piecewise_fold(gy1.rewrite(Piecewise)
+        ).simplify() == Piecewise(
+            (1, y <= -1),
+            (-y**2/2 - y + S.Half, y <= 0),
+            (y**2/2 - y + S.Half, y < 1),
+            (0, True))
+    assert piecewise_fold(g1y.rewrite(Piecewise)
+        ).simplify() == Piecewise(
+            (-1, y <= -1),
+            (y**2/2 + y - S.Half, y <= 0),
+            (-y**2/2 + y - S.Half, y < 1),
+            (0, True))
+
+    # g1y and gy1 should simplify if the condition that y < 1
+    # is applied, e.g. Min(1, Max(-1, y)) --> Max(-1, y)
+    assert gy1 == Piecewise(
+        (
+            -Min(1, Max(-1, y))**2/2 - Min(1, Max(-1, y)) +
+            Min(1, Max(0, y))**2 + S.Half, y < 1),
+        (0, True)
+        )
+    assert g1y == Piecewise(
+        (
+            Min(1, Max(-1, y))**2/2 + Min(1, Max(-1, y)) -
+            Min(1, Max(0, y))**2 - S.Half, y < 1),
+        (0, True))
+
+
+def test_piecewise_integrate2():
+    from itertools import permutations
+    lim = Tuple(x, c, d)
+    p = Piecewise((1, x < a), (2, x > b), (3, True))
+    q = p.integrate(lim)
+    assert q == Piecewise(
+        (-c + 2*d - 2*Min(d, Max(a, c)) + Min(d, Max(a, b, c)), c < d),
+        (-2*c + d + 2*Min(c, Max(a, d)) - Min(c, Max(a, b, d)), True))
+    for v in permutations((1, 2, 3, 4)):
+        r = dict(zip((a, b, c, d), v))
+        assert p.subs(r).integrate(lim.subs(r)) == q.subs(r)
+
+
+def test_meijer_bypass():
+    # totally bypass meijerg machinery when dealing
+    # with Piecewise in integrate
+    assert Piecewise((1, x < 4), (0, True)).integrate((x, oo, 1)) == -3
+
+
+def test_piecewise_integrate3_inequality_conditions():
+    from sympy.utilities.iterables import cartes
+    lim = (x, 0, 5)
+    # set below includes two pts below range, 2 pts in range,
+    # 2 pts above range, and the boundaries
+    N = (-2, -1, 0, 1, 2, 5, 6, 7)
+
+    p = Piecewise((1, x > a), (2, x > b), (0, True))
+    ans = p.integrate(lim)
+    for i, j in cartes(N, repeat=2):
+        reps = dict(zip((a, b), (i, j)))
+        assert ans.subs(reps) == p.subs(reps).integrate(lim)
+    assert ans.subs(a, 4).subs(b, 1) == 0 + 2*3 + 1
+
+    p = Piecewise((1, x > a), (2, x < b), (0, True))
+    ans = p.integrate(lim)
+    for i, j in cartes(N, repeat=2):
+        reps = dict(zip((a, b), (i, j)))
+        assert ans.subs(reps) == p.subs(reps).integrate(lim)
+
+    # delete old tests that involved c1 and c2 since those
+    # reduce to the above except that a value of 0 was used
+    # for two expressions whereas the above uses 3 different
+    # values
+
+
+@slow
+def test_piecewise_integrate4_symbolic_conditions():
+    a = Symbol('a', real=True)
+    b = Symbol('b', real=True)
+    x = Symbol('x', real=True)
+    y = Symbol('y', real=True)
+    p0 = Piecewise((0, Or(x < a, x > b)), (1, True))
+    p1 = Piecewise((0, x < a), (0, x > b), (1, True))
+    p2 = Piecewise((0, x > b), (0, x < a), (1, True))
+    p3 = Piecewise((0, x < a), (1, x < b), (0, True))
+    p4 = Piecewise((0, x > b), (1, x > a), (0, True))
+    p5 = Piecewise((1, And(a < x, x < b)), (0, True))
+
+    # check values of a=1, b=3 (and reversed) with values
+    # of y of 0, 1, 2, 3, 4
+    lim = Tuple(x, -oo, y)
+    for p in (p0, p1, p2, p3, p4, p5):
+        ans = p.integrate(lim)
+        for i in range(5):
+            reps = {a:1, b:3, y:i}
+            assert ans.subs(reps) == p.subs(reps).integrate(lim.subs(reps))
+            reps = {a: 3, b:1, y:i}
+            assert ans.subs(reps) == p.subs(reps).integrate(lim.subs(reps))
+    lim = Tuple(x, y, oo)
+    for p in (p0, p1, p2, p3, p4, p5):
+        ans = p.integrate(lim)
+        for i in range(5):
+            reps = {a:1, b:3, y:i}
+            assert ans.subs(reps) == p.subs(reps).integrate(lim.subs(reps))
+            reps = {a:3, b:1, y:i}
+            assert ans.subs(reps) == p.subs(reps).integrate(lim.subs(reps))
+
+    ans = Piecewise(
+        (0, x <= Min(a, b)),
+        (x - Min(a, b), x <= b),
+        (b - Min(a, b), True))
+    for i in (p0, p1, p2, p4):
+        assert i.integrate(x) == ans
+    assert p3.integrate(x) == Piecewise(
+        (0, x < a),
+        (-a + x, x <= Max(a, b)),
+        (-a + Max(a, b), True))
+    assert p5.integrate(x) == Piecewise(
+        (0, x <= a),
+        (-a + x, x <= Max(a, b)),
+        (-a + Max(a, b), True))
+
+    p1 = Piecewise((0, x < a), (S.Half, x > b), (1, True))
+    p2 = Piecewise((S.Half, x > b), (0, x < a), (1, True))
+    p3 = Piecewise((0, x < a), (1, x < b), (S.Half, True))
+    p4 = Piecewise((S.Half, x > b), (1, x > a), (0, True))
+    p5 = Piecewise((1, And(a < x, x < b)), (S.Half, x > b), (0, True))
+
+    # check values of a=1, b=3 (and reversed) with values
+    # of y of 0, 1, 2, 3, 4
+    lim = Tuple(x, -oo, y)
+    for p in (p1, p2, p3, p4, p5):
+        ans = p.integrate(lim)
+        for i in range(5):
+            reps = {a:1, b:3, y:i}
+            assert ans.subs(reps) == p.subs(reps).integrate(lim.subs(reps))
+            reps = {a: 3, b:1, y:i}
+            assert ans.subs(reps) == p.subs(reps).integrate(lim.subs(reps))
+
+
+def test_piecewise_integrate5_independent_conditions():
+    p = Piecewise((0, Eq(y, 0)), (x*y, True))
+    assert integrate(p, (x, 1, 3)) == Piecewise((0, Eq(y, 0)), (4*y, True))
+
+
+def test_issue_22917():
+    p = (Piecewise((0, ITE((x - y > 1) | (2 * x - 2 * y > 1), False,
+                           ITE(x - y > 1, 2 * y - 2 < -1, 2 * x - 2 * y > 1))),
+                   (Piecewise((0, ITE(x - y > 1, True, 2 * x - 2 * y > 1)),
+                              (2 * Piecewise((0, x - y > 1), (y, True)), True)), True))
+         + 2 * Piecewise((1, ITE((x - y > 1) | (2 * x - 2 * y > 1), False,
+                                 ITE(x - y > 1, 2 * y - 2 < -1, 2 * x - 2 * y > 1))),
+                         (Piecewise((1, ITE(x - y > 1, True, 2 * x - 2 * y > 1)),
+                                    (2 * Piecewise((1, x - y > 1), (x, True)), True)), True)))
+    assert piecewise_fold(p) == Piecewise((2, (x - y > S.Half) | (x - y > 1)),
+                                          (2*y + 4, x - y > 1),
+                                          (4*x + 2*y, True))
+    assert piecewise_fold(p > 1).rewrite(ITE) == ITE((x - y > S.Half) | (x - y > 1), True,
+                                                     ITE(x - y > 1, 2*y + 4 > 1, 4*x + 2*y > 1))
+
+
+def test_piecewise_simplify():
+    p = Piecewise(((x**2 + 1)/x**2, Eq(x*(1 + x) - x**2, 0)),
+                  ((-1)**x*(-1), True))
+    assert p.simplify() == \
+        Piecewise((zoo, Eq(x, 0)), ((-1)**(x + 1), True))
+    # simplify when there are Eq in conditions
+    assert Piecewise(
+        (a, And(Eq(a, 0), Eq(a + b, 0))), (1, True)).simplify(
+        ) == Piecewise(
+        (0, And(Eq(a, 0), Eq(b, 0))), (1, True))
+    assert Piecewise((2*x*factorial(a)/(factorial(y)*factorial(-y + a)),
+        Eq(y, 0) & Eq(-y + a, 0)), (2*factorial(a)/(factorial(y)*factorial(-y
+        + a)), Eq(y, 0) & Eq(-y + a, 1)), (0, True)).simplify(
+        ) == Piecewise(
+            (2*x, And(Eq(a, 0), Eq(y, 0))),
+            (2, And(Eq(a, 1), Eq(y, 0))),
+            (0, True))
+    args = (2, And(Eq(x, 2), Ge(y, 0))), (x, True)
+    assert Piecewise(*args).simplify() == Piecewise(*args)
+    args = (1, Eq(x, 0)), (sin(x)/x, True)
+    assert Piecewise(*args).simplify() == Piecewise(*args)
+    assert Piecewise((2 + y, And(Eq(x, 2), Eq(y, 0))), (x, True)
+        ).simplify() == x
+    # check that x or f(x) are recognized as being Symbol-like for lhs
+    args = Tuple((1, Eq(x, 0)), (sin(x) + 1 + x, True))
+    ans = x + sin(x) + 1
+    f = Function('f')
+    assert Piecewise(*args).simplify() == ans
+    assert Piecewise(*args.subs(x, f(x))).simplify() == ans.subs(x, f(x))
+
+    # issue 18634
+    d = Symbol("d", integer=True)
+    n = Symbol("n", integer=True)
+    t = Symbol("t", positive=True)
+    expr = Piecewise((-d + 2*n, Eq(1/t, 1)), (t**(1 - 4*n)*t**(4*n - 1)*(-d + 2*n), True))
+    assert expr.simplify() == -d + 2*n
+
+    # issue 22747
+    p = Piecewise((0, (t < -2) & (t < -1) & (t < 0)), ((t/2 + 1)*(t +
+        1)*(t + 2), (t < -1) & (t < 0)), ((S.Half - t/2)*(1 - t)*(t + 1),
+        (t < -2) & (t < -1) & (t < 1)), ((t + 1)*(-t*(t/2 + 1) + (S.Half
+        - t/2)*(1 - t)), (t < -2) & (t < -1) & (t < 0) & (t < 1)), ((t +
+        1)*((S.Half - t/2)*(1 - t) + (t/2 + 1)*(t + 2)), (t < -1) & (t <
+        1)), ((t + 1)*(-t*(t/2 + 1) + (S.Half - t/2)*(1 - t)), (t < -1) &
+        (t < 0) & (t < 1)), (0, (t < -2) & (t < -1)), ((t/2 + 1)*(t +
+        1)*(t + 2), t < -1), ((t + 1)*(-t*(t/2 + 1) + (S.Half - t/2)*(t +
+        1)), (t < 0) & ((t < -2) | (t < 0))), ((S.Half - t/2)*(1 - t)*(t
+        + 1), (t < 1) & ((t < -2) | (t < 1))), (0, True)) + Piecewise((0,
+        (t < -1) & (t < 0) & (t < 1)), ((1 - t)*(t/2 + S.Half)*(t + 1),
+        (t < 0) & (t < 1)), ((1 - t)*(1 - t/2)*(2 - t), (t < -1) & (t <
+        0) & (t < 2)), ((1 - t)*((1 - t)*(t/2 + S.Half) + (1 - t/2)*(2 -
+        t)), (t < -1) & (t < 0) & (t < 1) & (t < 2)), ((1 - t)*((1 -
+        t/2)*(2 - t) + (t/2 + S.Half)*(t + 1)), (t < 0) & (t < 2)), ((1 -
+        t)*((1 - t)*(t/2 + S.Half) + (1 - t/2)*(2 - t)), (t < 0) & (t <
+        1) & (t < 2)), (0, (t < -1) & (t < 0)), ((1 - t)*(t/2 +
+        S.Half)*(t + 1), t < 0), ((1 - t)*(t*(1 - t/2) + (1 - t)*(t/2 +
+        S.Half)), (t < 1) & ((t < -1) | (t < 1))), ((1 - t)*(1 - t/2)*(2
+        - t), (t < 2) & ((t < -1) | (t < 2))), (0, True))
+    assert p.simplify() == Piecewise(
+        (0, t < -2), ((t + 1)*(t + 2)**2/2, t < -1), (-3*t**3/2
+        - 5*t**2/2 + 1, t < 0), (3*t**3/2 - 5*t**2/2 + 1, t < 1), ((1 -
+        t)*(t - 2)**2/2, t < 2), (0, True))
+
+    # coverage
+    nan = Undefined
+    covered = Piecewise((1, x > 3), (2, x < 2), (3, x > 1))
+    assert covered.simplify().args  == covered.args
+    assert Piecewise((1, x < 2), (2, x < 1), (3, True)).simplify(
+        ) == Piecewise((1, x < 2), (3, True))
+    assert Piecewise((1, x > 2)).simplify() == Piecewise((1, x > 2),
+        (nan, True))
+    assert Piecewise((1, (x >= 2) & (x < oo))
+        ).simplify() == Piecewise((1, (x >= 2) & (x < oo)), (nan, True))
+    assert Piecewise((1, x < 2), (2, (x > 1) & (x < 3)), (3, True)
+        ). simplify() == Piecewise((1, x < 2), (2, x < 3), (3, True))
+    assert Piecewise((1, x < 2), (2, (x <= 3) & (x > 1)), (3, True)
+        ).simplify() == Piecewise((1, x < 2), (2, x <= 3), (3, True))
+    assert Piecewise((1, x < 2), (2, (x > 2) & (x < 3)), (3, True)
+        ).simplify() == Piecewise((1, x < 2), (2, (x > 2) & (x < 3)),
+        (3, True))
+    assert Piecewise((1, x < 2), (2, (x >= 1) & (x <= 3)), (3, True)
+        ).simplify() == Piecewise((1, x < 2), (2, x <= 3), (3, True))
+    assert Piecewise((1, x < 1), (2, (x >= 2) & (x <= 3)), (3, True)
+        ).simplify() == Piecewise((1, x < 1), (2, (x >= 2) & (x <= 3)),
+        (3, True))
+
+
+def test_piecewise_solve():
+    abs2 = Piecewise((-x, x <= 0), (x, x > 0))
+    f = abs2.subs(x, x - 2)
+    assert solve(f, x) == [2]
+    assert solve(f - 1, x) == [1, 3]
+
+    f = Piecewise(((x - 2)**2, x >= 0), (1, True))
+    assert solve(f, x) == [2]
+
+    g = Piecewise(((x - 5)**5, x >= 4), (f, True))
+    assert solve(g, x) == [2, 5]
+
+    g = Piecewise(((x - 5)**5, x >= 4), (f, x < 4))
+    assert solve(g, x) == [2, 5]
+
+    g = Piecewise(((x - 5)**5, x >= 2), (f, x < 2))
+    assert solve(g, x) == [5]
+
+    g = Piecewise(((x - 5)**5, x >= 2), (f, True))
+    assert solve(g, x) == [5]
+
+    g = Piecewise(((x - 5)**5, x >= 2), (f, True), (10, False))
+    assert solve(g, x) == [5]
+
+    g = Piecewise(((x - 5)**5, x >= 2),
+                  (-x + 2, x - 2 <= 0), (x - 2, x - 2 > 0))
+    assert solve(g, x) == [5]
+
+    # if no symbol is given the piecewise detection must still work
+    assert solve(Piecewise((x - 2, x > 2), (2 - x, True)) - 3) == [-1, 5]
+
+    f = Piecewise(((x - 2)**2, x >= 0), (0, True))
+    raises(NotImplementedError, lambda: solve(f, x))
+
+    def nona(ans):
+        return list(filter(lambda x: x is not S.NaN, ans))
+    p = Piecewise((x**2 - 4, x < y), (x - 2, True))
+    ans = solve(p, x)
+    assert nona([i.subs(y, -2) for i in ans]) == [2]
+    assert nona([i.subs(y, 2) for i in ans]) == [-2, 2]
+    assert nona([i.subs(y, 3) for i in ans]) == [-2, 2]
+    assert ans == [
+        Piecewise((-2, y > -2), (S.NaN, True)),
+        Piecewise((2, y <= 2), (S.NaN, True)),
+        Piecewise((2, y > 2), (S.NaN, True))]
+
+    # issue 6060
+    absxm3 = Piecewise(
+        (x - 3, 0 <= x - 3),
+        (3 - x, 0 > x - 3)
+    )
+    assert solve(absxm3 - y, x) == [
+        Piecewise((-y + 3, -y < 0), (S.NaN, True)),
+        Piecewise((y + 3, y >= 0), (S.NaN, True))]
+    p = Symbol('p', positive=True)
+    assert solve(absxm3 - p, x) == [-p + 3, p + 3]
+
+    # issue 6989
+    f = Function('f')
+    assert solve(Eq(-f(x), Piecewise((1, x > 0), (0, True))), f(x)) == \
+        [Piecewise((-1, x > 0), (0, True))]
+
+    # issue 8587
+    f = Piecewise((2*x**2, And(0 < x, x < 1)), (2, True))
+    assert solve(f - 1) == [1/sqrt(2)]
+
+
+def test_piecewise_fold():
+    p = Piecewise((x, x < 1), (1, 1 <= x))
+
+    assert piecewise_fold(x*p) == Piecewise((x**2, x < 1), (x, 1 <= x))
+    assert piecewise_fold(p + p) == Piecewise((2*x, x < 1), (2, 1 <= x))
+    assert piecewise_fold(Piecewise((1, x < 0), (2, True))
+                          + Piecewise((10, x < 0), (-10, True))) == \
+        Piecewise((11, x < 0), (-8, True))
+
+    p1 = Piecewise((0, x < 0), (x, x <= 1), (0, True))
+    p2 = Piecewise((0, x < 0), (1 - x, x <= 1), (0, True))
+
+    p = 4*p1 + 2*p2
+    assert integrate(
+        piecewise_fold(p), (x, -oo, oo)) == integrate(2*x + 2, (x, 0, 1))
+
+    assert piecewise_fold(
+        Piecewise((1, y <= 0), (-Piecewise((2, y >= 0)), True)
+        )) == Piecewise((1, y <= 0), (-2, y >= 0))
+
+    assert piecewise_fold(Piecewise((x, ITE(x > 0, y < 1, y > 1)))
+        ) == Piecewise((x, ((x <= 0) | (y < 1)) & ((x > 0) | (y > 1))))
+
+    a, b = (Piecewise((2, Eq(x, 0)), (0, True)),
+        Piecewise((x, Eq(-x + y, 0)), (1, Eq(-x + y, 1)), (0, True)))
+    assert piecewise_fold(Mul(a, b, evaluate=False)
+        ) == piecewise_fold(Mul(b, a, evaluate=False))
+
+
+def test_piecewise_fold_piecewise_in_cond():
+    p1 = Piecewise((cos(x), x < 0), (0, True))
+    p2 = Piecewise((0, Eq(p1, 0)), (p1 / Abs(p1), True))
+    assert p2.subs(x, -pi/2) == 0
+    assert p2.subs(x, 1) == 0
+    assert p2.subs(x, -pi/4) == 1
+    p4 = Piecewise((0, Eq(p1, 0)), (1,True))
+    ans = piecewise_fold(p4)
+    for i in range(-1, 1):
+        assert ans.subs(x, i) == p4.subs(x, i)
+
+    r1 = 1 < Piecewise((1, x < 1), (3, True))
+    ans = piecewise_fold(r1)
+    for i in range(2):
+        assert ans.subs(x, i) == r1.subs(x, i)
+
+    p5 = Piecewise((1, x < 0), (3, True))
+    p6 = Piecewise((1, x < 1), (3, True))
+    p7 = Piecewise((1, p5 < p6), (0, True))
+    ans = piecewise_fold(p7)
+    for i in range(-1, 2):
+        assert ans.subs(x, i) == p7.subs(x, i)
+
+
+def test_piecewise_fold_piecewise_in_cond_2():
+    p1 = Piecewise((cos(x), x < 0), (0, True))
+    p2 = Piecewise((0, Eq(p1, 0)), (1 / p1, True))
+    p3 = Piecewise(
+        (0, (x >= 0) | Eq(cos(x), 0)),
+        (1/cos(x), x < 0),
+        (zoo, True))  # redundant b/c all x are already covered
+    assert(piecewise_fold(p2) == p3)
+
+
+def test_piecewise_fold_expand():
+    p1 = Piecewise((1, Interval(0, 1, False, True).contains(x)), (0, True))
+
+    p2 = piecewise_fold(expand((1 - x)*p1))
+    cond = ((x >= 0) & (x < 1))
+    assert piecewise_fold(expand((1 - x)*p1), evaluate=False
+        ) == Piecewise((1 - x, cond), (-x, cond), (1, cond), (0, True), evaluate=False)
+    assert piecewise_fold(expand((1 - x)*p1), evaluate=None
+        ) == Piecewise((1 - x, cond), (0, True))
+    assert p2 == Piecewise((1 - x, cond), (0, True))
+    assert p2 == expand(piecewise_fold((1 - x)*p1))
+
+
+def test_piecewise_duplicate():
+    p = Piecewise((x, x < -10), (x**2, x <= -1), (x, 1 < x))
+    assert p == Piecewise(*p.args)
+
+
+def test_doit():
+    p1 = Piecewise((x, x < 1), (x**2, -1 <= x), (x, 3 < x))
+    p2 = Piecewise((x, x < 1), (Integral(2 * x), -1 <= x), (x, 3 < x))
+    assert p2.doit() == p1
+    assert p2.doit(deep=False) == p2
+    # issue 17165
+    p1 = Sum(y**x, (x, -1, oo)).doit()
+    assert p1.doit() == p1
+
+
+def test_piecewise_interval():
+    p1 = Piecewise((x, Interval(0, 1).contains(x)), (0, True))
+    assert p1.subs(x, -0.5) == 0
+    assert p1.subs(x, 0.5) == 0.5
+    assert p1.diff(x) == Piecewise((1, Interval(0, 1).contains(x)), (0, True))
+    assert integrate(p1, x) == Piecewise(
+        (0, x <= 0),
+        (x**2/2, x <= 1),
+        (S.Half, True))
+
+
+def test_piecewise_exclusive():
+    p = Piecewise((0, x < 0), (S.Half, x <= 0), (1, True))
+    assert piecewise_exclusive(p) == Piecewise((0, x < 0), (S.Half, Eq(x, 0)),
+                                               (1, x > 0), evaluate=False)
+    assert piecewise_exclusive(p + 2) == Piecewise((0, x < 0), (S.Half, Eq(x, 0)),
+                                               (1, x > 0), evaluate=False) + 2
+    assert piecewise_exclusive(Piecewise((1, y <= 0),
+                                         (-Piecewise((2, y >= 0)), True))) == \
+        Piecewise((1, y <= 0),
+                  (-Piecewise((2, y >= 0),
+                              (S.NaN, y < 0), evaluate=False), y > 0), evaluate=False)
+    assert piecewise_exclusive(Piecewise((1, x > y))) == Piecewise((1, x > y),
+                                                                  (S.NaN, x <= y),
+                                                                  evaluate=False)
+    assert piecewise_exclusive(Piecewise((1, x > y)),
+                               skip_nan=True) == Piecewise((1, x > y))
+
+    xr, yr = symbols('xr, yr', real=True)
+
+    p1 = Piecewise((1, xr < 0), (2, True), evaluate=False)
+    p1x = Piecewise((1, xr < 0), (2, xr >= 0), evaluate=False)
+
+    p2 = Piecewise((p1, yr < 0), (3, True), evaluate=False)
+    p2x = Piecewise((p1, yr < 0), (3, yr >= 0), evaluate=False)
+    p2xx = Piecewise((p1x, yr < 0), (3, yr >= 0), evaluate=False)
+
+    assert piecewise_exclusive(p2) == p2xx
+    assert piecewise_exclusive(p2, deep=False) == p2x
+
+
+def test_piecewise_collapse():
+    assert Piecewise((x, True)) == x
+    a = x < 1
+    assert Piecewise((x, a), (x + 1, a)) == Piecewise((x, a))
+    assert Piecewise((x, a), (x + 1, a.reversed)) == Piecewise((x, a))
+    b = x < 5
+    def canonical(i):
+        if isinstance(i, Piecewise):
+            return Piecewise(*i.args)
+        return i
+    for args in [
+        ((1, a), (Piecewise((2, a), (3, b)), b)),
+        ((1, a), (Piecewise((2, a), (3, b.reversed)), b)),
+        ((1, a), (Piecewise((2, a), (3, b)), b), (4, True)),
+        ((1, a), (Piecewise((2, a), (3, b), (4, True)), b)),
+        ((1, a), (Piecewise((2, a), (3, b), (4, True)), b), (5, True))]:
+        for i in (0, 2, 10):
+            assert canonical(
+                Piecewise(*args, evaluate=False).subs(x, i)
+                ) == canonical(Piecewise(*args).subs(x, i))
+    r1, r2, r3, r4 = symbols('r1:5')
+    a = x < r1
+    b = x < r2
+    c = x < r3
+    d = x < r4
+    assert Piecewise((1, a), (Piecewise(
+        (2, a), (3, b), (4, c)), b), (5, c)
+        ) == Piecewise((1, a), (3, b), (5, c))
+    assert Piecewise((1, a), (Piecewise(
+        (2, a), (3, b), (4, c), (6, True)), c), (5, d)
+        ) == Piecewise((1, a), (Piecewise(
+        (3, b), (4, c)), c), (5, d))
+    assert Piecewise((1, Or(a, d)), (Piecewise(
+        (2, d), (3, b), (4, c)), b), (5, c)
+        ) == Piecewise((1, Or(a, d)), (Piecewise(
+        (2, d), (3, b)), b), (5, c))
+    assert Piecewise((1, c), (2, ~c), (3, S.true)
+        ) == Piecewise((1, c), (2, S.true))
+    assert Piecewise((1, c), (2, And(~c, b)), (3,True)
+        ) == Piecewise((1, c), (2, b), (3, True))
+    assert Piecewise((1, c), (2, Or(~c, b)), (3,True)
+        ).subs(dict(zip((r1, r2, r3, r4, x), (1, 2, 3, 4, 3.5))))  == 2
+    assert Piecewise((1, c), (2, ~c)) == Piecewise((1, c), (2, True))
+
+
+def test_piecewise_lambdify():
+    p = Piecewise(
+        (x**2, x < 0),
+        (x, Interval(0, 1, False, True).contains(x)),
+        (2 - x, x >= 1),
+        (0, True)
+    )
+
+    f = lambdify(x, p)
+    assert f(-2.0) == 4.0
+    assert f(0.0) == 0.0
+    assert f(0.5) == 0.5
+    assert f(2.0) == 0.0
+
+
+def test_piecewise_series():
+    from sympy.series.order import O
+    p1 = Piecewise((sin(x), x < 0), (cos(x), x > 0))
+    p2 = Piecewise((x + O(x**2), x < 0), (1 + O(x**2), x > 0))
+    assert p1.nseries(x, n=2) == p2
+
+
+def test_piecewise_as_leading_term():
+    p1 = Piecewise((1/x, x > 1), (0, True))
+    p2 = Piecewise((x, x > 1), (0, True))
+    p3 = Piecewise((1/x, x > 1), (x, True))
+    p4 = Piecewise((x, x > 1), (1/x, True))
+    p5 = Piecewise((1/x, x > 1), (x, True))
+    p6 = Piecewise((1/x, x < 1), (x, True))
+    p7 = Piecewise((x, x < 1), (1/x, True))
+    p8 = Piecewise((x, x > 1), (1/x, True))
+    assert p1.as_leading_term(x) == 0
+    assert p2.as_leading_term(x) == 0
+    assert p3.as_leading_term(x) == x
+    assert p4.as_leading_term(x) == 1/x
+    assert p5.as_leading_term(x) == x
+    assert p6.as_leading_term(x) == 1/x
+    assert p7.as_leading_term(x) == x
+    assert p8.as_leading_term(x) == 1/x
+
+
+def test_piecewise_complex():
+    p1 = Piecewise((2, x < 0), (1, 0 <= x))
+    p2 = Piecewise((2*I, x < 0), (I, 0 <= x))
+    p3 = Piecewise((I*x, x > 1), (1 + I, True))
+    p4 = Piecewise((-I*conjugate(x), x > 1), (1 - I, True))
+
+    assert conjugate(p1) == p1
+    assert conjugate(p2) == piecewise_fold(-p2)
+    assert conjugate(p3) == p4
+
+    assert p1.is_imaginary is False
+    assert p1.is_real is True
+    assert p2.is_imaginary is True
+    assert p2.is_real is False
+    assert p3.is_imaginary is None
+    assert p3.is_real is None
+
+    assert p1.as_real_imag() == (p1, 0)
+    assert p2.as_real_imag() == (0, -I*p2)
+
+
+def test_conjugate_transpose():
+    A, B = symbols("A B", commutative=False)
+    p = Piecewise((A*B**2, x > 0), (A**2*B, True))
+    assert p.adjoint() == \
+        Piecewise((adjoint(A*B**2), x > 0), (adjoint(A**2*B), True))
+    assert p.conjugate() == \
+        Piecewise((conjugate(A*B**2), x > 0), (conjugate(A**2*B), True))
+    assert p.transpose() == \
+        Piecewise((transpose(A*B**2), x > 0), (transpose(A**2*B), True))
+
+
+def test_piecewise_evaluate():
+    assert Piecewise((x, True)) == x
+    assert Piecewise((x, True), evaluate=True) == x
+    assert Piecewise((1, Eq(1, x))).args == ((1, Eq(x, 1)),)
+    assert Piecewise((1, Eq(1, x)), evaluate=False).args == (
+        (1, Eq(1, x)),)
+    # like the additive and multiplicative identities that
+    # cannot be kept in Add/Mul, we also do not keep a single True
+    p = Piecewise((x, True), evaluate=False)
+    assert p == x
+
+
+def test_as_expr_set_pairs():
+    assert Piecewise((x, x > 0), (-x, x <= 0)).as_expr_set_pairs() == \
+        [(x, Interval(0, oo, True, True)), (-x, Interval(-oo, 0))]
+
+    assert Piecewise(((x - 2)**2, x >= 0), (0, True)).as_expr_set_pairs() == \
+        [((x - 2)**2, Interval(0, oo)), (0, Interval(-oo, 0, True, True))]
+
+
+def test_S_srepr_is_identity():
+    p = Piecewise((10, Eq(x, 0)), (12, True))
+    q = S(srepr(p))
+    assert p == q
+
+
+def test_issue_12587():
+    # sort holes into intervals
+    p = Piecewise((1, x > 4), (2, Not((x <= 3) & (x > -1))), (3, True))
+    assert p.integrate((x, -5, 5)) == 23
+    p = Piecewise((1, x > 1), (2, x < y), (3, True))
+    lim = x, -3, 3
+    ans = p.integrate(lim)
+    for i in range(-1, 3):
+        assert ans.subs(y, i) == p.subs(y, i).integrate(lim)
+
+
+def test_issue_11045():
+    assert integrate(1/(x*sqrt(x**2 - 1)), (x, 1, 2)) == pi/3
+
+    # handle And with Or arguments
+    assert Piecewise((1, And(Or(x < 1, x > 3), x < 2)), (0, True)
+        ).integrate((x, 0, 3)) == 1
+
+    # hidden false
+    assert Piecewise((1, x > 1), (2, x > x + 1), (3, True)
+        ).integrate((x, 0, 3)) == 5
+    # targetcond is Eq
+    assert Piecewise((1, x > 1), (2, Eq(1, x)), (3, True)
+        ).integrate((x, 0, 4)) == 6
+    # And has Relational needing to be solved
+    assert Piecewise((1, And(2*x > x + 1, x < 2)), (0, True)
+        ).integrate((x, 0, 3)) == 1
+    # Or has Relational needing to be solved
+    assert Piecewise((1, Or(2*x > x + 2, x < 1)), (0, True)
+        ).integrate((x, 0, 3)) == 2
+    # ignore hidden false (handled in canonicalization)
+    assert Piecewise((1, x > 1), (2, x > x + 1), (3, True)
+        ).integrate((x, 0, 3)) == 5
+    # watch for hidden True Piecewise
+    assert Piecewise((2, Eq(1 - x, x*(1/x - 1))), (0, True)
+        ).integrate((x, 0, 3)) == 6
+
+    # overlapping conditions of targetcond are recognized and ignored;
+    # the condition x > 3 will be pre-empted by the first condition
+    assert Piecewise((1, Or(x < 1, x > 2)), (2, x > 3), (3, True)
+        ).integrate((x, 0, 4)) == 6
+
+    # convert Ne to Or
+    assert Piecewise((1, Ne(x, 0)), (2, True)
+        ).integrate((x, -1, 1)) == 2
+
+    # no default but well defined
+    assert Piecewise((x, (x > 1) & (x < 3)), (1, (x < 4))
+        ).integrate((x, 1, 4)) == 5
+
+    p = Piecewise((x, (x > 1) & (x < 3)), (1, (x < 4)))
+    nan = Undefined
+    i = p.integrate((x, 1, y))
+    assert i == Piecewise(
+        (y - 1, y < 1),
+        (Min(3, y)**2/2 - Min(3, y) + Min(4, y) - S.Half,
+            y <= Min(4, y)),
+        (nan, True))
+    assert p.integrate((x, 1, -1)) == i.subs(y, -1)
+    assert p.integrate((x, 1, 4)) == 5
+    assert p.integrate((x, 1, 5)) is nan
+
+    # handle Not
+    p = Piecewise((1, x > 1), (2, Not(And(x > 1, x< 3))), (3, True))
+    assert p.integrate((x, 0, 3)) == 4
+
+    # handle updating of int_expr when there is overlap
+    p = Piecewise(
+        (1, And(5 > x, x > 1)),
+        (2, Or(x < 3, x > 7)),
+        (4, x < 8))
+    assert p.integrate((x, 0, 10)) == 20
+
+    # And with Eq arg handling
+    assert Piecewise((1, x < 1), (2, And(Eq(x, 3), x > 1))
+        ).integrate((x, 0, 3)) is S.NaN
+    assert Piecewise((1, x < 1), (2, And(Eq(x, 3), x > 1)), (3, True)
+        ).integrate((x, 0, 3)) == 7
+    assert Piecewise((1, x < 0), (2, And(Eq(x, 3), x < 1)), (3, True)
+        ).integrate((x, -1, 1)) == 4
+    # middle condition doesn't matter: it's a zero width interval
+    assert Piecewise((1, x < 1), (2, Eq(x, 3) & (y < x)), (3, True)
+        ).integrate((x, 0, 3)) == 7
+
+
+def test_holes():
+    nan = Undefined
+    assert Piecewise((1, x < 2)).integrate(x) == Piecewise(
+        (x, x < 2), (nan, True))
+    assert Piecewise((1, And(x > 1, x < 2))).integrate(x) == Piecewise(
+        (nan, x < 1), (x, x < 2), (nan, True))
+    assert Piecewise((1, And(x > 1, x < 2))).integrate((x, 0, 3)) is nan
+    assert Piecewise((1, And(x > 0, x < 4))).integrate((x, 1, 3)) == 2
+
+    # this also tests that the integrate method is used on non-Piecwise
+    # arguments in _eval_integral
+    A, B = symbols("A B")
+    a, b = symbols('a b', real=True)
+    assert Piecewise((A, And(x < 0, a < 1)), (B, Or(x < 1, a > 2))
+        ).integrate(x) == Piecewise(
+        (B*x, (a > 2)),
+        (Piecewise((A*x, x < 0), (B*x, x < 1), (nan, True)), a < 1),
+        (Piecewise((B*x, x < 1), (nan, True)), True))
+
+
+def test_issue_11922():
+    def f(x):
+        return Piecewise((0, x < -1), (1 - x**2, x < 1), (0, True))
+    autocorr = lambda k: (
+        f(x) * f(x + k)).integrate((x, -1, 1))
+    assert autocorr(1.9) > 0
+    k = symbols('k')
+    good_autocorr = lambda k: (
+        (1 - x**2) * f(x + k)).integrate((x, -1, 1))
+    a = good_autocorr(k)
+    assert a.subs(k, 3) == 0
+    k = symbols('k', positive=True)
+    a = good_autocorr(k)
+    assert a.subs(k, 3) == 0
+    assert Piecewise((0, x < 1), (10, (x >= 1))
+        ).integrate() == Piecewise((0, x < 1), (10*x - 10, True))
+
+
+def test_issue_5227():
+    f = 0.0032513612725229*Piecewise((0, x < -80.8461538461539),
+        (-0.0160799238820171*x + 1.33215984776403, x < 2),
+        (Piecewise((0.3, x > 123), (0.7, True)) +
+        Piecewise((0.4, x > 2), (0.6, True)), x <=
+        123), (-0.00817409766454352*x + 2.10541401273885, x <
+        380.571428571429), (0, True))
+    i = integrate(f, (x, -oo, oo))
+    assert i == Integral(f, (x, -oo, oo)).doit()
+    assert str(i) == '1.00195081676351'
+    assert Piecewise((1, x - y < 0), (0, True)
+        ).integrate(y) == Piecewise((0, y <= x), (-x + y, True))
+
+
+def test_issue_10137():
+    a = Symbol('a', real=True)
+    b = Symbol('b', real=True)
+    x = Symbol('x', real=True)
+    y = Symbol('y', real=True)
+    p0 = Piecewise((0, Or(x < a, x > b)), (1, True))
+    p1 = Piecewise((0, Or(a > x, b < x)), (1, True))
+    assert integrate(p0, (x, y, oo)) == integrate(p1, (x, y, oo))
+    p3 = Piecewise((1, And(0 < x, x < a)), (0, True))
+    p4 = Piecewise((1, And(a > x, x > 0)), (0, True))
+    ip3 = integrate(p3, x)
+    assert ip3 == Piecewise(
+        (0, x <= 0),
+        (x, x <= Max(0, a)),
+        (Max(0, a), True))
+    ip4 = integrate(p4, x)
+    assert ip4 == ip3
+    assert p3.integrate((x, 2, 4)) == Min(4, Max(2, a)) - 2
+    assert p4.integrate((x, 2, 4)) == Min(4, Max(2, a)) - 2
+
+
+def test_stackoverflow_43852159():
+    f = lambda x: Piecewise((1, (x >= -1) & (x <= 1)), (0, True))
+    Conv = lambda x: integrate(f(x - y)*f(y), (y, -oo, +oo))
+    cx = Conv(x)
+    assert cx.subs(x, -1.5) == cx.subs(x, 1.5)
+    assert cx.subs(x, 3) == 0
+    assert piecewise_fold(f(x - y)*f(y)) == Piecewise(
+        (1, (y >= -1) & (y <= 1) & (x - y >= -1) & (x - y <= 1)),
+        (0, True))
+
+
+def test_issue_12557():
+    '''
+    # 3200 seconds to compute the fourier part of issue
+    import sympy as sym
+    x,y,z,t = sym.symbols('x y z t')
+    k = sym.symbols("k", integer=True)
+    fourier = sym.fourier_series(sym.cos(k*x)*sym.sqrt(x**2),
+                                 (x, -sym.pi, sym.pi))
+    assert fourier == FourierSeries(
+    sqrt(x**2)*cos(k*x), (x, -pi, pi), (Piecewise((pi**2,
+    Eq(k, 0)), (2*(-1)**k/k**2 - 2/k**2, True))/(2*pi),
+    SeqFormula(Piecewise((pi**2, (Eq(_n, 0) & Eq(k, 0)) | (Eq(_n, 0) &
+    Eq(_n, k) & Eq(k, 0)) | (Eq(_n, 0) & Eq(k, 0) & Eq(_n, -k)) | (Eq(_n,
+    0) & Eq(_n, k) & Eq(k, 0) & Eq(_n, -k))), (pi**2/2, Eq(_n, k) | Eq(_n,
+    -k) | (Eq(_n, 0) & Eq(_n, k)) | (Eq(_n, k) & Eq(k, 0)) | (Eq(_n, 0) &
+    Eq(_n, -k)) | (Eq(_n, k) & Eq(_n, -k)) | (Eq(k, 0) & Eq(_n, -k)) |
+    (Eq(_n, 0) & Eq(_n, k) & Eq(_n, -k)) | (Eq(_n, k) & Eq(k, 0) & Eq(_n,
+    -k))), ((-1)**k*pi**2*_n**3*sin(pi*_n)/(pi*_n**4 - 2*pi*_n**2*k**2 +
+    pi*k**4) - (-1)**k*pi**2*_n**3*sin(pi*_n)/(-pi*_n**4 + 2*pi*_n**2*k**2
+    - pi*k**4) + (-1)**k*pi*_n**2*cos(pi*_n)/(pi*_n**4 - 2*pi*_n**2*k**2 +
+    pi*k**4) - (-1)**k*pi*_n**2*cos(pi*_n)/(-pi*_n**4 + 2*pi*_n**2*k**2 -
+    pi*k**4) - (-1)**k*pi**2*_n*k**2*sin(pi*_n)/(pi*_n**4 -
+    2*pi*_n**2*k**2 + pi*k**4) +
+    (-1)**k*pi**2*_n*k**2*sin(pi*_n)/(-pi*_n**4 + 2*pi*_n**2*k**2 -
+    pi*k**4) + (-1)**k*pi*k**2*cos(pi*_n)/(pi*_n**4 - 2*pi*_n**2*k**2 +
+    pi*k**4) - (-1)**k*pi*k**2*cos(pi*_n)/(-pi*_n**4 + 2*pi*_n**2*k**2 -
+    pi*k**4) - (2*_n**2 + 2*k**2)/(_n**4 - 2*_n**2*k**2 + k**4),
+    True))*cos(_n*x)/pi, (_n, 1, oo)), SeqFormula(0, (_k, 1, oo))))
+    '''
+    x = symbols("x", real=True)
+    k = symbols('k', integer=True, finite=True)
+    abs2 = lambda x: Piecewise((-x, x <= 0), (x, x > 0))
+    assert integrate(abs2(x), (x, -pi, pi)) == pi**2
+    func = cos(k*x)*sqrt(x**2)
+    assert integrate(func, (x, -pi, pi)) == Piecewise(
+        (2*(-1)**k/k**2 - 2/k**2, Ne(k, 0)), (pi**2, True))
+
+def test_issue_6900():
+    from itertools import permutations
+    t0, t1, T, t = symbols('t0, t1 T t')
+    f = Piecewise((0, t < t0), (x, And(t0 <= t, t < t1)), (0, t >= t1))
+    g = f.integrate(t)
+    assert g == Piecewise(
+        (0, t <= t0),
+        (t*x - t0*x, t <= Max(t0, t1)),
+        (-t0*x + x*Max(t0, t1), True))
+    for i in permutations(range(2)):
+        reps = dict(zip((t0,t1), i))
+        for tt in range(-1,3):
+            assert (g.xreplace(reps).subs(t,tt) ==
+                f.xreplace(reps).integrate(t).subs(t,tt))
+    lim = Tuple(t, t0, T)
+    g = f.integrate(lim)
+    ans = Piecewise(
+        (-t0*x + x*Min(T, Max(t0, t1)), T > t0),
+        (0, True))
+    for i in permutations(range(3)):
+        reps = dict(zip((t0,t1,T), i))
+        tru = f.xreplace(reps).integrate(lim.xreplace(reps))
+        assert tru == ans.xreplace(reps)
+    assert g == ans
+
+
+def test_issue_10122():
+    assert solve(abs(x) + abs(x - 1) - 1 > 0, x
+        ) == Or(And(-oo < x, x < S.Zero), And(S.One < x, x < oo))
+
+
+def test_issue_4313():
+    u = Piecewise((0, x <= 0), (1, x >= a), (x/a, True))
+    e = (u - u.subs(x, y))**2/(x - y)**2
+    M = Max(0, a)
+    assert integrate(e,  x).expand() == Piecewise(
+        (Piecewise(
+            (0, x <= 0),
+            (-y**2/(a**2*x - a**2*y) + x/a**2 - 2*y*log(-y)/a**2 +
+                2*y*log(x - y)/a**2 - y/a**2, x <= M),
+            (-y**2/(-a**2*y + a**2*M) + 1/(-y + M) -
+                1/(x - y) - 2*y*log(-y)/a**2 + 2*y*log(-y +
+                M)/a**2 - y/a**2 + M/a**2, True)),
+        ((a <= y) & (y <= 0)) | ((y <= 0) & (y > -oo))),
+        (Piecewise(
+            (-1/(x - y), x <= 0),
+            (-a**2/(a**2*x - a**2*y) + 2*a*y/(a**2*x - a**2*y) -
+                y**2/(a**2*x - a**2*y) + 2*log(-y)/a - 2*log(x - y)/a +
+                2/a + x/a**2 - 2*y*log(-y)/a**2 + 2*y*log(x - y)/a**2 -
+                y/a**2, x <= M),
+            (-a**2/(-a**2*y + a**2*M) + 2*a*y/(-a**2*y +
+                a**2*M) - y**2/(-a**2*y + a**2*M) +
+                2*log(-y)/a - 2*log(-y + M)/a + 2/a -
+                2*y*log(-y)/a**2 + 2*y*log(-y + M)/a**2 -
+                y/a**2 + M/a**2, True)),
+        a <= y),
+        (Piecewise(
+            (-y**2/(a**2*x - a**2*y), x <= 0),
+            (x/a**2 + y/a**2, x <= M),
+            (a**2/(-a**2*y + a**2*M) -
+                a**2/(a**2*x - a**2*y) - 2*a*y/(-a**2*y + a**2*M) +
+                2*a*y/(a**2*x - a**2*y) + y**2/(-a**2*y + a**2*M) -
+                y**2/(a**2*x - a**2*y) + y/a**2 + M/a**2, True)),
+        True))
+
+
+def test__intervals():
+    assert Piecewise((x + 2, Eq(x, 3)))._intervals(x) == (True, [])
+    assert Piecewise(
+        (1, x > x + 1),
+        (Piecewise((1, x < x + 1)), 2*x < 2*x + 1),
+        (1, True))._intervals(x) == (True, [(-oo, oo, 1, 1)])
+    assert Piecewise((1, Ne(x, I)), (0, True))._intervals(x) == (True,
+        [(-oo, oo, 1, 0)])
+    assert Piecewise((-cos(x), sin(x) >= 0), (cos(x), True)
+        )._intervals(x) == (True,
+        [(0, pi, -cos(x), 0), (-oo, oo, cos(x), 1)])
+    # the following tests that duplicates are removed and that non-Eq
+    # generated zero-width intervals are removed
+    assert Piecewise((1, Abs(x**(-2)) > 1), (0, True)
+        )._intervals(x) == (True,
+        [(-1, 0, 1, 0), (0, 1, 1, 0), (-oo, oo, 0, 1)])
+
+
+def test_containment():
+    a, b, c, d, e = [1, 2, 3, 4, 5]
+    p = (Piecewise((d, x > 1), (e, True))*
+        Piecewise((a, Abs(x - 1) < 1), (b, Abs(x - 2) < 2), (c, True)))
+    assert p.integrate(x).diff(x) == Piecewise(
+        (c*e, x <= 0),
+        (a*e, x <= 1),
+        (a*d, x < 2),  # this is what we want to get right
+        (b*d, x < 4),
+        (c*d, True))
+
+
+def test_piecewise_with_DiracDelta():
+    d1 = DiracDelta(x - 1)
+    assert integrate(d1, (x, -oo, oo)) == 1
+    assert integrate(d1, (x, 0, 2)) == 1
+    assert Piecewise((d1, Eq(x, 2)), (0, True)).integrate(x) == 0
+    assert Piecewise((d1, x < 2), (0, True)).integrate(x) == Piecewise(
+        (Heaviside(x - 1), x < 2), (1, True))
+    # TODO raise error if function is discontinuous at limit of
+    # integration, e.g. integrate(d1, (x, -2, 1)) or Piecewise(
+    # (d1, Eq(x, 1)
+
+
+def test_issue_10258():
+    assert Piecewise((0, x < 1), (1, True)).is_zero is None
+    assert Piecewise((-1, x < 1), (1, True)).is_zero is False
+    a = Symbol('a', zero=True)
+    assert Piecewise((0, x < 1), (a, True)).is_zero
+    assert Piecewise((1, x < 1), (a, x < 3)).is_zero is None
+    a = Symbol('a')
+    assert Piecewise((0, x < 1), (a, True)).is_zero is None
+    assert Piecewise((0, x < 1), (1, True)).is_nonzero is None
+    assert Piecewise((1, x < 1), (2, True)).is_nonzero
+    assert Piecewise((0, x < 1), (oo, True)).is_finite is None
+    assert Piecewise((0, x < 1), (1, True)).is_finite
+    b = Basic()
+    assert Piecewise((b, x < 1)).is_finite is None
+
+    # 10258
+    c = Piecewise((1, x < 0), (2, True)) < 3
+    assert c != True
+    assert piecewise_fold(c) == True
+
+
+def test_issue_10087():
+    a, b = Piecewise((x, x > 1), (2, True)), Piecewise((x, x > 3), (3, True))
+    m = a*b
+    f = piecewise_fold(m)
+    for i in (0, 2, 4):
+        assert m.subs(x, i) == f.subs(x, i)
+    m = a + b
+    f = piecewise_fold(m)
+    for i in (0, 2, 4):
+        assert m.subs(x, i) == f.subs(x, i)
+
+
+def test_issue_8919():
+    c = symbols('c:5')
+    x = symbols("x")
+    f1 = Piecewise((c[1], x < 1), (c[2], True))
+    f2 = Piecewise((c[3], x < Rational(1, 3)), (c[4], True))
+    assert integrate(f1*f2, (x, 0, 2)
+        ) == c[1]*c[3]/3 + 2*c[1]*c[4]/3 + c[2]*c[4]
+    f1 = Piecewise((0, x < 1), (2, True))
+    f2 = Piecewise((3, x < 2), (0, True))
+    assert integrate(f1*f2, (x, 0, 3)) == 6
+
+    y = symbols("y", positive=True)
+    a, b, c, x, z = symbols("a,b,c,x,z", real=True)
+    I = Integral(Piecewise(
+        (0, (x >= y) | (x < 0) | (b > c)),
+        (a, True)), (x, 0, z))
+    ans = I.doit()
+    assert ans == Piecewise((0, b > c), (a*Min(y, z) - a*Min(0, z), True))
+    for cond in (True, False):
+         for yy in range(1, 3):
+             for zz in range(-yy, 0, yy):
+                 reps = [(b > c, cond), (y, yy), (z, zz)]
+                 assert ans.subs(reps) == I.subs(reps).doit()
+
+
+def test_unevaluated_integrals():
+    f = Function('f')
+    p = Piecewise((1, Eq(f(x) - 1, 0)), (2, x - 10 < 0), (0, True))
+    assert p.integrate(x) == Integral(p, x)
+    assert p.integrate((x, 0, 5)) == Integral(p, (x, 0, 5))
+    # test it by replacing f(x) with x%2 which will not
+    # affect the answer: the integrand is essentially 2 over
+    # the domain of integration
+    assert Integral(p, (x, 0, 5)).subs(f(x), x%2).n() == 10.0
+
+    # this is a test of using _solve_inequality when
+    # solve_univariate_inequality fails
+    assert p.integrate(y) == Piecewise(
+        (y, Eq(f(x), 1) | ((x < 10) & Eq(f(x), 1))),
+        (2*y, (x > -oo) & (x < 10)), (0, True))
+
+
+def test_conditions_as_alternate_booleans():
+    a, b, c = symbols('a:c')
+    assert Piecewise((x, Piecewise((y < 1, x > 0), (y > 1, True)))
+        ) == Piecewise((x, ITE(x > 0, y < 1, y > 1)))
+
+
+def test_Piecewise_rewrite_as_ITE():
+    a, b, c, d = symbols('a:d')
+
+    def _ITE(*args):
+        return Piecewise(*args).rewrite(ITE)
+
+    assert _ITE((a, x < 1), (b, x >= 1)) == ITE(x < 1, a, b)
+    assert _ITE((a, x < 1), (b, x < oo)) == ITE(x < 1, a, b)
+    assert _ITE((a, x < 1), (b, Or(y < 1, x < oo)), (c, y > 0)
+               ) == ITE(x < 1, a, b)
+    assert _ITE((a, x < 1), (b, True)) == ITE(x < 1, a, b)
+    assert _ITE((a, x < 1), (b, x < 2), (c, True)
+               ) == ITE(x < 1, a, ITE(x < 2, b, c))
+    assert _ITE((a, x < 1), (b, y < 2), (c, True)
+               ) == ITE(x < 1, a, ITE(y < 2, b, c))
+    assert _ITE((a, x < 1), (b, x < oo), (c, y < 1)
+               ) == ITE(x < 1, a, b)
+    assert _ITE((a, x < 1), (c, y < 1), (b, x < oo), (d, True)
+               ) == ITE(x < 1, a, ITE(y < 1, c, b))
+    assert _ITE((a, x < 0), (b, Or(x < oo, y < 1))
+               ) == ITE(x < 0, a, b)
+    raises(TypeError, lambda: _ITE((x + 1, x < 1), (x, True)))
+    # if `a` in the following were replaced with y then the coverage
+    # is complete but something other than as_set would need to be
+    # used to detect this
+    raises(NotImplementedError, lambda: _ITE((x, x < y), (y, x >= a)))
+    raises(ValueError, lambda: _ITE((a, x < 2), (b, x > 3)))
+
+
+def test_issue_14052():
+    assert integrate(abs(sin(x)), (x, 0, 2*pi)) == 4
+
+
+def test_issue_14240():
+    assert piecewise_fold(
+        Piecewise((1, a), (2, b), (4, True)) +
+        Piecewise((8, a), (16, True))
+        ) == Piecewise((9, a), (18, b), (20, True))
+    assert piecewise_fold(
+        Piecewise((2, a), (3, b), (5, True)) *
+        Piecewise((7, a), (11, True))
+        ) == Piecewise((14, a), (33, b), (55, True))
+    # these will hang if naive folding is used
+    assert piecewise_fold(Add(*[
+        Piecewise((i, a), (0, True)) for i in range(40)])
+        ) == Piecewise((780, a), (0, True))
+    assert piecewise_fold(Mul(*[
+        Piecewise((i, a), (0, True)) for i in range(1, 41)])
+        ) == Piecewise((factorial(40), a), (0, True))
+
+
+def test_issue_14787():
+    x = Symbol('x')
+    f = Piecewise((x, x < 1), ((S(58) / 7), True))
+    assert str(f.evalf()) == "Piecewise((x, x < 1), (8.28571428571429, True))"
+
+def test_issue_21481():
+    b, e = symbols('b e')
+    C = Piecewise(
+        (2,
+        ((b > 1) & (e > 0)) |
+        ((b > 0) & (b < 1) & (e < 0)) |
+        ((e >= 2) & (b < -1) & Eq(Mod(e, 2), 0)) |
+        ((e <= -2) & (b > -1) & (b < 0) & Eq(Mod(e, 2), 0))),
+        (S.Half,
+        ((b > 1) & (e < 0)) |
+        ((b > 0) & (e > 0) & (b < 1)) |
+        ((e <= -2) & (b < -1) & Eq(Mod(e, 2), 0)) |
+        ((e >= 2) & (b > -1) & (b < 0) & Eq(Mod(e, 2), 0))),
+        (-S.Half,
+        Eq(Mod(e, 2), 1) &
+        (((e <= -1) & (b < -1)) | ((e >= 1) & (b > -1) & (b < 0)))),
+        (-2,
+        ((e >= 1) & (b < -1) & Eq(Mod(e, 2), 1)) |
+        ((e <= -1) & (b > -1) & (b < 0) & Eq(Mod(e, 2), 1)))
+    )
+    A = Piecewise(
+        (1, Eq(b, 1) | Eq(e, 0) | (Eq(b, -1) & Eq(Mod(e, 2), 0))),
+        (0, Eq(b, 0) & (e > 0)),
+        (-1, Eq(b, -1) & Eq(Mod(e, 2), 1)),
+        (C, Eq(im(b), 0) & Eq(im(e), 0))
+    )
+
+    B = piecewise_fold(A)
+    sa = A.simplify()
+    sb = B.simplify()
+    v = (-2, -1, -S.Half, 0, S.Half, 1, 2)
+    for i in v:
+        for j in v:
+            r = {b:i, e:j}
+            ok = [k.xreplace(r) for k in (A, B, sa, sb)]
+            assert len(set(ok)) == 1
+
+
+def test_issue_8458():
+    x, y = symbols('x y')
+    # Original issue
+    p1 = Piecewise((0, Eq(x, 0)), (sin(x), True))
+    assert p1.simplify() == sin(x)
+    # Slightly larger variant
+    p2 = Piecewise((x, Eq(x, 0)), (4*x + (y-2)**4, Eq(x, 0) & Eq(x+y, 2)), (sin(x), True))
+    assert p2.simplify() == sin(x)
+    # Test for problem highlighted during review
+    p3 = Piecewise((x+1, Eq(x, -1)), (4*x + (y-2)**4, Eq(x, 0) & Eq(x+y, 2)), (sin(x), True))
+    assert p3.simplify() == Piecewise((0, Eq(x, -1)), (sin(x), True))
+
+
+def test_issue_16417():
+    z = Symbol('z')
+    assert unchanged(Piecewise, (1, Or(Eq(im(z), 0), Gt(re(z), 0))), (2, True))
+
+    x = Symbol('x')
+    assert unchanged(Piecewise, (S.Pi, re(x) < 0),
+                 (0, Or(re(x) > 0, Ne(im(x), 0))),
+                 (S.NaN, True))
+    r = Symbol('r', real=True)
+    p = Piecewise((S.Pi, re(r) < 0),
+                 (0, Or(re(r) > 0, Ne(im(r), 0))),
+                 (S.NaN, True))
+    assert p == Piecewise((S.Pi, r < 0),
+                 (0, r > 0),
+                 (S.NaN, True), evaluate=False)
+    # Does not work since imaginary != 0...
+    #i = Symbol('i', imaginary=True)
+    #p = Piecewise((S.Pi, re(i) < 0),
+    #              (0, Or(re(i) > 0, Ne(im(i), 0))),
+    #              (S.NaN, True))
+    #assert p == Piecewise((0, Ne(im(i), 0)),
+    #                      (S.NaN, True), evaluate=False)
+    i = I*r
+    p = Piecewise((S.Pi, re(i) < 0),
+                  (0, Or(re(i) > 0, Ne(im(i), 0))),
+                  (S.NaN, True))
+    assert p == Piecewise((0, Ne(im(i), 0)),
+                          (S.NaN, True), evaluate=False)
+    assert p == Piecewise((0, Ne(r, 0)),
+                          (S.NaN, True), evaluate=False)
+
+
+def test_eval_rewrite_as_KroneckerDelta():
+    x, y, z, n, t, m = symbols('x y z n t m')
+    K = KroneckerDelta
+    f = lambda p: expand(p.rewrite(K))
+
+    p1 = Piecewise((0, Eq(x, y)), (1, True))
+    assert f(p1) == 1 - K(x, y)
+
+    p2 = Piecewise((x, Eq(y,0)), (z, Eq(t,0)), (n, True))
+    assert f(p2) == n*K(0, t)*K(0, y) - n*K(0, t) - n*K(0, y) + n + \
+           x*K(0, y) - z*K(0, t)*K(0, y) + z*K(0, t)
+
+    p3 = Piecewise((1, Ne(x, y)), (0, True))
+    assert f(p3) == 1 - K(x, y)
+
+    p4 = Piecewise((1, Eq(x, 3)), (4, True), (5, True))
+    assert f(p4) == 4 - 3*K(3, x)
+
+    p5 = Piecewise((3, Ne(x, 2)), (4, Eq(y, 2)), (5, True))
+    assert f(p5) == -K(2, x)*K(2, y) + 2*K(2, x) + 3
+
+    p6 = Piecewise((0, Ne(x, 1) & Ne(y, 4)), (1, True))
+    assert f(p6) == -K(1, x)*K(4, y) + K(1, x) + K(4, y)
+
+    p7 = Piecewise((2, Eq(y, 3) & Ne(x, 2)), (1, True))
+    assert f(p7) == -K(2, x)*K(3, y) + K(3, y) + 1
+
+    p8 = Piecewise((4, Eq(x, 3) & Ne(y, 2)), (1, True))
+    assert f(p8) == -3*K(2, y)*K(3, x) + 3*K(3, x) + 1
+
+    p9 = Piecewise((6, Eq(x, 4) & Eq(y, 1)), (1, True))
+    assert f(p9) == 5 * K(1, y) * K(4, x) + 1
+
+    p10 = Piecewise((4, Ne(x, -4) | Ne(y, 1)), (1, True))
+    assert f(p10) == -3 * K(-4, x) * K(1, y) + 4
+
+    p11 = Piecewise((1, Eq(y, 2) | Ne(x, -3)), (2, True))
+    assert f(p11) == -K(-3, x)*K(2, y) + K(-3, x) + 1
+
+    p12 = Piecewise((-1, Eq(x, 1) | Ne(y, 3)), (1, True))
+    assert f(p12) == -2*K(1, x)*K(3, y) + 2*K(3, y) - 1
+
+    p13 = Piecewise((3, Eq(x, 2) | Eq(y, 4)), (1, True))
+    assert f(p13) == -2*K(2, x)*K(4, y) + 2*K(2, x) + 2*K(4, y) + 1
+
+    p14 = Piecewise((1, Ne(x, 0) | Ne(y, 1)), (3, True))
+    assert f(p14) == 2 * K(0, x) * K(1, y) + 1
+
+    p15 = Piecewise((2, Eq(x, 3) | Ne(y, 2)), (3, Eq(x, 4) & Eq(y, 5)), (1, True))
+    assert f(p15) == -2*K(2, y)*K(3, x)*K(4, x)*K(5, y) + K(2, y)*K(3, x) + \
+           2*K(2, y)*K(4, x)*K(5, y) - K(2, y) + 2
+
+    p16 = Piecewise((0, Ne(m, n)), (1, True))*Piecewise((0, Ne(n, t)), (1, True))\
+          *Piecewise((0, Ne(n, x)), (1, True)) - Piecewise((0, Ne(t, x)), (1, True))
+    assert f(p16) == K(m, n)*K(n, t)*K(n, x) - K(t, x)
+
+    p17 = Piecewise((0, Ne(t, x) & (Ne(m, n) | Ne(n, t) | Ne(n, x))),
+                    (1, Ne(t, x)), (-1, Ne(m, n) | Ne(n, t) | Ne(n, x)), (0, True))
+    assert f(p17) == K(m, n)*K(n, t)*K(n, x) - K(t, x)
+
+    p18 = Piecewise((-4, Eq(y, 1) | (Eq(x, -5) & Eq(x, z))), (4, True))
+    assert f(p18) == 8*K(-5, x)*K(1, y)*K(x, z) - 8*K(-5, x)*K(x, z) - 8*K(1, y) + 4
+
+    p19 = Piecewise((0, x > 2), (1, True))
+    assert f(p19) == p19
+
+    p20 = Piecewise((0, And(x < 2, x > -5)), (1, True))
+    assert f(p20) == p20
+
+    p21 = Piecewise((0, Or(x > 1, x < 0)), (1, True))
+    assert f(p21) == p21
+
+    p22 = Piecewise((0, ~((Eq(y, -1) | Ne(x, 0)) & (Ne(x, 1) | Ne(y, -1)))), (1, True))
+    assert f(p22) == K(-1, y)*K(0, x) - K(-1, y)*K(1, x) - K(0, x) + 1
+
+
+@slow
+def test_identical_conds_issue():
+    from sympy.stats import Uniform, density
+    u1 = Uniform('u1', 0, 1)
+    u2 = Uniform('u2', 0, 1)
+    # Result is quite big, so not really important here (and should ideally be
+    # simpler). Should not give an exception though.
+    density(u1 + u2)
+
+
+def test_issue_7370():
+    f = Piecewise((1, x <= 2400))
+    v = integrate(f, (x, 0, Float("252.4", 30)))
+    assert str(v) == '252.400000000000000000000000000'
+
+
+def test_issue_14933():
+    x = Symbol('x')
+    y = Symbol('y')
+
+    inp = MatrixSymbol('inp', 1, 1)
+    rep_dict = {y: inp[0, 0], x: inp[0, 0]}
+
+    p = Piecewise((1, ITE(y > 0, x < 0, True)))
+    assert p.xreplace(rep_dict) == Piecewise((1, ITE(inp[0, 0] > 0, inp[0, 0] < 0, True)))
+
+
+def test_issue_16715():
+    raises(NotImplementedError, lambda: Piecewise((x, x<0), (0, y>1)).as_expr_set_pairs())
+
+
+def test_issue_20360():
+    t, tau = symbols("t tau", real=True)
+    n = symbols("n", integer=True)
+    lam = pi * (n - S.Half)
+    eq = integrate(exp(lam * tau), (tau, 0, t))
+    assert eq.simplify() == (2*exp(pi*t*(2*n - 1)/2) - 2)/(pi*(2*n - 1))
+
+
+def test_piecewise_eval():
+    # XXX these tests might need modification if this
+    # simplification is moved out of eval and into
+    # boolalg or Piecewise simplification functions
+    f = lambda x: x.args[0].cond
+    # unsimplified
+    assert f(Piecewise((x, (x > -oo) & (x < 3)))
+        ) == ((x > -oo) & (x < 3))
+    assert f(Piecewise((x, (x > -oo) & (x < oo)))
+        ) == ((x > -oo) & (x < oo))
+    assert f(Piecewise((x, (x > -3) & (x < 3)))
+        ) == ((x > -3) & (x < 3))
+    assert f(Piecewise((x, (x > -3) & (x < oo)))
+        ) == ((x > -3) & (x < oo))
+    assert f(Piecewise((x, (x <= 3) & (x > -oo)))
+        ) == ((x <= 3) & (x > -oo))
+    assert f(Piecewise((x, (x <= 3) & (x > -3)))
+        ) == ((x <= 3) & (x > -3))
+    assert f(Piecewise((x, (x >= -3) & (x < 3)))
+        ) == ((x >= -3) & (x < 3))
+    assert f(Piecewise((x, (x >= -3) & (x < oo)))
+        ) == ((x >= -3) & (x < oo))
+    assert f(Piecewise((x, (x >= -3) & (x <= 3)))
+        ) == ((x >= -3) & (x <= 3))
+    # could simplify by keeping only the first
+    # arg of result
+    assert f(Piecewise((x, (x <= oo) & (x > -oo)))
+        ) == (x > -oo) & (x <= oo)
+    assert f(Piecewise((x, (x <= oo) & (x > -3)))
+        ) == (x > -3) & (x <= oo)
+    assert f(Piecewise((x, (x >= -oo) & (x < 3)))
+        ) == (x < 3) & (x >= -oo)
+    assert f(Piecewise((x, (x >= -oo) & (x < oo)))
+        ) == (x < oo) & (x >= -oo)
+    assert f(Piecewise((x, (x >= -oo) & (x <= 3)))
+        ) == (x <= 3) & (x >= -oo)
+    assert f(Piecewise((x, (x >= -oo) & (x <= oo)))
+        ) == (x <= oo) & (x >= -oo)  # but cannot be True unless x is real
+    assert f(Piecewise((x, (x >= -3) & (x <= oo)))
+        ) == (x >= -3) & (x <= oo)
+    assert f(Piecewise((x, (Abs(arg(a)) <= 1) | (Abs(arg(a)) < 1)))
+        ) == (Abs(arg(a)) <= 1) | (Abs(arg(a)) < 1)
+
+
+def test_issue_22533():
+    x = Symbol('x', real=True)
+    f = Piecewise((-1 / x, x <= 0), (1 / x, True))
+    assert integrate(f, x) == Piecewise((-log(x), x <= 0), (log(x), True))
+
+
+def test_issue_24072():
+    assert Piecewise((1, x > 1), (2, x <= 1), (3, x <= 1)
+        ) == Piecewise((1, x > 1), (2, True))
+
+
+def test_piecewise__eval_is_meromorphic():
+    """ Issue 24127: Tests eval_is_meromorphic auxiliary method """
+    x = symbols('x', real=True)
+    f = Piecewise((1, x < 0), (sqrt(1 - x), True))
+    assert f.is_meromorphic(x, I) is None
+    assert f.is_meromorphic(x, -1) == True
+    assert f.is_meromorphic(x, 0) == None
+    assert f.is_meromorphic(x, 1) == False
+    assert f.is_meromorphic(x, 2) == True
+    assert f.is_meromorphic(x, Symbol('a')) == None
+    assert f.is_meromorphic(x, Symbol('a', real=True)) == None

llmeval-env/lib/python3.10/site-packages/sympy/functions/elementary/tests/test_trigonometric.py

@@ -0,0 +1,2162 @@
+from sympy.calculus.accumulationbounds import AccumBounds
+from sympy.core.add import Add
+from sympy.core.function import (Lambda, diff)
+from sympy.core.mod import Mod
+from sympy.core.mul import Mul
+from sympy.core.numbers import (E, Float, I, Rational, nan, oo, pi, zoo)
+from sympy.core.power import Pow
+from sympy.core.singleton import S
+from sympy.core.symbol import (Symbol, symbols)
+from sympy.functions.elementary.complexes import (arg, conjugate, im, re)
+from sympy.functions.elementary.exponential import (exp, log)
+from sympy.functions.elementary.hyperbolic import (acoth, asinh, atanh, cosh, coth, sinh, tanh)
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.trigonometric import (acos, acot, acsc, asec, asin, atan, atan2,
+                                                      cos, cot, csc, sec, sin, sinc, tan)
+from sympy.functions.special.bessel import (besselj, jn)
+from sympy.functions.special.delta_functions import Heaviside
+from sympy.matrices.dense import Matrix
+from sympy.polys.polytools import (cancel, gcd)
+from sympy.series.limits import limit
+from sympy.series.order import O
+from sympy.series.series import series
+from sympy.sets.fancysets import ImageSet
+from sympy.sets.sets import (FiniteSet, Interval)
+from sympy.simplify.simplify import simplify
+from sympy.core.expr import unchanged
+from sympy.core.function import ArgumentIndexError
+from sympy.core.relational import Ne, Eq
+from sympy.functions.elementary.piecewise import Piecewise
+from sympy.sets.setexpr import SetExpr
+from sympy.testing.pytest import XFAIL, slow, raises
+
+
+x, y, z = symbols('x y z')
+r = Symbol('r', real=True)
+k, m = symbols('k m', integer=True)
+p = Symbol('p', positive=True)
+n = Symbol('n', negative=True)
+np = Symbol('p', nonpositive=True)
+nn = Symbol('n', nonnegative=True)
+nz = Symbol('nz', nonzero=True)
+ep = Symbol('ep', extended_positive=True)
+en = Symbol('en', extended_negative=True)
+enp = Symbol('ep', extended_nonpositive=True)
+enn = Symbol('en', extended_nonnegative=True)
+enz = Symbol('enz', extended_nonzero=True)
+a = Symbol('a', algebraic=True)
+na = Symbol('na', nonzero=True, algebraic=True)
+
+
+def test_sin():
+    x, y = symbols('x y')
+    z = symbols('z', imaginary=True)
+
+    assert sin.nargs == FiniteSet(1)
+    assert sin(nan) is nan
+    assert sin(zoo) is nan
+
+    assert sin(oo) == AccumBounds(-1, 1)
+    assert sin(oo) - sin(oo) == AccumBounds(-2, 2)
+    assert sin(oo*I) == oo*I
+    assert sin(-oo*I) == -oo*I
+    assert 0*sin(oo) is S.Zero
+    assert 0/sin(oo) is S.Zero
+    assert 0 + sin(oo) == AccumBounds(-1, 1)
+    assert 5 + sin(oo) == AccumBounds(4, 6)
+
+    assert sin(0) == 0
+
+    assert sin(z*I) == I*sinh(z)
+    assert sin(asin(x)) == x
+    assert sin(atan(x)) == x / sqrt(1 + x**2)
+    assert sin(acos(x)) == sqrt(1 - x**2)
+    assert sin(acot(x)) == 1 / (sqrt(1 + 1 / x**2) * x)
+    assert sin(acsc(x)) == 1 / x
+    assert sin(asec(x)) == sqrt(1 - 1 / x**2)
+    assert sin(atan2(y, x)) == y / sqrt(x**2 + y**2)
+
+    assert sin(pi*I) == sinh(pi)*I
+    assert sin(-pi*I) == -sinh(pi)*I
+    assert sin(-2*I) == -sinh(2)*I
+
+    assert sin(pi) == 0
+    assert sin(-pi) == 0
+    assert sin(2*pi) == 0
+    assert sin(-2*pi) == 0
+    assert sin(-3*10**73*pi) == 0
+    assert sin(7*10**103*pi) == 0
+
+    assert sin(pi/2) == 1
+    assert sin(-pi/2) == -1
+    assert sin(pi*Rational(5, 2)) == 1
+    assert sin(pi*Rational(7, 2)) == -1
+
+    ne = symbols('ne', integer=True, even=False)
+    e = symbols('e', even=True)
+    assert sin(pi*ne/2) == (-1)**(ne/2 - S.Half)
+    assert sin(pi*k/2).func == sin
+    assert sin(pi*e/2) == 0
+    assert sin(pi*k) == 0
+    assert sin(pi*k).subs(k, 3) == sin(pi*k/2).subs(k, 6)  # issue 8298
+
+    assert sin(pi/3) == S.Half*sqrt(3)
+    assert sin(pi*Rational(-2, 3)) == Rational(-1, 2)*sqrt(3)
+
+    assert sin(pi/4) == S.Half*sqrt(2)
+    assert sin(-pi/4) == Rational(-1, 2)*sqrt(2)
+    assert sin(pi*Rational(17, 4)) == S.Half*sqrt(2)
+    assert sin(pi*Rational(-3, 4)) == Rational(-1, 2)*sqrt(2)
+
+    assert sin(pi/6) == S.Half
+    assert sin(-pi/6) == Rational(-1, 2)
+    assert sin(pi*Rational(7, 6)) == Rational(-1, 2)
+    assert sin(pi*Rational(-5, 6)) == Rational(-1, 2)
+
+    assert sin(pi*Rational(1, 5)) == sqrt((5 - sqrt(5)) / 8)
+    assert sin(pi*Rational(2, 5)) == sqrt((5 + sqrt(5)) / 8)
+    assert sin(pi*Rational(3, 5)) == sin(pi*Rational(2, 5))
+    assert sin(pi*Rational(4, 5)) == sin(pi*Rational(1, 5))
+    assert sin(pi*Rational(6, 5)) == -sin(pi*Rational(1, 5))
+    assert sin(pi*Rational(8, 5)) == -sin(pi*Rational(2, 5))
+
+    assert sin(pi*Rational(-1273, 5)) == -sin(pi*Rational(2, 5))
+
+    assert sin(pi/8) == sqrt((2 - sqrt(2))/4)
+
+    assert sin(pi/10) == Rational(-1, 4) + sqrt(5)/4
+
+    assert sin(pi/12) == -sqrt(2)/4 + sqrt(6)/4
+    assert sin(pi*Rational(5, 12)) == sqrt(2)/4 + sqrt(6)/4
+    assert sin(pi*Rational(-7, 12)) == -sqrt(2)/4 - sqrt(6)/4
+    assert sin(pi*Rational(-11, 12)) == sqrt(2)/4 - sqrt(6)/4
+
+    assert sin(pi*Rational(104, 105)) == sin(pi/105)
+    assert sin(pi*Rational(106, 105)) == -sin(pi/105)
+
+    assert sin(pi*Rational(-104, 105)) == -sin(pi/105)
+    assert sin(pi*Rational(-106, 105)) == sin(pi/105)
+
+    assert sin(x*I) == sinh(x)*I
+
+    assert sin(k*pi) == 0
+    assert sin(17*k*pi) == 0
+    assert sin(2*k*pi + 4) == sin(4)
+    assert sin(2*k*pi + m*pi + 1) == (-1)**(m + 2*k)*sin(1)
+
+    assert sin(k*pi*I) == sinh(k*pi)*I
+
+    assert sin(r).is_real is True
+
+    assert sin(0, evaluate=False).is_algebraic
+    assert sin(a).is_algebraic is None
+    assert sin(na).is_algebraic is False
+    q = Symbol('q', rational=True)
+    assert sin(pi*q).is_algebraic
+    qn = Symbol('qn', rational=True, nonzero=True)
+    assert sin(qn).is_rational is False
+    assert sin(q).is_rational is None  # issue 8653
+
+    assert isinstance(sin( re(x) - im(y)), sin) is True
+    assert isinstance(sin(-re(x) + im(y)), sin) is False
+
+    assert sin(SetExpr(Interval(0, 1))) == SetExpr(ImageSet(Lambda(x, sin(x)),
+                       Interval(0, 1)))
+
+    for d in list(range(1, 22)) + [60, 85]:
+        for n in range(d*2 + 1):
+            x = n*pi/d
+            e = abs( float(sin(x)) - sin(float(x)) )
+            assert e < 1e-12
+
+    assert sin(0, evaluate=False).is_zero is True
+    assert sin(k*pi, evaluate=False).is_zero is True
+
+    assert sin(Add(1, -1, evaluate=False), evaluate=False).is_zero is True
+
+
+def test_sin_cos():
+    for d in [1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 24, 30, 40, 60, 120]:  # list is not exhaustive...
+        for n in range(-2*d, d*2):
+            x = n*pi/d
+            assert sin(x + pi/2) == cos(x), "fails for %d*pi/%d" % (n, d)
+            assert sin(x - pi/2) == -cos(x), "fails for %d*pi/%d" % (n, d)
+            assert sin(x) == cos(x - pi/2), "fails for %d*pi/%d" % (n, d)
+            assert -sin(x) == cos(x + pi/2), "fails for %d*pi/%d" % (n, d)
+
+
+def test_sin_series():
+    assert sin(x).series(x, 0, 9) == \
+        x - x**3/6 + x**5/120 - x**7/5040 + O(x**9)
+
+
+def test_sin_rewrite():
+    assert sin(x).rewrite(exp) == -I*(exp(I*x) - exp(-I*x))/2
+    assert sin(x).rewrite(tan) == 2*tan(x/2)/(1 + tan(x/2)**2)
+    assert sin(x).rewrite(cot) == \
+        Piecewise((0, Eq(im(x), 0) & Eq(Mod(x, pi), 0)),
+                  (2*cot(x/2)/(cot(x/2)**2 + 1), True))
+    assert sin(sinh(x)).rewrite(
+        exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, sinh(3)).n()
+    assert sin(cosh(x)).rewrite(
+        exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, cosh(3)).n()
+    assert sin(tanh(x)).rewrite(
+        exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, tanh(3)).n()
+    assert sin(coth(x)).rewrite(
+        exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, coth(3)).n()
+    assert sin(sin(x)).rewrite(
+        exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, sin(3)).n()
+    assert sin(cos(x)).rewrite(
+        exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, cos(3)).n()
+    assert sin(tan(x)).rewrite(
+        exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, tan(3)).n()
+    assert sin(cot(x)).rewrite(
+        exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, cot(3)).n()
+    assert sin(log(x)).rewrite(Pow) == I*x**-I / 2 - I*x**I /2
+    assert sin(x).rewrite(csc) == 1/csc(x)
+    assert sin(x).rewrite(cos) == cos(x - pi / 2, evaluate=False)
+    assert sin(x).rewrite(sec) == 1 / sec(x - pi / 2, evaluate=False)
+    assert sin(cos(x)).rewrite(Pow) == sin(cos(x))
+
+
+def _test_extrig(f, i, e):
+    from sympy.core.function import expand_trig
+    assert unchanged(f, i)
+    assert expand_trig(f(i)) == f(i)
+    # testing directly instead of with .expand(trig=True)
+    # because the other expansions undo the unevaluated Mul
+    assert expand_trig(f(Mul(i, 1, evaluate=False))) == e
+    assert abs(f(i) - e).n() < 1e-10
+
+
+def test_sin_expansion():
+    # Note: these formulas are not unique.  The ones here come from the
+    # Chebyshev formulas.
+    assert sin(x + y).expand(trig=True) == sin(x)*cos(y) + cos(x)*sin(y)
+    assert sin(x - y).expand(trig=True) == sin(x)*cos(y) - cos(x)*sin(y)
+    assert sin(y - x).expand(trig=True) == cos(x)*sin(y) - sin(x)*cos(y)
+    assert sin(2*x).expand(trig=True) == 2*sin(x)*cos(x)
+    assert sin(3*x).expand(trig=True) == -4*sin(x)**3 + 3*sin(x)
+    assert sin(4*x).expand(trig=True) == -8*sin(x)**3*cos(x) + 4*sin(x)*cos(x)
+    _test_extrig(sin, 2, 2*sin(1)*cos(1))
+    _test_extrig(sin, 3, -4*sin(1)**3 + 3*sin(1))
+
+
+def test_sin_AccumBounds():
+    assert sin(AccumBounds(-oo, oo)) == AccumBounds(-1, 1)
+    assert sin(AccumBounds(0, oo)) == AccumBounds(-1, 1)
+    assert sin(AccumBounds(-oo, 0)) == AccumBounds(-1, 1)
+    assert sin(AccumBounds(0, 2*S.Pi)) == AccumBounds(-1, 1)
+    assert sin(AccumBounds(0, S.Pi*Rational(3, 4))) == AccumBounds(0, 1)
+    assert sin(AccumBounds(S.Pi*Rational(3, 4), S.Pi*Rational(7, 4))) == AccumBounds(-1, sin(S.Pi*Rational(3, 4)))
+    assert sin(AccumBounds(S.Pi/4, S.Pi/3)) == AccumBounds(sin(S.Pi/4), sin(S.Pi/3))
+    assert sin(AccumBounds(S.Pi*Rational(3, 4), S.Pi*Rational(5, 6))) == AccumBounds(sin(S.Pi*Rational(5, 6)), sin(S.Pi*Rational(3, 4)))
+
+
+def test_sin_fdiff():
+    assert sin(x).fdiff() == cos(x)
+    raises(ArgumentIndexError, lambda: sin(x).fdiff(2))
+
+
+def test_trig_symmetry():
+    assert sin(-x) == -sin(x)
+    assert cos(-x) == cos(x)
+    assert tan(-x) == -tan(x)
+    assert cot(-x) == -cot(x)
+    assert sin(x + pi) == -sin(x)
+    assert sin(x + 2*pi) == sin(x)
+    assert sin(x + 3*pi) == -sin(x)
+    assert sin(x + 4*pi) == sin(x)
+    assert sin(x - 5*pi) == -sin(x)
+    assert cos(x + pi) == -cos(x)
+    assert cos(x + 2*pi) == cos(x)
+    assert cos(x + 3*pi) == -cos(x)
+    assert cos(x + 4*pi) == cos(x)
+    assert cos(x - 5*pi) == -cos(x)
+    assert tan(x + pi) == tan(x)
+    assert tan(x - 3*pi) == tan(x)
+    assert cot(x + pi) == cot(x)
+    assert cot(x - 3*pi) == cot(x)
+    assert sin(pi/2 - x) == cos(x)
+    assert sin(pi*Rational(3, 2) - x) == -cos(x)
+    assert sin(pi*Rational(5, 2) - x) == cos(x)
+    assert cos(pi/2 - x) == sin(x)
+    assert cos(pi*Rational(3, 2) - x) == -sin(x)
+    assert cos(pi*Rational(5, 2) - x) == sin(x)
+    assert tan(pi/2 - x) == cot(x)
+    assert tan(pi*Rational(3, 2) - x) == cot(x)
+    assert tan(pi*Rational(5, 2) - x) == cot(x)
+    assert cot(pi/2 - x) == tan(x)
+    assert cot(pi*Rational(3, 2) - x) == tan(x)
+    assert cot(pi*Rational(5, 2) - x) == tan(x)
+    assert sin(pi/2 + x) == cos(x)
+    assert cos(pi/2 + x) == -sin(x)
+    assert tan(pi/2 + x) == -cot(x)
+    assert cot(pi/2 + x) == -tan(x)
+
+
+def test_cos():
+    x, y = symbols('x y')
+
+    assert cos.nargs == FiniteSet(1)
+    assert cos(nan) is nan
+
+    assert cos(oo) == AccumBounds(-1, 1)
+    assert cos(oo) - cos(oo) == AccumBounds(-2, 2)
+    assert cos(oo*I) is oo
+    assert cos(-oo*I) is oo
+    assert cos(zoo) is nan
+
+    assert cos(0) == 1
+
+    assert cos(acos(x)) == x
+    assert cos(atan(x)) == 1 / sqrt(1 + x**2)
+    assert cos(asin(x)) == sqrt(1 - x**2)
+    assert cos(acot(x)) == 1 / sqrt(1 + 1 / x**2)
+    assert cos(acsc(x)) == sqrt(1 - 1 / x**2)
+    assert cos(asec(x)) == 1 / x
+    assert cos(atan2(y, x)) == x / sqrt(x**2 + y**2)
+
+    assert cos(pi*I) == cosh(pi)
+    assert cos(-pi*I) == cosh(pi)
+    assert cos(-2*I) == cosh(2)
+
+    assert cos(pi/2) == 0
+    assert cos(-pi/2) == 0
+    assert cos(pi/2) == 0
+    assert cos(-pi/2) == 0
+    assert cos((-3*10**73 + 1)*pi/2) == 0
+    assert cos((7*10**103 + 1)*pi/2) == 0
+
+    n = symbols('n', integer=True, even=False)
+    e = symbols('e', even=True)
+    assert cos(pi*n/2) == 0
+    assert cos(pi*e/2) == (-1)**(e/2)
+
+    assert cos(pi) == -1
+    assert cos(-pi) == -1
+    assert cos(2*pi) == 1
+    assert cos(5*pi) == -1
+    assert cos(8*pi) == 1
+
+    assert cos(pi/3) == S.Half
+    assert cos(pi*Rational(-2, 3)) == Rational(-1, 2)
+
+    assert cos(pi/4) == S.Half*sqrt(2)
+    assert cos(-pi/4) == S.Half*sqrt(2)
+    assert cos(pi*Rational(11, 4)) == Rational(-1, 2)*sqrt(2)
+    assert cos(pi*Rational(-3, 4)) == Rational(-1, 2)*sqrt(2)
+
+    assert cos(pi/6) == S.Half*sqrt(3)
+    assert cos(-pi/6) == S.Half*sqrt(3)
+    assert cos(pi*Rational(7, 6)) == Rational(-1, 2)*sqrt(3)
+    assert cos(pi*Rational(-5, 6)) == Rational(-1, 2)*sqrt(3)
+
+    assert cos(pi*Rational(1, 5)) == (sqrt(5) + 1)/4
+    assert cos(pi*Rational(2, 5)) == (sqrt(5) - 1)/4
+    assert cos(pi*Rational(3, 5)) == -cos(pi*Rational(2, 5))
+    assert cos(pi*Rational(4, 5)) == -cos(pi*Rational(1, 5))
+    assert cos(pi*Rational(6, 5)) == -cos(pi*Rational(1, 5))
+    assert cos(pi*Rational(8, 5)) == cos(pi*Rational(2, 5))
+
+    assert cos(pi*Rational(-1273, 5)) == -cos(pi*Rational(2, 5))
+
+    assert cos(pi/8) == sqrt((2 + sqrt(2))/4)
+
+    assert cos(pi/12) == sqrt(2)/4 + sqrt(6)/4
+    assert cos(pi*Rational(5, 12)) == -sqrt(2)/4 + sqrt(6)/4
+    assert cos(pi*Rational(7, 12)) == sqrt(2)/4 - sqrt(6)/4
+    assert cos(pi*Rational(11, 12)) == -sqrt(2)/4 - sqrt(6)/4
+
+    assert cos(pi*Rational(104, 105)) == -cos(pi/105)
+    assert cos(pi*Rational(106, 105)) == -cos(pi/105)
+
+    assert cos(pi*Rational(-104, 105)) == -cos(pi/105)
+    assert cos(pi*Rational(-106, 105)) == -cos(pi/105)
+
+    assert cos(x*I) == cosh(x)
+    assert cos(k*pi*I) == cosh(k*pi)
+
+    assert cos(r).is_real is True
+
+    assert cos(0, evaluate=False).is_algebraic
+    assert cos(a).is_algebraic is None
+    assert cos(na).is_algebraic is False
+    q = Symbol('q', rational=True)
+    assert cos(pi*q).is_algebraic
+    assert cos(pi*Rational(2, 7)).is_algebraic
+
+    assert cos(k*pi) == (-1)**k
+    assert cos(2*k*pi) == 1
+    assert cos(0, evaluate=False).is_zero is False
+    assert cos(Rational(1, 2)).is_zero is False
+    # The following test will return None as the result, but really it should
+    # be True even if it is not always possible to resolve an assumptions query.
+    assert cos(asin(-1, evaluate=False), evaluate=False).is_zero is None
+    for d in list(range(1, 22)) + [60, 85]:
+        for n in range(2*d + 1):
+            x = n*pi/d
+            e = abs( float(cos(x)) - cos(float(x)) )
+            assert e < 1e-12
+
+
+def test_issue_6190():
+    c = Float('123456789012345678901234567890.25', '')
+    for cls in [sin, cos, tan, cot]:
+        assert cls(c*pi) == cls(pi/4)
+        assert cls(4.125*pi) == cls(pi/8)
+        assert cls(4.7*pi) == cls((4.7 % 2)*pi)
+
+
+def test_cos_series():
+    assert cos(x).series(x, 0, 9) == \
+        1 - x**2/2 + x**4/24 - x**6/720 + x**8/40320 + O(x**9)
+
+
+def test_cos_rewrite():
+    assert cos(x).rewrite(exp) == exp(I*x)/2 + exp(-I*x)/2
+    assert cos(x).rewrite(tan) == (1 - tan(x/2)**2)/(1 + tan(x/2)**2)
+    assert cos(x).rewrite(cot) == \
+        Piecewise((1, Eq(im(x), 0) & Eq(Mod(x, 2*pi), 0)),
+                  ((cot(x/2)**2 - 1)/(cot(x/2)**2 + 1), True))
+    assert cos(sinh(x)).rewrite(
+        exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, sinh(3)).n()
+    assert cos(cosh(x)).rewrite(
+        exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, cosh(3)).n()
+    assert cos(tanh(x)).rewrite(
+        exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, tanh(3)).n()
+    assert cos(coth(x)).rewrite(
+        exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, coth(3)).n()
+    assert cos(sin(x)).rewrite(
+        exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, sin(3)).n()
+    assert cos(cos(x)).rewrite(
+        exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, cos(3)).n()
+    assert cos(tan(x)).rewrite(
+        exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, tan(3)).n()
+    assert cos(cot(x)).rewrite(
+        exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, cot(3)).n()
+    assert cos(log(x)).rewrite(Pow) == x**I/2 + x**-I/2
+    assert cos(x).rewrite(sec) == 1/sec(x)
+    assert cos(x).rewrite(sin) == sin(x + pi/2, evaluate=False)
+    assert cos(x).rewrite(csc) == 1/csc(-x + pi/2, evaluate=False)
+    assert cos(sin(x)).rewrite(Pow) == cos(sin(x))
+
+
+def test_cos_expansion():
+    assert cos(x + y).expand(trig=True) == cos(x)*cos(y) - sin(x)*sin(y)
+    assert cos(x - y).expand(trig=True) == cos(x)*cos(y) + sin(x)*sin(y)
+    assert cos(y - x).expand(trig=True) == cos(x)*cos(y) + sin(x)*sin(y)
+    assert cos(2*x).expand(trig=True) == 2*cos(x)**2 - 1
+    assert cos(3*x).expand(trig=True) == 4*cos(x)**3 - 3*cos(x)
+    assert cos(4*x).expand(trig=True) == 8*cos(x)**4 - 8*cos(x)**2 + 1
+    _test_extrig(cos, 2, 2*cos(1)**2 - 1)
+    _test_extrig(cos, 3, 4*cos(1)**3 - 3*cos(1))
+
+
+def test_cos_AccumBounds():
+    assert cos(AccumBounds(-oo, oo)) == AccumBounds(-1, 1)
+    assert cos(AccumBounds(0, oo)) == AccumBounds(-1, 1)
+    assert cos(AccumBounds(-oo, 0)) == AccumBounds(-1, 1)
+    assert cos(AccumBounds(0, 2*S.Pi)) == AccumBounds(-1, 1)
+    assert cos(AccumBounds(-S.Pi/3, S.Pi/4)) == AccumBounds(cos(-S.Pi/3), 1)
+    assert cos(AccumBounds(S.Pi*Rational(3, 4), S.Pi*Rational(5, 4))) == AccumBounds(-1, cos(S.Pi*Rational(3, 4)))
+    assert cos(AccumBounds(S.Pi*Rational(5, 4), S.Pi*Rational(4, 3))) == AccumBounds(cos(S.Pi*Rational(5, 4)), cos(S.Pi*Rational(4, 3)))
+    assert cos(AccumBounds(S.Pi/4, S.Pi/3)) == AccumBounds(cos(S.Pi/3), cos(S.Pi/4))
+
+
+def test_cos_fdiff():
+    assert cos(x).fdiff() == -sin(x)
+    raises(ArgumentIndexError, lambda: cos(x).fdiff(2))
+
+
+def test_tan():
+    assert tan(nan) is nan
+
+    assert tan(zoo) is nan
+    assert tan(oo) == AccumBounds(-oo, oo)
+    assert tan(oo) - tan(oo) == AccumBounds(-oo, oo)
+    assert tan.nargs == FiniteSet(1)
+    assert tan(oo*I) == I
+    assert tan(-oo*I) == -I
+
+    assert tan(0) == 0
+
+    assert tan(atan(x)) == x
+    assert tan(asin(x)) == x / sqrt(1 - x**2)
+    assert tan(acos(x)) == sqrt(1 - x**2) / x
+    assert tan(acot(x)) == 1 / x
+    assert tan(acsc(x)) == 1 / (sqrt(1 - 1 / x**2) * x)
+    assert tan(asec(x)) == sqrt(1 - 1 / x**2) * x
+    assert tan(atan2(y, x)) == y/x
+
+    assert tan(pi*I) == tanh(pi)*I
+    assert tan(-pi*I) == -tanh(pi)*I
+    assert tan(-2*I) == -tanh(2)*I
+
+    assert tan(pi) == 0
+    assert tan(-pi) == 0
+    assert tan(2*pi) == 0
+    assert tan(-2*pi) == 0
+    assert tan(-3*10**73*pi) == 0
+
+    assert tan(pi/2) is zoo
+    assert tan(pi*Rational(3, 2)) is zoo
+
+    assert tan(pi/3) == sqrt(3)
+    assert tan(pi*Rational(-2, 3)) == sqrt(3)
+
+    assert tan(pi/4) is S.One
+    assert tan(-pi/4) is S.NegativeOne
+    assert tan(pi*Rational(17, 4)) is S.One
+    assert tan(pi*Rational(-3, 4)) is S.One
+
+    assert tan(pi/5) == sqrt(5 - 2*sqrt(5))
+    assert tan(pi*Rational(2, 5)) == sqrt(5 + 2*sqrt(5))
+    assert tan(pi*Rational(18, 5)) == -sqrt(5 + 2*sqrt(5))
+    assert tan(pi*Rational(-16, 5)) == -sqrt(5 - 2*sqrt(5))
+
+    assert tan(pi/6) == 1/sqrt(3)
+    assert tan(-pi/6) == -1/sqrt(3)
+    assert tan(pi*Rational(7, 6)) == 1/sqrt(3)
+    assert tan(pi*Rational(-5, 6)) == 1/sqrt(3)
+
+    assert tan(pi/8) == -1 + sqrt(2)
+    assert tan(pi*Rational(3, 8)) == 1 + sqrt(2)  # issue 15959
+    assert tan(pi*Rational(5, 8)) == -1 - sqrt(2)
+    assert tan(pi*Rational(7, 8)) == 1 - sqrt(2)
+
+    assert tan(pi/10) == sqrt(1 - 2*sqrt(5)/5)
+    assert tan(pi*Rational(3, 10)) == sqrt(1 + 2*sqrt(5)/5)
+    assert tan(pi*Rational(17, 10)) == -sqrt(1 + 2*sqrt(5)/5)
+    assert tan(pi*Rational(-31, 10)) == -sqrt(1 - 2*sqrt(5)/5)
+
+    assert tan(pi/12) == -sqrt(3) + 2
+    assert tan(pi*Rational(5, 12)) == sqrt(3) + 2
+    assert tan(pi*Rational(7, 12)) == -sqrt(3) - 2
+    assert tan(pi*Rational(11, 12)) == sqrt(3) - 2
+
+    assert tan(pi/24).radsimp() == -2 - sqrt(3) + sqrt(2) + sqrt(6)
+    assert tan(pi*Rational(5, 24)).radsimp() == -2 + sqrt(3) - sqrt(2) + sqrt(6)
+    assert tan(pi*Rational(7, 24)).radsimp() == 2 - sqrt(3) - sqrt(2) + sqrt(6)
+    assert tan(pi*Rational(11, 24)).radsimp() == 2 + sqrt(3) + sqrt(2) + sqrt(6)
+    assert tan(pi*Rational(13, 24)).radsimp() == -2 - sqrt(3) - sqrt(2) - sqrt(6)
+    assert tan(pi*Rational(17, 24)).radsimp() == -2 + sqrt(3) + sqrt(2) - sqrt(6)
+    assert tan(pi*Rational(19, 24)).radsimp() == 2 - sqrt(3) + sqrt(2) - sqrt(6)
+    assert tan(pi*Rational(23, 24)).radsimp() == 2 + sqrt(3) - sqrt(2) - sqrt(6)
+
+    assert tan(x*I) == tanh(x)*I
+
+    assert tan(k*pi) == 0
+    assert tan(17*k*pi) == 0
+
+    assert tan(k*pi*I) == tanh(k*pi)*I
+
+    assert tan(r).is_real is None
+    assert tan(r).is_extended_real is True
+
+    assert tan(0, evaluate=False).is_algebraic
+    assert tan(a).is_algebraic is None
+    assert tan(na).is_algebraic is False
+
+    assert tan(pi*Rational(10, 7)) == tan(pi*Rational(3, 7))
+    assert tan(pi*Rational(11, 7)) == -tan(pi*Rational(3, 7))
+    assert tan(pi*Rational(-11, 7)) == tan(pi*Rational(3, 7))
+
+    assert tan(pi*Rational(15, 14)) == tan(pi/14)
+    assert tan(pi*Rational(-15, 14)) == -tan(pi/14)
+
+    assert tan(r).is_finite is None
+    assert tan(I*r).is_finite is True
+
+    # https://github.com/sympy/sympy/issues/21177
+    f = tan(pi*(x + S(3)/2))/(3*x)
+    assert f.as_leading_term(x) == -1/(3*pi*x**2)
+
+
+def test_tan_series():
+    assert tan(x).series(x, 0, 9) == \
+        x + x**3/3 + 2*x**5/15 + 17*x**7/315 + O(x**9)
+
+
+def test_tan_rewrite():
+    neg_exp, pos_exp = exp(-x*I), exp(x*I)
+    assert tan(x).rewrite(exp) == I*(neg_exp - pos_exp)/(neg_exp + pos_exp)
+    assert tan(x).rewrite(sin) == 2*sin(x)**2/sin(2*x)
+    assert tan(x).rewrite(cos) == cos(x - S.Pi/2, evaluate=False)/cos(x)
+    assert tan(x).rewrite(cot) == 1/cot(x)
+    assert tan(sinh(x)).rewrite(exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, sinh(3)).n()
+    assert tan(cosh(x)).rewrite(exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, cosh(3)).n()
+    assert tan(tanh(x)).rewrite(exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, tanh(3)).n()
+    assert tan(coth(x)).rewrite(exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, coth(3)).n()
+    assert tan(sin(x)).rewrite(exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, sin(3)).n()
+    assert tan(cos(x)).rewrite(exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, cos(3)).n()
+    assert tan(tan(x)).rewrite(exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, tan(3)).n()
+    assert tan(cot(x)).rewrite(exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, cot(3)).n()
+    assert tan(log(x)).rewrite(Pow) == I*(x**-I - x**I)/(x**-I + x**I)
+    assert tan(x).rewrite(sec) == sec(x)/sec(x - pi/2, evaluate=False)
+    assert tan(x).rewrite(csc) == csc(-x + pi/2, evaluate=False)/csc(x)
+    assert tan(sin(x)).rewrite(Pow) == tan(sin(x))
+
+
+@slow
+def test_tan_rewrite_slow():
+    assert 0 == (cos(pi/34)*tan(pi/34) - sin(pi/34)).rewrite(pow)
+    assert 0 == (cos(pi/17)*tan(pi/17) - sin(pi/17)).rewrite(pow)
+    assert tan(pi/19).rewrite(pow) == tan(pi/19)
+    assert tan(pi*Rational(8, 19)).rewrite(sqrt) == tan(pi*Rational(8, 19))
+    assert tan(pi*Rational(2, 5), evaluate=False).rewrite(sqrt) == sqrt(sqrt(5)/8 +
+               Rational(5, 8))/(Rational(-1, 4) + sqrt(5)/4)
+
+
+def test_tan_subs():
+    assert tan(x).subs(tan(x), y) == y
+    assert tan(x).subs(x, y) == tan(y)
+    assert tan(x).subs(x, S.Pi/2) is zoo
+    assert tan(x).subs(x, S.Pi*Rational(3, 2)) is zoo
+
+
+def test_tan_expansion():
+    assert tan(x + y).expand(trig=True) == ((tan(x) + tan(y))/(1 - tan(x)*tan(y))).expand()
+    assert tan(x - y).expand(trig=True) == ((tan(x) - tan(y))/(1 + tan(x)*tan(y))).expand()
+    assert tan(x + y + z).expand(trig=True) == (
+        (tan(x) + tan(y) + tan(z) - tan(x)*tan(y)*tan(z))/
+        (1 - tan(x)*tan(y) - tan(x)*tan(z) - tan(y)*tan(z))).expand()
+    assert 0 == tan(2*x).expand(trig=True).rewrite(tan).subs([(tan(x), Rational(1, 7))])*24 - 7
+    assert 0 == tan(3*x).expand(trig=True).rewrite(tan).subs([(tan(x), Rational(1, 5))])*55 - 37
+    assert 0 == tan(4*x - pi/4).expand(trig=True).rewrite(tan).subs([(tan(x), Rational(1, 5))])*239 - 1
+    _test_extrig(tan, 2, 2*tan(1)/(1 - tan(1)**2))
+    _test_extrig(tan, 3, (-tan(1)**3 + 3*tan(1))/(1 - 3*tan(1)**2))
+
+
+def test_tan_AccumBounds():
+    assert tan(AccumBounds(-oo, oo)) == AccumBounds(-oo, oo)
+    assert tan(AccumBounds(S.Pi/3, S.Pi*Rational(2, 3))) == AccumBounds(-oo, oo)
+    assert tan(AccumBounds(S.Pi/6, S.Pi/3)) == AccumBounds(tan(S.Pi/6), tan(S.Pi/3))
+
+
+def test_tan_fdiff():
+    assert tan(x).fdiff() == tan(x)**2 + 1
+    raises(ArgumentIndexError, lambda: tan(x).fdiff(2))
+
+
+def test_cot():
+    assert cot(nan) is nan
+
+    assert cot.nargs == FiniteSet(1)
+    assert cot(oo*I) == -I
+    assert cot(-oo*I) == I
+    assert cot(zoo) is nan
+
+    assert cot(0) is zoo
+    assert cot(2*pi) is zoo
+
+    assert cot(acot(x)) == x
+    assert cot(atan(x)) == 1 / x
+    assert cot(asin(x)) == sqrt(1 - x**2) / x
+    assert cot(acos(x)) == x / sqrt(1 - x**2)
+    assert cot(acsc(x)) == sqrt(1 - 1 / x**2) * x
+    assert cot(asec(x)) == 1 / (sqrt(1 - 1 / x**2) * x)
+    assert cot(atan2(y, x)) == x/y
+
+    assert cot(pi*I) == -coth(pi)*I
+    assert cot(-pi*I) == coth(pi)*I
+    assert cot(-2*I) == coth(2)*I
+
+    assert cot(pi) == cot(2*pi) == cot(3*pi)
+    assert cot(-pi) == cot(-2*pi) == cot(-3*pi)
+
+    assert cot(pi/2) == 0
+    assert cot(-pi/2) == 0
+    assert cot(pi*Rational(5, 2)) == 0
+    assert cot(pi*Rational(7, 2)) == 0
+
+    assert cot(pi/3) == 1/sqrt(3)
+    assert cot(pi*Rational(-2, 3)) == 1/sqrt(3)
+
+    assert cot(pi/4) is S.One
+    assert cot(-pi/4) is S.NegativeOne
+    assert cot(pi*Rational(17, 4)) is S.One
+    assert cot(pi*Rational(-3, 4)) is S.One
+
+    assert cot(pi/6) == sqrt(3)
+    assert cot(-pi/6) == -sqrt(3)
+    assert cot(pi*Rational(7, 6)) == sqrt(3)
+    assert cot(pi*Rational(-5, 6)) == sqrt(3)
+
+    assert cot(pi/8) == 1 + sqrt(2)
+    assert cot(pi*Rational(3, 8)) == -1 + sqrt(2)
+    assert cot(pi*Rational(5, 8)) == 1 - sqrt(2)
+    assert cot(pi*Rational(7, 8)) == -1 - sqrt(2)
+
+    assert cot(pi/12) == sqrt(3) + 2
+    assert cot(pi*Rational(5, 12)) == -sqrt(3) + 2
+    assert cot(pi*Rational(7, 12)) == sqrt(3) - 2
+    assert cot(pi*Rational(11, 12)) == -sqrt(3) - 2
+
+    assert cot(pi/24).radsimp() == sqrt(2) + sqrt(3) + 2 + sqrt(6)
+    assert cot(pi*Rational(5, 24)).radsimp() == -sqrt(2) - sqrt(3) + 2 + sqrt(6)
+    assert cot(pi*Rational(7, 24)).radsimp() == -sqrt(2) + sqrt(3) - 2 + sqrt(6)
+    assert cot(pi*Rational(11, 24)).radsimp() == sqrt(2) - sqrt(3) - 2 + sqrt(6)
+    assert cot(pi*Rational(13, 24)).radsimp() == -sqrt(2) + sqrt(3) + 2 - sqrt(6)
+    assert cot(pi*Rational(17, 24)).radsimp() == sqrt(2) - sqrt(3) + 2 - sqrt(6)
+    assert cot(pi*Rational(19, 24)).radsimp() == sqrt(2) + sqrt(3) - 2 - sqrt(6)
+    assert cot(pi*Rational(23, 24)).radsimp() == -sqrt(2) - sqrt(3) - 2 - sqrt(6)
+
+    assert cot(x*I) == -coth(x)*I
+    assert cot(k*pi*I) == -coth(k*pi)*I
+
+    assert cot(r).is_real is None
+    assert cot(r).is_extended_real is True
+
+    assert cot(a).is_algebraic is None
+    assert cot(na).is_algebraic is False
+
+    assert cot(pi*Rational(10, 7)) == cot(pi*Rational(3, 7))
+    assert cot(pi*Rational(11, 7)) == -cot(pi*Rational(3, 7))
+    assert cot(pi*Rational(-11, 7)) == cot(pi*Rational(3, 7))
+
+    assert cot(pi*Rational(39, 34)) == cot(pi*Rational(5, 34))
+    assert cot(pi*Rational(-41, 34)) == -cot(pi*Rational(7, 34))
+
+    assert cot(x).is_finite is None
+    assert cot(r).is_finite is None
+    i = Symbol('i', imaginary=True)
+    assert cot(i).is_finite is True
+
+    assert cot(x).subs(x, 3*pi) is zoo
+
+    # https://github.com/sympy/sympy/issues/21177
+    f = cot(pi*(x + 4))/(3*x)
+    assert f.as_leading_term(x) == 1/(3*pi*x**2)
+
+
+def test_tan_cot_sin_cos_evalf():
+    assert abs((tan(pi*Rational(8, 15))*cos(pi*Rational(8, 15))/sin(pi*Rational(8, 15)) - 1).evalf()) < 1e-14
+    assert abs((cot(pi*Rational(4, 15))*sin(pi*Rational(4, 15))/cos(pi*Rational(4, 15)) - 1).evalf()) < 1e-14
+
+@XFAIL
+def test_tan_cot_sin_cos_ratsimp():
+    assert 1 == (tan(pi*Rational(8, 15))*cos(pi*Rational(8, 15))/sin(pi*Rational(8, 15))).ratsimp()
+    assert 1 == (cot(pi*Rational(4, 15))*sin(pi*Rational(4, 15))/cos(pi*Rational(4, 15))).ratsimp()
+
+
+def test_cot_series():
+    assert cot(x).series(x, 0, 9) == \
+        1/x - x/3 - x**3/45 - 2*x**5/945 - x**7/4725 + O(x**9)
+    # issue 6210
+    assert cot(x**4 + x**5).series(x, 0, 1) == \
+        x**(-4) - 1/x**3 + x**(-2) - 1/x + 1 + O(x)
+    assert cot(pi*(1-x)).series(x, 0, 3) == -1/(pi*x) + pi*x/3 + O(x**3)
+    assert cot(x).taylor_term(0, x) == 1/x
+    assert cot(x).taylor_term(2, x) is S.Zero
+    assert cot(x).taylor_term(3, x) == -x**3/45
+
+
+def test_cot_rewrite():
+    neg_exp, pos_exp = exp(-x*I), exp(x*I)
+    assert cot(x).rewrite(exp) == I*(pos_exp + neg_exp)/(pos_exp - neg_exp)
+    assert cot(x).rewrite(sin) == sin(2*x)/(2*(sin(x)**2))
+    assert cot(x).rewrite(cos) == cos(x)/cos(x - pi/2, evaluate=False)
+    assert cot(x).rewrite(tan) == 1/tan(x)
+    def check(func):
+        z = cot(func(x)).rewrite(exp) - cot(x).rewrite(exp).subs(x, func(x))
+        assert z.rewrite(exp).expand() == 0
+    check(sinh)
+    check(cosh)
+    check(tanh)
+    check(coth)
+    check(sin)
+    check(cos)
+    check(tan)
+    assert cot(log(x)).rewrite(Pow) == -I*(x**-I + x**I)/(x**-I - x**I)
+    assert cot(x).rewrite(sec) == sec(x - pi / 2, evaluate=False) / sec(x)
+    assert cot(x).rewrite(csc) == csc(x) / csc(- x + pi / 2, evaluate=False)
+    assert cot(sin(x)).rewrite(Pow) == cot(sin(x))
+
+
+@slow
+def test_cot_rewrite_slow():
+    assert cot(pi*Rational(4, 34)).rewrite(pow).ratsimp() == \
+        (cos(pi*Rational(4, 34))/sin(pi*Rational(4, 34))).rewrite(pow).ratsimp()
+    assert cot(pi*Rational(4, 17)).rewrite(pow) == \
+        (cos(pi*Rational(4, 17))/sin(pi*Rational(4, 17))).rewrite(pow)
+    assert cot(pi/19).rewrite(pow) == cot(pi/19)
+    assert cot(pi/19).rewrite(sqrt) == cot(pi/19)
+    assert cot(pi*Rational(2, 5), evaluate=False).rewrite(sqrt) == \
+        (Rational(-1, 4) + sqrt(5)/4) / sqrt(sqrt(5)/8 + Rational(5, 8))
+
+
+def test_cot_subs():
+    assert cot(x).subs(cot(x), y) == y
+    assert cot(x).subs(x, y) == cot(y)
+    assert cot(x).subs(x, 0) is zoo
+    assert cot(x).subs(x, S.Pi) is zoo
+
+
+def test_cot_expansion():
+    assert cot(x + y).expand(trig=True).together() == (
+        (cot(x)*cot(y) - 1)/(cot(x) + cot(y)))
+    assert cot(x - y).expand(trig=True).together() == (
+        cot(x)*cot(-y) - 1)/(cot(x) + cot(-y))
+    assert cot(x + y + z).expand(trig=True).together() == (
+        (cot(x)*cot(y)*cot(z) - cot(x) - cot(y) - cot(z))/
+        (-1 + cot(x)*cot(y) + cot(x)*cot(z) + cot(y)*cot(z)))
+    assert cot(3*x).expand(trig=True).together() == (
+        (cot(x)**2 - 3)*cot(x)/(3*cot(x)**2 - 1))
+    assert cot(2*x).expand(trig=True) == cot(x)/2 - 1/(2*cot(x))
+    assert cot(3*x).expand(trig=True).together() == (
+        cot(x)**2 - 3)*cot(x)/(3*cot(x)**2 - 1)
+    assert cot(4*x - pi/4).expand(trig=True).cancel() == (
+        -tan(x)**4 + 4*tan(x)**3 + 6*tan(x)**2 - 4*tan(x) - 1
+        )/(tan(x)**4 + 4*tan(x)**3 - 6*tan(x)**2 - 4*tan(x) + 1)
+    _test_extrig(cot, 2, (-1 + cot(1)**2)/(2*cot(1)))
+    _test_extrig(cot, 3, (-3*cot(1) + cot(1)**3)/(-1 + 3*cot(1)**2))
+
+
+def test_cot_AccumBounds():
+    assert cot(AccumBounds(-oo, oo)) == AccumBounds(-oo, oo)
+    assert cot(AccumBounds(-S.Pi/3, S.Pi/3)) == AccumBounds(-oo, oo)
+    assert cot(AccumBounds(S.Pi/6, S.Pi/3)) == AccumBounds(cot(S.Pi/3), cot(S.Pi/6))
+
+
+def test_cot_fdiff():
+    assert cot(x).fdiff() == -cot(x)**2 - 1
+    raises(ArgumentIndexError, lambda: cot(x).fdiff(2))
+
+
+def test_sinc():
+    assert isinstance(sinc(x), sinc)
+
+    s = Symbol('s', zero=True)
+    assert sinc(s) is S.One
+    assert sinc(S.Infinity) is S.Zero
+    assert sinc(S.NegativeInfinity) is S.Zero
+    assert sinc(S.NaN) is S.NaN
+    assert sinc(S.ComplexInfinity) is S.NaN
+
+    n = Symbol('n', integer=True, nonzero=True)
+    assert sinc(n*pi) is S.Zero
+    assert sinc(-n*pi) is S.Zero
+    assert sinc(pi/2) == 2 / pi
+    assert sinc(-pi/2) == 2 / pi
+    assert sinc(pi*Rational(5, 2)) == 2 / (5*pi)
+    assert sinc(pi*Rational(7, 2)) == -2 / (7*pi)
+
+    assert sinc(-x) == sinc(x)
+
+    assert sinc(x).diff(x) == cos(x)/x - sin(x)/x**2
+    assert sinc(x).diff(x) == (sin(x)/x).diff(x)
+    assert sinc(x).diff(x, x) == (-sin(x) - 2*cos(x)/x + 2*sin(x)/x**2)/x
+    assert sinc(x).diff(x, x) == (sin(x)/x).diff(x, x)
+    assert limit(sinc(x).diff(x), x, 0) == 0
+    assert limit(sinc(x).diff(x, x), x, 0) == -S(1)/3
+
+    # https://github.com/sympy/sympy/issues/11402
+    #
+    # assert sinc(x).diff(x) == Piecewise(((x*cos(x) - sin(x)) / x**2, Ne(x, 0)), (0, True))
+    #
+    # assert sinc(x).diff(x).equals(sinc(x).rewrite(sin).diff(x))
+    #
+    # assert sinc(x).diff(x).subs(x, 0) is S.Zero
+
+    assert sinc(x).series() == 1 - x**2/6 + x**4/120 + O(x**6)
+
+    assert sinc(x).rewrite(jn) == jn(0, x)
+    assert sinc(x).rewrite(sin) == Piecewise((sin(x)/x, Ne(x, 0)), (1, True))
+    assert sinc(pi, evaluate=False).is_zero is True
+    assert sinc(0, evaluate=False).is_zero is False
+    assert sinc(n*pi, evaluate=False).is_zero is True
+    assert sinc(x).is_zero is None
+    xr = Symbol('xr', real=True, nonzero=True)
+    assert sinc(x).is_real is None
+    assert sinc(xr).is_real is True
+    assert sinc(I*xr).is_real is True
+    assert sinc(I*100).is_real is True
+    assert sinc(x).is_finite is None
+    assert sinc(xr).is_finite is True
+
+
+def test_asin():
+    assert asin(nan) is nan
+
+    assert asin.nargs == FiniteSet(1)
+    assert asin(oo) == -I*oo
+    assert asin(-oo) == I*oo
+    assert asin(zoo) is zoo
+
+    # Note: asin(-x) = - asin(x)
+    assert asin(0) == 0
+    assert asin(1) == pi/2
+    assert asin(-1) == -pi/2
+    assert asin(sqrt(3)/2) == pi/3
+    assert asin(-sqrt(3)/2) == -pi/3
+    assert asin(sqrt(2)/2) == pi/4
+    assert asin(-sqrt(2)/2) == -pi/4
+    assert asin(sqrt((5 - sqrt(5))/8)) == pi/5
+    assert asin(-sqrt((5 - sqrt(5))/8)) == -pi/5
+    assert asin(S.Half) == pi/6
+    assert asin(Rational(-1, 2)) == -pi/6
+    assert asin((sqrt(2 - sqrt(2)))/2) == pi/8
+    assert asin(-(sqrt(2 - sqrt(2)))/2) == -pi/8
+    assert asin((sqrt(5) - 1)/4) == pi/10
+    assert asin(-(sqrt(5) - 1)/4) == -pi/10
+    assert asin((sqrt(3) - 1)/sqrt(2**3)) == pi/12
+    assert asin(-(sqrt(3) - 1)/sqrt(2**3)) == -pi/12
+
+    # check round-trip for exact values:
+    for d in [5, 6, 8, 10, 12]:
+        for n in range(-(d//2), d//2 + 1):
+            if gcd(n, d) == 1:
+                assert asin(sin(n*pi/d)) == n*pi/d
+
+    assert asin(x).diff(x) == 1/sqrt(1 - x**2)
+
+    assert asin(0.2, evaluate=False).is_real is True
+    assert asin(-2).is_real is False
+    assert asin(r).is_real is None
+
+    assert asin(-2*I) == -I*asinh(2)
+
+    assert asin(Rational(1, 7), evaluate=False).is_positive is True
+    assert asin(Rational(-1, 7), evaluate=False).is_positive is False
+    assert asin(p).is_positive is None
+    assert asin(sin(Rational(7, 2))) == Rational(-7, 2) + pi
+    assert asin(sin(Rational(-7, 4))) == Rational(7, 4) - pi
+    assert unchanged(asin, cos(x))
+
+
+def test_asin_series():
+    assert asin(x).series(x, 0, 9) == \
+        x + x**3/6 + 3*x**5/40 + 5*x**7/112 + O(x**9)
+    t5 = asin(x).taylor_term(5, x)
+    assert t5 == 3*x**5/40
+    assert asin(x).taylor_term(7, x, t5, 0) == 5*x**7/112
+
+
+def test_asin_leading_term():
+    assert asin(x).as_leading_term(x) == x
+    # Tests concerning branch points
+    assert asin(x + 1).as_leading_term(x) == pi/2
+    assert asin(x - 1).as_leading_term(x) == -pi/2
+    assert asin(1/x).as_leading_term(x, cdir=1) == I*log(x) + pi/2 - I*log(2)
+    assert asin(1/x).as_leading_term(x, cdir=-1) == -I*log(x) - 3*pi/2 + I*log(2)
+    # Tests concerning points lying on branch cuts
+    assert asin(I*x + 2).as_leading_term(x, cdir=1) == pi - asin(2)
+    assert asin(-I*x + 2).as_leading_term(x, cdir=1) == asin(2)
+    assert asin(I*x - 2).as_leading_term(x, cdir=1) == -asin(2)
+    assert asin(-I*x - 2).as_leading_term(x, cdir=1) == -pi + asin(2)
+    # Tests concerning im(ndir) == 0
+    assert asin(-I*x**2 + x - 2).as_leading_term(x, cdir=1) == -pi/2 + I*log(2 - sqrt(3))
+    assert asin(-I*x**2 + x - 2).as_leading_term(x, cdir=-1) == -pi/2 + I*log(2 - sqrt(3))
+
+
+def test_asin_rewrite():
+    assert asin(x).rewrite(log) == -I*log(I*x + sqrt(1 - x**2))
+    assert asin(x).rewrite(atan) == 2*atan(x/(1 + sqrt(1 - x**2)))
+    assert asin(x).rewrite(acos) == S.Pi/2 - acos(x)
+    assert asin(x).rewrite(acot) == 2*acot((sqrt(-x**2 + 1) + 1)/x)
+    assert asin(x).rewrite(asec) == -asec(1/x) + pi/2
+    assert asin(x).rewrite(acsc) == acsc(1/x)
+
+
+def test_asin_fdiff():
+    assert asin(x).fdiff() == 1/sqrt(1 - x**2)
+    raises(ArgumentIndexError, lambda: asin(x).fdiff(2))
+
+
+def test_acos():
+    assert acos(nan) is nan
+    assert acos(zoo) is zoo
+
+    assert acos.nargs == FiniteSet(1)
+    assert acos(oo) == I*oo
+    assert acos(-oo) == -I*oo
+
+    # Note: acos(-x) = pi - acos(x)
+    assert acos(0) == pi/2
+    assert acos(S.Half) == pi/3
+    assert acos(Rational(-1, 2)) == pi*Rational(2, 3)
+    assert acos(1) == 0
+    assert acos(-1) == pi
+    assert acos(sqrt(2)/2) == pi/4
+    assert acos(-sqrt(2)/2) == pi*Rational(3, 4)
+
+    # check round-trip for exact values:
+    for d in [5, 6, 8, 10, 12]:
+        for num in range(d):
+            if gcd(num, d) == 1:
+                assert acos(cos(num*pi/d)) == num*pi/d
+
+    assert acos(2*I) == pi/2 - asin(2*I)
+
+    assert acos(x).diff(x) == -1/sqrt(1 - x**2)
+
+    assert acos(0.2).is_real is True
+    assert acos(-2).is_real is False
+    assert acos(r).is_real is None
+
+    assert acos(Rational(1, 7), evaluate=False).is_positive is True
+    assert acos(Rational(-1, 7), evaluate=False).is_positive is True
+    assert acos(Rational(3, 2), evaluate=False).is_positive is False
+    assert acos(p).is_positive is None
+
+    assert acos(2 + p).conjugate() != acos(10 + p)
+    assert acos(-3 + n).conjugate() != acos(-3 + n)
+    assert acos(Rational(1, 3)).conjugate() == acos(Rational(1, 3))
+    assert acos(Rational(-1, 3)).conjugate() == acos(Rational(-1, 3))
+    assert acos(p + n*I).conjugate() == acos(p - n*I)
+    assert acos(z).conjugate() != acos(conjugate(z))
+
+
+def test_acos_leading_term():
+    assert acos(x).as_leading_term(x) == pi/2
+    # Tests concerning branch points
+    assert acos(x + 1).as_leading_term(x) == sqrt(2)*sqrt(-x)
+    assert acos(x - 1).as_leading_term(x) == pi
+    assert acos(1/x).as_leading_term(x, cdir=1) == -I*log(x) + I*log(2)
+    assert acos(1/x).as_leading_term(x, cdir=-1) == I*log(x) + 2*pi - I*log(2)
+    # Tests concerning points lying on branch cuts
+    assert acos(I*x + 2).as_leading_term(x, cdir=1) == -acos(2)
+    assert acos(-I*x + 2).as_leading_term(x, cdir=1) == acos(2)
+    assert acos(I*x - 2).as_leading_term(x, cdir=1) == acos(-2)
+    assert acos(-I*x - 2).as_leading_term(x, cdir=1) == 2*pi - acos(-2)
+    # Tests concerning im(ndir) == 0
+    assert acos(-I*x**2 + x - 2).as_leading_term(x, cdir=1) == pi + I*log(sqrt(3) + 2)
+    assert acos(-I*x**2 + x - 2).as_leading_term(x, cdir=-1) == pi + I*log(sqrt(3) + 2)
+
+
+def test_acos_series():
+    assert acos(x).series(x, 0, 8) == \
+        pi/2 - x - x**3/6 - 3*x**5/40 - 5*x**7/112 + O(x**8)
+    assert acos(x).series(x, 0, 8) == pi/2 - asin(x).series(x, 0, 8)
+    t5 = acos(x).taylor_term(5, x)
+    assert t5 == -3*x**5/40
+    assert acos(x).taylor_term(7, x, t5, 0) == -5*x**7/112
+    assert acos(x).taylor_term(0, x) == pi/2
+    assert acos(x).taylor_term(2, x) is S.Zero
+
+
+def test_acos_rewrite():
+    assert acos(x).rewrite(log) == pi/2 + I*log(I*x + sqrt(1 - x**2))
+    assert acos(x).rewrite(atan) == pi*(-x*sqrt(x**(-2)) + 1)/2 + atan(sqrt(1 - x**2)/x)
+    assert acos(0).rewrite(atan) == S.Pi/2
+    assert acos(0.5).rewrite(atan) == acos(0.5).rewrite(log)
+    assert acos(x).rewrite(asin) == S.Pi/2 - asin(x)
+    assert acos(x).rewrite(acot) == -2*acot((sqrt(-x**2 + 1) + 1)/x) + pi/2
+    assert acos(x).rewrite(asec) == asec(1/x)
+    assert acos(x).rewrite(acsc) == -acsc(1/x) + pi/2
+
+
+def test_acos_fdiff():
+    assert acos(x).fdiff() == -1/sqrt(1 - x**2)
+    raises(ArgumentIndexError, lambda: acos(x).fdiff(2))
+
+
+def test_atan():
+    assert atan(nan) is nan
+
+    assert atan.nargs == FiniteSet(1)
+    assert atan(oo) == pi/2
+    assert atan(-oo) == -pi/2
+    assert atan(zoo) == AccumBounds(-pi/2, pi/2)
+
+    assert atan(0) == 0
+    assert atan(1) == pi/4
+    assert atan(sqrt(3)) == pi/3
+    assert atan(-(1 + sqrt(2))) == pi*Rational(-3, 8)
+    assert atan(sqrt(5 - 2 * sqrt(5))) == pi/5
+    assert atan(-sqrt(1 - 2 * sqrt(5)/ 5)) == -pi/10
+    assert atan(sqrt(1 + 2 * sqrt(5) / 5)) == pi*Rational(3, 10)
+    assert atan(-2 + sqrt(3)) == -pi/12
+    assert atan(2 + sqrt(3)) == pi*Rational(5, 12)
+    assert atan(-2 - sqrt(3)) == pi*Rational(-5, 12)
+
+    # check round-trip for exact values:
+    for d in [5, 6, 8, 10, 12]:
+        for num in range(-(d//2), d//2 + 1):
+            if gcd(num, d) == 1:
+                assert atan(tan(num*pi/d)) == num*pi/d
+
+    assert atan(oo) == pi/2
+    assert atan(x).diff(x) == 1/(1 + x**2)
+
+    assert atan(r).is_real is True
+
+    assert atan(-2*I) == -I*atanh(2)
+    assert unchanged(atan, cot(x))
+    assert atan(cot(Rational(1, 4))) == Rational(-1, 4) + pi/2
+    assert acot(Rational(1, 4)).is_rational is False
+
+    for s in (x, p, n, np, nn, nz, ep, en, enp, enn, enz):
+        if s.is_real or s.is_extended_real is None:
+            assert s.is_nonzero is atan(s).is_nonzero
+            assert s.is_positive is atan(s).is_positive
+            assert s.is_negative is atan(s).is_negative
+            assert s.is_nonpositive is atan(s).is_nonpositive
+            assert s.is_nonnegative is atan(s).is_nonnegative
+        else:
+            assert s.is_extended_nonzero is atan(s).is_nonzero
+            assert s.is_extended_positive is atan(s).is_positive
+            assert s.is_extended_negative is atan(s).is_negative
+            assert s.is_extended_nonpositive is atan(s).is_nonpositive
+            assert s.is_extended_nonnegative is atan(s).is_nonnegative
+        assert s.is_extended_nonzero is atan(s).is_extended_nonzero
+        assert s.is_extended_positive is atan(s).is_extended_positive
+        assert s.is_extended_negative is atan(s).is_extended_negative
+        assert s.is_extended_nonpositive is atan(s).is_extended_nonpositive
+        assert s.is_extended_nonnegative is atan(s).is_extended_nonnegative
+
+
+def test_atan_rewrite():
+    assert atan(x).rewrite(log) == I*(log(1 - I*x)-log(1 + I*x))/2
+    assert atan(x).rewrite(asin) == (-asin(1/sqrt(x**2 + 1)) + pi/2)*sqrt(x**2)/x
+    assert atan(x).rewrite(acos) == sqrt(x**2)*acos(1/sqrt(x**2 + 1))/x
+    assert atan(x).rewrite(acot) == acot(1/x)
+    assert atan(x).rewrite(asec) == sqrt(x**2)*asec(sqrt(x**2 + 1))/x
+    assert atan(x).rewrite(acsc) == (-acsc(sqrt(x**2 + 1)) + pi/2)*sqrt(x**2)/x
+
+    assert atan(-5*I).evalf() == atan(x).rewrite(log).evalf(subs={x:-5*I})
+    assert atan(5*I).evalf() == atan(x).rewrite(log).evalf(subs={x:5*I})
+
+
+def test_atan_fdiff():
+    assert atan(x).fdiff() == 1/(x**2 + 1)
+    raises(ArgumentIndexError, lambda: atan(x).fdiff(2))
+
+
+def test_atan_leading_term():
+    assert atan(x).as_leading_term(x) == x
+    assert atan(1/x).as_leading_term(x, cdir=1) == pi/2
+    assert atan(1/x).as_leading_term(x, cdir=-1) == -pi/2
+    # Tests concerning branch points
+    assert atan(x + I).as_leading_term(x, cdir=1) == -I*log(x)/2 + pi/4 + I*log(2)/2
+    assert atan(x + I).as_leading_term(x, cdir=-1) == -I*log(x)/2 - 3*pi/4 + I*log(2)/2
+    assert atan(x - I).as_leading_term(x, cdir=1) == I*log(x)/2 + pi/4 - I*log(2)/2
+    assert atan(x - I).as_leading_term(x, cdir=-1) == I*log(x)/2 + pi/4 - I*log(2)/2
+    # Tests concerning points lying on branch cuts
+    assert atan(x + 2*I).as_leading_term(x, cdir=1) == I*atanh(2)
+    assert atan(x + 2*I).as_leading_term(x, cdir=-1) == -pi + I*atanh(2)
+    assert atan(x - 2*I).as_leading_term(x, cdir=1) == pi - I*atanh(2)
+    assert atan(x - 2*I).as_leading_term(x, cdir=-1) == -I*atanh(2)
+    # Tests concerning re(ndir) == 0
+    assert atan(2*I - I*x - x**2).as_leading_term(x, cdir=1) == -pi/2 + I*log(3)/2
+    assert atan(2*I - I*x - x**2).as_leading_term(x, cdir=-1) == -pi/2 + I*log(3)/2
+
+
+def test_atan2():
+    assert atan2.nargs == FiniteSet(2)
+    assert atan2(0, 0) is S.NaN
+    assert atan2(0, 1) == 0
+    assert atan2(1, 1) == pi/4
+    assert atan2(1, 0) == pi/2
+    assert atan2(1, -1) == pi*Rational(3, 4)
+    assert atan2(0, -1) == pi
+    assert atan2(-1, -1) == pi*Rational(-3, 4)
+    assert atan2(-1, 0) == -pi/2
+    assert atan2(-1, 1) == -pi/4
+    i = symbols('i', imaginary=True)
+    r = symbols('r', real=True)
+    eq = atan2(r, i)
+    ans = -I*log((i + I*r)/sqrt(i**2 + r**2))
+    reps = ((r, 2), (i, I))
+    assert eq.subs(reps) == ans.subs(reps)
+
+    x = Symbol('x', negative=True)
+    y = Symbol('y', negative=True)
+    assert atan2(y, x) == atan(y/x) - pi
+    y = Symbol('y', nonnegative=True)
+    assert atan2(y, x) == atan(y/x) + pi
+    y = Symbol('y')
+    assert atan2(y, x) == atan2(y, x, evaluate=False)
+
+    u = Symbol("u", positive=True)
+    assert atan2(0, u) == 0
+    u = Symbol("u", negative=True)
+    assert atan2(0, u) == pi
+
+    assert atan2(y, oo) ==  0
+    assert atan2(y, -oo)==  2*pi*Heaviside(re(y), S.Half) - pi
+
+    assert atan2(y, x).rewrite(log) == -I*log((x + I*y)/sqrt(x**2 + y**2))
+    assert atan2(0, 0) is S.NaN
+
+    ex = atan2(y, x) - arg(x + I*y)
+    assert ex.subs({x:2, y:3}).rewrite(arg) == 0
+    assert ex.subs({x:2, y:3*I}).rewrite(arg) == -pi - I*log(sqrt(5)*I/5)
+    assert ex.subs({x:2*I, y:3}).rewrite(arg) == -pi/2 - I*log(sqrt(5)*I)
+    assert ex.subs({x:2*I, y:3*I}).rewrite(arg) == -pi + atan(Rational(2, 3)) + atan(Rational(3, 2))
+    i = symbols('i', imaginary=True)
+    r = symbols('r', real=True)
+    e = atan2(i, r)
+    rewrite = e.rewrite(arg)
+    reps = {i: I, r: -2}
+    assert rewrite == -I*log(abs(I*i + r)/sqrt(abs(i**2 + r**2))) + arg((I*i + r)/sqrt(i**2 + r**2))
+    assert (e - rewrite).subs(reps).equals(0)
+
+    assert atan2(0, x).rewrite(atan) == Piecewise((pi, re(x) < 0),
+                                            (0, Ne(x, 0)),
+                                            (nan, True))
+    assert atan2(0, r).rewrite(atan) == Piecewise((pi, r < 0), (0, Ne(r, 0)), (S.NaN, True))
+    assert atan2(0, i),rewrite(atan) == 0
+    assert atan2(0, r + i).rewrite(atan) == Piecewise((pi, r < 0), (0, True))
+
+    assert atan2(y, x).rewrite(atan) == Piecewise(
+            (2*atan(y/(x + sqrt(x**2 + y**2))), Ne(y, 0)),
+            (pi, re(x) < 0),
+            (0, (re(x) > 0) | Ne(im(x), 0)),
+            (nan, True))
+    assert conjugate(atan2(x, y)) == atan2(conjugate(x), conjugate(y))
+
+    assert diff(atan2(y, x), x) == -y/(x**2 + y**2)
+    assert diff(atan2(y, x), y) == x/(x**2 + y**2)
+
+    assert simplify(diff(atan2(y, x).rewrite(log), x)) == -y/(x**2 + y**2)
+    assert simplify(diff(atan2(y, x).rewrite(log), y)) ==  x/(x**2 + y**2)
+
+    assert str(atan2(1, 2).evalf(5)) == '0.46365'
+    raises(ArgumentIndexError, lambda: atan2(x, y).fdiff(3))
+
+def test_issue_17461():
+    class A(Symbol):
+        is_extended_real = True
+
+        def _eval_evalf(self, prec):
+            return Float(5.0)
+
+    x = A('X')
+    y = A('Y')
+    assert abs(atan2(x, y).evalf() - 0.785398163397448) <= 1e-10
+
+def test_acot():
+    assert acot(nan) is nan
+
+    assert acot.nargs == FiniteSet(1)
+    assert acot(-oo) == 0
+    assert acot(oo) == 0
+    assert acot(zoo) == 0
+    assert acot(1) == pi/4
+    assert acot(0) == pi/2
+    assert acot(sqrt(3)/3) == pi/3
+    assert acot(1/sqrt(3)) == pi/3
+    assert acot(-1/sqrt(3)) == -pi/3
+    assert acot(x).diff(x) == -1/(1 + x**2)
+
+    assert acot(r).is_extended_real is True
+
+    assert acot(I*pi) == -I*acoth(pi)
+    assert acot(-2*I) == I*acoth(2)
+    assert acot(x).is_positive is None
+    assert acot(n).is_positive is False
+    assert acot(p).is_positive is True
+    assert acot(I).is_positive is False
+    assert acot(Rational(1, 4)).is_rational is False
+    assert unchanged(acot, cot(x))
+    assert unchanged(acot, tan(x))
+    assert acot(cot(Rational(1, 4))) == Rational(1, 4)
+    assert acot(tan(Rational(-1, 4))) == Rational(1, 4) - pi/2
+
+
+def test_acot_rewrite():
+    assert acot(x).rewrite(log) == I*(log(1 - I/x)-log(1 + I/x))/2
+    assert acot(x).rewrite(asin) == x*(-asin(sqrt(-x**2)/sqrt(-x**2 - 1)) + pi/2)*sqrt(x**(-2))
+    assert acot(x).rewrite(acos) == x*sqrt(x**(-2))*acos(sqrt(-x**2)/sqrt(-x**2 - 1))
+    assert acot(x).rewrite(atan) == atan(1/x)
+    assert acot(x).rewrite(asec) == x*sqrt(x**(-2))*asec(sqrt((x**2 + 1)/x**2))
+    assert acot(x).rewrite(acsc) == x*(-acsc(sqrt((x**2 + 1)/x**2)) + pi/2)*sqrt(x**(-2))
+
+    assert acot(-I/5).evalf() == acot(x).rewrite(log).evalf(subs={x:-I/5})
+    assert acot(I/5).evalf() == acot(x).rewrite(log).evalf(subs={x:I/5})
+
+
+def test_acot_fdiff():
+    assert acot(x).fdiff() == -1/(x**2 + 1)
+    raises(ArgumentIndexError, lambda: acot(x).fdiff(2))
+
+def test_acot_leading_term():
+    assert acot(1/x).as_leading_term(x) == x
+    # Tests concerning branch points
+    assert acot(x + I).as_leading_term(x, cdir=1) == I*log(x)/2 + pi/4 - I*log(2)/2
+    assert acot(x + I).as_leading_term(x, cdir=-1) == I*log(x)/2 + pi/4 - I*log(2)/2
+    assert acot(x - I).as_leading_term(x, cdir=1) == -I*log(x)/2 + pi/4 + I*log(2)/2
+    assert acot(x - I).as_leading_term(x, cdir=-1) == -I*log(x)/2 - 3*pi/4 + I*log(2)/2
+    # Tests concerning points lying on branch cuts
+    assert acot(x).as_leading_term(x, cdir=1) == pi/2
+    assert acot(x).as_leading_term(x, cdir=-1) == -pi/2
+    assert acot(x + I/2).as_leading_term(x, cdir=1) == pi - I*acoth(S(1)/2)
+    assert acot(x + I/2).as_leading_term(x, cdir=-1) == -I*acoth(S(1)/2)
+    assert acot(x - I/2).as_leading_term(x, cdir=1) == I*acoth(S(1)/2)
+    assert acot(x - I/2).as_leading_term(x, cdir=-1) == -pi + I*acoth(S(1)/2)
+    # Tests concerning re(ndir) == 0
+    assert acot(I/2 - I*x - x**2).as_leading_term(x, cdir=1) == -pi/2 - I*log(3)/2
+    assert acot(I/2 - I*x - x**2).as_leading_term(x, cdir=-1) == -pi/2 - I*log(3)/2
+
+
+def test_attributes():
+    assert sin(x).args == (x,)
+
+
+def test_sincos_rewrite():
+    assert sin(pi/2 - x) == cos(x)
+    assert sin(pi - x) == sin(x)
+    assert cos(pi/2 - x) == sin(x)
+    assert cos(pi - x) == -cos(x)
+
+
+def _check_even_rewrite(func, arg):
+    """Checks that the expr has been rewritten using f(-x) -> f(x)
+    arg : -x
+    """
+    return func(arg).args[0] == -arg
+
+
+def _check_odd_rewrite(func, arg):
+    """Checks that the expr has been rewritten using f(-x) -> -f(x)
+    arg : -x
+    """
+    return func(arg).func.is_Mul
+
+
+def _check_no_rewrite(func, arg):
+    """Checks that the expr is not rewritten"""
+    return func(arg).args[0] == arg
+
+
+def test_evenodd_rewrite():
+    a = cos(2)  # negative
+    b = sin(1)  # positive
+    even = [cos]
+    odd = [sin, tan, cot, asin, atan, acot]
+    with_minus = [-1, -2**1024 * E, -pi/105, -x*y, -x - y]
+    for func in even:
+        for expr in with_minus:
+            assert _check_even_rewrite(func, expr)
+        assert _check_no_rewrite(func, a*b)
+        assert func(
+            x - y) == func(y - x)  # it doesn't matter which form is canonical
+    for func in odd:
+        for expr in with_minus:
+            assert _check_odd_rewrite(func, expr)
+        assert _check_no_rewrite(func, a*b)
+        assert func(
+            x - y) == -func(y - x)  # it doesn't matter which form is canonical
+
+
+def test_as_leading_term_issue_5272():
+    assert sin(x).as_leading_term(x) == x
+    assert cos(x).as_leading_term(x) == 1
+    assert tan(x).as_leading_term(x) == x
+    assert cot(x).as_leading_term(x) == 1/x
+
+
+def test_leading_terms():
+    assert sin(1/x).as_leading_term(x) == AccumBounds(-1, 1)
+    assert sin(S.Half).as_leading_term(x) == sin(S.Half)
+    assert cos(1/x).as_leading_term(x) == AccumBounds(-1, 1)
+    assert cos(S.Half).as_leading_term(x) == cos(S.Half)
+    assert sec(1/x).as_leading_term(x) == AccumBounds(S.NegativeInfinity, S.Infinity)
+    assert csc(1/x).as_leading_term(x) == AccumBounds(S.NegativeInfinity, S.Infinity)
+    assert tan(1/x).as_leading_term(x) == AccumBounds(S.NegativeInfinity, S.Infinity)
+    assert cot(1/x).as_leading_term(x) == AccumBounds(S.NegativeInfinity, S.Infinity)
+
+    # https://github.com/sympy/sympy/issues/21038
+    f = sin(pi*(x + 4))/(3*x)
+    assert f.as_leading_term(x) == pi/3
+
+
+def test_atan2_expansion():
+    assert cancel(atan2(x**2, x + 1).diff(x) - atan(x**2/(x + 1)).diff(x)) == 0
+    assert cancel(atan(y/x).series(y, 0, 5) - atan2(y, x).series(y, 0, 5)
+                  + atan2(0, x) - atan(0)) == O(y**5)
+    assert cancel(atan(y/x).series(x, 1, 4) - atan2(y, x).series(x, 1, 4)
+                  + atan2(y, 1) - atan(y)) == O((x - 1)**4, (x, 1))
+    assert cancel(atan((y + x)/x).series(x, 1, 3) - atan2(y + x, x).series(x, 1, 3)
+                  + atan2(1 + y, 1) - atan(1 + y)) == O((x - 1)**3, (x, 1))
+    assert Matrix([atan2(y, x)]).jacobian([y, x]) == \
+        Matrix([[x/(y**2 + x**2), -y/(y**2 + x**2)]])
+
+
+def test_aseries():
+    def t(n, v, d, e):
+        assert abs(
+            n(1/v).evalf() - n(1/x).series(x, dir=d).removeO().subs(x, v)) < e
+    t(atan, 0.1, '+', 1e-5)
+    t(atan, -0.1, '-', 1e-5)
+    t(acot, 0.1, '+', 1e-5)
+    t(acot, -0.1, '-', 1e-5)
+
+
+def test_issue_4420():
+    i = Symbol('i', integer=True)
+    e = Symbol('e', even=True)
+    o = Symbol('o', odd=True)
+
+    # unknown parity for variable
+    assert cos(4*i*pi) == 1
+    assert sin(4*i*pi) == 0
+    assert tan(4*i*pi) == 0
+    assert cot(4*i*pi) is zoo
+
+    assert cos(3*i*pi) == cos(pi*i)  # +/-1
+    assert sin(3*i*pi) == 0
+    assert tan(3*i*pi) == 0
+    assert cot(3*i*pi) is zoo
+
+    assert cos(4.0*i*pi) == 1
+    assert sin(4.0*i*pi) == 0
+    assert tan(4.0*i*pi) == 0
+    assert cot(4.0*i*pi) is zoo
+
+    assert cos(3.0*i*pi) == cos(pi*i)  # +/-1
+    assert sin(3.0*i*pi) == 0
+    assert tan(3.0*i*pi) == 0
+    assert cot(3.0*i*pi) is zoo
+
+    assert cos(4.5*i*pi) == cos(0.5*pi*i)
+    assert sin(4.5*i*pi) == sin(0.5*pi*i)
+    assert tan(4.5*i*pi) == tan(0.5*pi*i)
+    assert cot(4.5*i*pi) == cot(0.5*pi*i)
+
+    # parity of variable is known
+    assert cos(4*e*pi) == 1
+    assert sin(4*e*pi) == 0
+    assert tan(4*e*pi) == 0
+    assert cot(4*e*pi) is zoo
+
+    assert cos(3*e*pi) == 1
+    assert sin(3*e*pi) == 0
+    assert tan(3*e*pi) == 0
+    assert cot(3*e*pi) is zoo
+
+    assert cos(4.0*e*pi) == 1
+    assert sin(4.0*e*pi) == 0
+    assert tan(4.0*e*pi) == 0
+    assert cot(4.0*e*pi) is zoo
+
+    assert cos(3.0*e*pi) == 1
+    assert sin(3.0*e*pi) == 0
+    assert tan(3.0*e*pi) == 0
+    assert cot(3.0*e*pi) is zoo
+
+    assert cos(4.5*e*pi) == cos(0.5*pi*e)
+    assert sin(4.5*e*pi) == sin(0.5*pi*e)
+    assert tan(4.5*e*pi) == tan(0.5*pi*e)
+    assert cot(4.5*e*pi) == cot(0.5*pi*e)
+
+    assert cos(4*o*pi) == 1
+    assert sin(4*o*pi) == 0
+    assert tan(4*o*pi) == 0
+    assert cot(4*o*pi) is zoo
+
+    assert cos(3*o*pi) == -1
+    assert sin(3*o*pi) == 0
+    assert tan(3*o*pi) == 0
+    assert cot(3*o*pi) is zoo
+
+    assert cos(4.0*o*pi) == 1
+    assert sin(4.0*o*pi) == 0
+    assert tan(4.0*o*pi) == 0
+    assert cot(4.0*o*pi) is zoo
+
+    assert cos(3.0*o*pi) == -1
+    assert sin(3.0*o*pi) == 0
+    assert tan(3.0*o*pi) == 0
+    assert cot(3.0*o*pi) is zoo
+
+    assert cos(4.5*o*pi) == cos(0.5*pi*o)
+    assert sin(4.5*o*pi) == sin(0.5*pi*o)
+    assert tan(4.5*o*pi) == tan(0.5*pi*o)
+    assert cot(4.5*o*pi) == cot(0.5*pi*o)
+
+    # x could be imaginary
+    assert cos(4*x*pi) == cos(4*pi*x)
+    assert sin(4*x*pi) == sin(4*pi*x)
+    assert tan(4*x*pi) == tan(4*pi*x)
+    assert cot(4*x*pi) == cot(4*pi*x)
+
+    assert cos(3*x*pi) == cos(3*pi*x)
+    assert sin(3*x*pi) == sin(3*pi*x)
+    assert tan(3*x*pi) == tan(3*pi*x)
+    assert cot(3*x*pi) == cot(3*pi*x)
+
+    assert cos(4.0*x*pi) == cos(4.0*pi*x)
+    assert sin(4.0*x*pi) == sin(4.0*pi*x)
+    assert tan(4.0*x*pi) == tan(4.0*pi*x)
+    assert cot(4.0*x*pi) == cot(4.0*pi*x)
+
+    assert cos(3.0*x*pi) == cos(3.0*pi*x)
+    assert sin(3.0*x*pi) == sin(3.0*pi*x)
+    assert tan(3.0*x*pi) == tan(3.0*pi*x)
+    assert cot(3.0*x*pi) == cot(3.0*pi*x)
+
+    assert cos(4.5*x*pi) == cos(4.5*pi*x)
+    assert sin(4.5*x*pi) == sin(4.5*pi*x)
+    assert tan(4.5*x*pi) == tan(4.5*pi*x)
+    assert cot(4.5*x*pi) == cot(4.5*pi*x)
+
+
+def test_inverses():
+    raises(AttributeError, lambda: sin(x).inverse())
+    raises(AttributeError, lambda: cos(x).inverse())
+    assert tan(x).inverse() == atan
+    assert cot(x).inverse() == acot
+    raises(AttributeError, lambda: csc(x).inverse())
+    raises(AttributeError, lambda: sec(x).inverse())
+    assert asin(x).inverse() == sin
+    assert acos(x).inverse() == cos
+    assert atan(x).inverse() == tan
+    assert acot(x).inverse() == cot
+
+
+def test_real_imag():
+    a, b = symbols('a b', real=True)
+    z = a + b*I
+    for deep in [True, False]:
+        assert sin(
+            z).as_real_imag(deep=deep) == (sin(a)*cosh(b), cos(a)*sinh(b))
+        assert cos(
+            z).as_real_imag(deep=deep) == (cos(a)*cosh(b), -sin(a)*sinh(b))
+        assert tan(z).as_real_imag(deep=deep) == (sin(2*a)/(cos(2*a) +
+            cosh(2*b)), sinh(2*b)/(cos(2*a) + cosh(2*b)))
+        assert cot(z).as_real_imag(deep=deep) == (-sin(2*a)/(cos(2*a) -
+            cosh(2*b)), sinh(2*b)/(cos(2*a) - cosh(2*b)))
+        assert sin(a).as_real_imag(deep=deep) == (sin(a), 0)
+        assert cos(a).as_real_imag(deep=deep) == (cos(a), 0)
+        assert tan(a).as_real_imag(deep=deep) == (tan(a), 0)
+        assert cot(a).as_real_imag(deep=deep) == (cot(a), 0)
+
+
+@XFAIL
+def test_sin_cos_with_infinity():
+    # Test for issue 5196
+    # https://github.com/sympy/sympy/issues/5196
+    assert sin(oo) is S.NaN
+    assert cos(oo) is S.NaN
+
+
+@slow
+def test_sincos_rewrite_sqrt():
+    # equivalent to testing rewrite(pow)
+    for p in [1, 3, 5, 17]:
+        for t in [1, 8]:
+            n = t*p
+            # The vertices `exp(i*pi/n)` of a regular `n`-gon can
+            # be expressed by means of nested square roots if and
+            # only if `n` is a product of Fermat primes, `p`, and
+            # powers of 2, `t'. The code aims to check all vertices
+            # not belonging to an `m`-gon for `m < n`(`gcd(i, n) == 1`).
+            # For large `n` this makes the test too slow, therefore
+            # the vertices are limited to those of index `i < 10`.
+            for i in range(1, min((n + 1)//2 + 1, 10)):
+                if 1 == gcd(i, n):
+                    x = i*pi/n
+                    s1 = sin(x).rewrite(sqrt)
+                    c1 = cos(x).rewrite(sqrt)
+                    assert not s1.has(cos, sin), "fails for %d*pi/%d" % (i, n)
+                    assert not c1.has(cos, sin), "fails for %d*pi/%d" % (i, n)
+                    assert 1e-3 > abs(sin(x.evalf(5)) - s1.evalf(2)), "fails for %d*pi/%d" % (i, n)
+                    assert 1e-3 > abs(cos(x.evalf(5)) - c1.evalf(2)), "fails for %d*pi/%d" % (i, n)
+    assert cos(pi/14).rewrite(sqrt) == sqrt(cos(pi/7)/2 + S.Half)
+    assert cos(pi*Rational(-15, 2)/11, evaluate=False).rewrite(
+        sqrt) == -sqrt(-cos(pi*Rational(4, 11))/2 + S.Half)
+    assert cos(Mul(2, pi, S.Half, evaluate=False), evaluate=False).rewrite(
+        sqrt) == -1
+    e = cos(pi/3/17)  # don't use pi/15 since that is caught at instantiation
+    a = (
+        -3*sqrt(-sqrt(17) + 17)*sqrt(sqrt(17) + 17)/64 -
+        3*sqrt(34)*sqrt(sqrt(17) + 17)/128 - sqrt(sqrt(17) +
+        17)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) + 17)
+        + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/64 - sqrt(-sqrt(17)
+        + 17)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) +
+        17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/128 - Rational(1, 32) +
+        sqrt(2)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) +
+        17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/64 +
+        3*sqrt(2)*sqrt(sqrt(17) + 17)/128 + sqrt(34)*sqrt(-sqrt(17) + 17)/128
+        + 13*sqrt(2)*sqrt(-sqrt(17) + 17)/128 + sqrt(17)*sqrt(-sqrt(17) +
+        17)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) + 17)
+        + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/128 + 5*sqrt(17)/32
+        + sqrt(3)*sqrt(-sqrt(2)*sqrt(sqrt(17) + 17)*sqrt(sqrt(17)/32 +
+        sqrt(2)*sqrt(-sqrt(17) + 17)/32 +
+        sqrt(2)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) +
+        17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/32 + Rational(15, 32))/8 -
+        5*sqrt(2)*sqrt(sqrt(17)/32 + sqrt(2)*sqrt(-sqrt(17) + 17)/32 +
+        sqrt(2)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) +
+        17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/32 +
+        Rational(15, 32))*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) +
+        17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/64 -
+        3*sqrt(2)*sqrt(-sqrt(17) + 17)*sqrt(sqrt(17)/32 +
+        sqrt(2)*sqrt(-sqrt(17) + 17)/32 +
+        sqrt(2)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) +
+        17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/32 + Rational(15, 32))/32
+        + sqrt(34)*sqrt(sqrt(17)/32 + sqrt(2)*sqrt(-sqrt(17) + 17)/32 +
+        sqrt(2)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) +
+        17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/32 +
+        Rational(15, 32))*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) +
+        17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/64 +
+        sqrt(sqrt(17)/32 + sqrt(2)*sqrt(-sqrt(17) + 17)/32 +
+        sqrt(2)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) +
+        17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/32 + Rational(15, 32))/2 +
+        S.Half + sqrt(-sqrt(17) + 17)*sqrt(sqrt(17)/32 + sqrt(2)*sqrt(-sqrt(17) +
+        17)/32 + sqrt(2)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) -
+        sqrt(2)*sqrt(-sqrt(17) + 17) + sqrt(34)*sqrt(-sqrt(17) + 17) +
+        6*sqrt(17) + 34)/32 + Rational(15, 32))*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) -
+        sqrt(2)*sqrt(-sqrt(17) + 17) + sqrt(34)*sqrt(-sqrt(17) + 17) +
+        6*sqrt(17) + 34)/32 + sqrt(34)*sqrt(-sqrt(17) + 17)*sqrt(sqrt(17)/32 +
+        sqrt(2)*sqrt(-sqrt(17) + 17)/32 +
+        sqrt(2)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) +
+        17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/32 +
+        Rational(15, 32))/32)/2)
+    assert e.rewrite(sqrt) == a
+    assert e.n() == a.n()
+    # coverage of fermatCoords: multiplicity > 1; the following could be
+    # different but that portion of the code should be tested in some way
+    assert cos(pi/9/17).rewrite(sqrt) == \
+        sin(pi/9)*sin(pi*Rational(2, 17)) + cos(pi/9)*cos(pi*Rational(2, 17))
+
+
+@slow
+def test_sincos_rewrite_sqrt_257():
+    assert cos(pi/257).rewrite(sqrt).evalf(64) == cos(pi/257).evalf(64)
+
+
+@slow
+def test_tancot_rewrite_sqrt():
+    # equivalent to testing rewrite(pow)
+    for p in [1, 3, 5, 17]:
+        for t in [1, 8]:
+            n = t*p
+            for i in range(1, min((n + 1)//2 + 1, 10)):
+                if 1 == gcd(i, n):
+                    x = i*pi/n
+                    if  2*i != n and 3*i != 2*n:
+                        t1 = tan(x).rewrite(sqrt)
+                        assert not t1.has(cot, tan), "fails for %d*pi/%d" % (i, n)
+                        assert 1e-3 > abs( tan(x.evalf(7)) - t1.evalf(4) ), "fails for %d*pi/%d" % (i, n)
+                    if  i != 0 and i != n:
+                        c1 = cot(x).rewrite(sqrt)
+                        assert not c1.has(cot, tan), "fails for %d*pi/%d" % (i, n)
+                        assert 1e-3 > abs( cot(x.evalf(7)) - c1.evalf(4) ), "fails for %d*pi/%d" % (i, n)
+
+
+def test_sec():
+    x = symbols('x', real=True)
+    z = symbols('z')
+
+    assert sec.nargs == FiniteSet(1)
+
+    assert sec(zoo) is nan
+    assert sec(0) == 1
+    assert sec(pi) == -1
+    assert sec(pi/2) is zoo
+    assert sec(-pi/2) is zoo
+    assert sec(pi/6) == 2*sqrt(3)/3
+    assert sec(pi/3) == 2
+    assert sec(pi*Rational(5, 2)) is zoo
+    assert sec(pi*Rational(9, 7)) == -sec(pi*Rational(2, 7))
+    assert sec(pi*Rational(3, 4)) == -sqrt(2)  # issue 8421
+    assert sec(I) == 1/cosh(1)
+    assert sec(x*I) == 1/cosh(x)
+    assert sec(-x) == sec(x)
+
+    assert sec(asec(x)) == x
+
+    assert sec(z).conjugate() == sec(conjugate(z))
+
+    assert (sec(z).as_real_imag() ==
+    (cos(re(z))*cosh(im(z))/(sin(re(z))**2*sinh(im(z))**2 +
+                             cos(re(z))**2*cosh(im(z))**2),
+     sin(re(z))*sinh(im(z))/(sin(re(z))**2*sinh(im(z))**2 +
+                             cos(re(z))**2*cosh(im(z))**2)))
+
+    assert sec(x).expand(trig=True) == 1/cos(x)
+    assert sec(2*x).expand(trig=True) == 1/(2*cos(x)**2 - 1)
+
+    assert sec(x).is_extended_real == True
+    assert sec(z).is_real == None
+
+    assert sec(a).is_algebraic is None
+    assert sec(na).is_algebraic is False
+
+    assert sec(x).as_leading_term() == sec(x)
+
+    assert sec(0, evaluate=False).is_finite == True
+    assert sec(x).is_finite == None
+    assert sec(pi/2, evaluate=False).is_finite == False
+
+    assert series(sec(x), x, x0=0, n=6) == 1 + x**2/2 + 5*x**4/24 + O(x**6)
+
+    # https://github.com/sympy/sympy/issues/7166
+    assert series(sqrt(sec(x))) == 1 + x**2/4 + 7*x**4/96 + O(x**6)
+
+    # https://github.com/sympy/sympy/issues/7167
+    assert (series(sqrt(sec(x)), x, x0=pi*3/2, n=4) ==
+            1/sqrt(x - pi*Rational(3, 2)) + (x - pi*Rational(3, 2))**Rational(3, 2)/12 +
+            (x - pi*Rational(3, 2))**Rational(7, 2)/160 + O((x - pi*Rational(3, 2))**4, (x, pi*Rational(3, 2))))
+
+    assert sec(x).diff(x) == tan(x)*sec(x)
+
+    # Taylor Term checks
+    assert sec(z).taylor_term(4, z) == 5*z**4/24
+    assert sec(z).taylor_term(6, z) == 61*z**6/720
+    assert sec(z).taylor_term(5, z) == 0
+
+
+def test_sec_rewrite():
+    assert sec(x).rewrite(exp) == 1/(exp(I*x)/2 + exp(-I*x)/2)
+    assert sec(x).rewrite(cos) == 1/cos(x)
+    assert sec(x).rewrite(tan) == (tan(x/2)**2 + 1)/(-tan(x/2)**2 + 1)
+    assert sec(x).rewrite(pow) == sec(x)
+    assert sec(x).rewrite(sqrt) == sec(x)
+    assert sec(z).rewrite(cot) == (cot(z/2)**2 + 1)/(cot(z/2)**2 - 1)
+    assert sec(x).rewrite(sin) == 1 / sin(x + pi / 2, evaluate=False)
+    assert sec(x).rewrite(tan) == (tan(x / 2)**2 + 1) / (-tan(x / 2)**2 + 1)
+    assert sec(x).rewrite(csc) == csc(-x + pi/2, evaluate=False)
+
+
+def test_sec_fdiff():
+    assert sec(x).fdiff() == tan(x)*sec(x)
+    raises(ArgumentIndexError, lambda: sec(x).fdiff(2))
+
+
+def test_csc():
+    x = symbols('x', real=True)
+    z = symbols('z')
+
+    # https://github.com/sympy/sympy/issues/6707
+    cosecant = csc('x')
+    alternate = 1/sin('x')
+    assert cosecant.equals(alternate) == True
+    assert alternate.equals(cosecant) == True
+
+    assert csc.nargs == FiniteSet(1)
+
+    assert csc(0) is zoo
+    assert csc(pi) is zoo
+    assert csc(zoo) is nan
+
+    assert csc(pi/2) == 1
+    assert csc(-pi/2) == -1
+    assert csc(pi/6) == 2
+    assert csc(pi/3) == 2*sqrt(3)/3
+    assert csc(pi*Rational(5, 2)) == 1
+    assert csc(pi*Rational(9, 7)) == -csc(pi*Rational(2, 7))
+    assert csc(pi*Rational(3, 4)) == sqrt(2)  # issue 8421
+    assert csc(I) == -I/sinh(1)
+    assert csc(x*I) == -I/sinh(x)
+    assert csc(-x) == -csc(x)
+
+    assert csc(acsc(x)) == x
+
+    assert csc(z).conjugate() == csc(conjugate(z))
+
+    assert (csc(z).as_real_imag() ==
+            (sin(re(z))*cosh(im(z))/(sin(re(z))**2*cosh(im(z))**2 +
+                                     cos(re(z))**2*sinh(im(z))**2),
+             -cos(re(z))*sinh(im(z))/(sin(re(z))**2*cosh(im(z))**2 +
+                          cos(re(z))**2*sinh(im(z))**2)))
+
+    assert csc(x).expand(trig=True) == 1/sin(x)
+    assert csc(2*x).expand(trig=True) == 1/(2*sin(x)*cos(x))
+
+    assert csc(x).is_extended_real == True
+    assert csc(z).is_real == None
+
+    assert csc(a).is_algebraic is None
+    assert csc(na).is_algebraic is False
+
+    assert csc(x).as_leading_term() == csc(x)
+
+    assert csc(0, evaluate=False).is_finite == False
+    assert csc(x).is_finite == None
+    assert csc(pi/2, evaluate=False).is_finite == True
+
+    assert series(csc(x), x, x0=pi/2, n=6) == \
+        1 + (x - pi/2)**2/2 + 5*(x - pi/2)**4/24 + O((x - pi/2)**6, (x, pi/2))
+    assert series(csc(x), x, x0=0, n=6) == \
+            1/x + x/6 + 7*x**3/360 + 31*x**5/15120 + O(x**6)
+
+    assert csc(x).diff(x) == -cot(x)*csc(x)
+
+    assert csc(x).taylor_term(2, x) == 0
+    assert csc(x).taylor_term(3, x) == 7*x**3/360
+    assert csc(x).taylor_term(5, x) == 31*x**5/15120
+    raises(ArgumentIndexError, lambda: csc(x).fdiff(2))
+
+
+def test_asec():
+    z = Symbol('z', zero=True)
+    assert asec(z) is zoo
+    assert asec(nan) is nan
+    assert asec(1) == 0
+    assert asec(-1) == pi
+    assert asec(oo) == pi/2
+    assert asec(-oo) == pi/2
+    assert asec(zoo) == pi/2
+
+    assert asec(sec(pi*Rational(13, 4))) == pi*Rational(3, 4)
+    assert asec(1 + sqrt(5)) == pi*Rational(2, 5)
+    assert asec(2/sqrt(3)) == pi/6
+    assert asec(sqrt(4 - 2*sqrt(2))) == pi/8
+    assert asec(-sqrt(4 + 2*sqrt(2))) == pi*Rational(5, 8)
+    assert asec(sqrt(2 + 2*sqrt(5)/5)) == pi*Rational(3, 10)
+    assert asec(-sqrt(2 + 2*sqrt(5)/5)) == pi*Rational(7, 10)
+    assert asec(sqrt(2) - sqrt(6)) == pi*Rational(11, 12)
+
+    assert asec(x).diff(x) == 1/(x**2*sqrt(1 - 1/x**2))
+
+    assert asec(x).rewrite(log) == I*log(sqrt(1 - 1/x**2) + I/x) + pi/2
+    assert asec(x).rewrite(asin) == -asin(1/x) + pi/2
+    assert asec(x).rewrite(acos) == acos(1/x)
+    assert asec(x).rewrite(atan) == \
+        pi*(1 - sqrt(x**2)/x)/2 + sqrt(x**2)*atan(sqrt(x**2 - 1))/x
+    assert asec(x).rewrite(acot) == \
+        pi*(1 - sqrt(x**2)/x)/2 + sqrt(x**2)*acot(1/sqrt(x**2 - 1))/x
+    assert asec(x).rewrite(acsc) == -acsc(x) + pi/2
+    raises(ArgumentIndexError, lambda: asec(x).fdiff(2))
+
+
+def test_asec_is_real():
+    assert asec(S.Half).is_real is False
+    n = Symbol('n', positive=True, integer=True)
+    assert asec(n).is_extended_real is True
+    assert asec(x).is_real is None
+    assert asec(r).is_real is None
+    t = Symbol('t', real=False, finite=True)
+    assert asec(t).is_real is False
+
+
+def test_asec_leading_term():
+    assert asec(1/x).as_leading_term(x) == pi/2
+    # Tests concerning branch points
+    assert asec(x + 1).as_leading_term(x) == sqrt(2)*sqrt(x)
+    assert asec(x - 1).as_leading_term(x) == pi
+    # Tests concerning points lying on branch cuts
+    assert asec(x).as_leading_term(x, cdir=1) == -I*log(x) + I*log(2)
+    assert asec(x).as_leading_term(x, cdir=-1) == I*log(x) + 2*pi - I*log(2)
+    assert asec(I*x + 1/2).as_leading_term(x, cdir=1) == asec(1/2)
+    assert asec(-I*x + 1/2).as_leading_term(x, cdir=1) == -asec(1/2)
+    assert asec(I*x - 1/2).as_leading_term(x, cdir=1) == 2*pi - asec(-1/2)
+    assert asec(-I*x - 1/2).as_leading_term(x, cdir=1) == asec(-1/2)
+    # Tests concerning im(ndir) == 0
+    assert asec(-I*x**2 + x - S(1)/2).as_leading_term(x, cdir=1) == pi + I*log(2 - sqrt(3))
+    assert asec(-I*x**2 + x - S(1)/2).as_leading_term(x, cdir=-1) == pi + I*log(2 - sqrt(3))
+
+
+def test_asec_series():
+    assert asec(x).series(x, 0, 9) == \
+        I*log(2) - I*log(x) - I*x**2/4 - 3*I*x**4/32 \
+        - 5*I*x**6/96 - 35*I*x**8/1024 + O(x**9)
+    t4 = asec(x).taylor_term(4, x)
+    assert t4 == -3*I*x**4/32
+    assert asec(x).taylor_term(6, x, t4, 0) == -5*I*x**6/96
+
+
+def test_acsc():
+    assert acsc(nan) is nan
+    assert acsc(1) == pi/2
+    assert acsc(-1) == -pi/2
+    assert acsc(oo) == 0
+    assert acsc(-oo) == 0
+    assert acsc(zoo) == 0
+    assert acsc(0) is zoo
+
+    assert acsc(csc(3)) == -3 + pi
+    assert acsc(csc(4)) == -4 + pi
+    assert acsc(csc(6)) == 6 - 2*pi
+    assert unchanged(acsc, csc(x))
+    assert unchanged(acsc, sec(x))
+
+    assert acsc(2/sqrt(3)) == pi/3
+    assert acsc(csc(pi*Rational(13, 4))) == -pi/4
+    assert acsc(sqrt(2 + 2*sqrt(5)/5)) == pi/5
+    assert acsc(-sqrt(2 + 2*sqrt(5)/5)) == -pi/5
+    assert acsc(-2) == -pi/6
+    assert acsc(-sqrt(4 + 2*sqrt(2))) == -pi/8
+    assert acsc(sqrt(4 - 2*sqrt(2))) == pi*Rational(3, 8)
+    assert acsc(1 + sqrt(5)) == pi/10
+    assert acsc(sqrt(2) - sqrt(6)) == pi*Rational(-5, 12)
+
+    assert acsc(x).diff(x) == -1/(x**2*sqrt(1 - 1/x**2))
+
+    assert acsc(x).rewrite(log) == -I*log(sqrt(1 - 1/x**2) + I/x)
+    assert acsc(x).rewrite(asin) == asin(1/x)
+    assert acsc(x).rewrite(acos) == -acos(1/x) + pi/2
+    assert acsc(x).rewrite(atan) == \
+        (-atan(sqrt(x**2 - 1)) + pi/2)*sqrt(x**2)/x
+    assert acsc(x).rewrite(acot) == (-acot(1/sqrt(x**2 - 1)) + pi/2)*sqrt(x**2)/x
+    assert acsc(x).rewrite(asec) == -asec(x) + pi/2
+    raises(ArgumentIndexError, lambda: acsc(x).fdiff(2))
+
+
+def test_csc_rewrite():
+    assert csc(x).rewrite(pow) == csc(x)
+    assert csc(x).rewrite(sqrt) == csc(x)
+
+    assert csc(x).rewrite(exp) == 2*I/(exp(I*x) - exp(-I*x))
+    assert csc(x).rewrite(sin) == 1/sin(x)
+    assert csc(x).rewrite(tan) == (tan(x/2)**2 + 1)/(2*tan(x/2))
+    assert csc(x).rewrite(cot) == (cot(x/2)**2 + 1)/(2*cot(x/2))
+    assert csc(x).rewrite(cos) == 1/cos(x - pi/2, evaluate=False)
+    assert csc(x).rewrite(sec) == sec(-x + pi/2, evaluate=False)
+
+    # issue 17349
+    assert csc(1 - exp(-besselj(I, I))).rewrite(cos) == \
+           -1/cos(-pi/2 - 1 + cos(I*besselj(I, I)) +
+                  I*cos(-pi/2 + I*besselj(I, I), evaluate=False), evaluate=False)
+
+
+def test_acsc_leading_term():
+    assert acsc(1/x).as_leading_term(x) == x
+    # Tests concerning branch points
+    assert acsc(x + 1).as_leading_term(x) == pi/2
+    assert acsc(x - 1).as_leading_term(x) == -pi/2
+    # Tests concerning points lying on branch cuts
+    assert acsc(x).as_leading_term(x, cdir=1) == I*log(x) + pi/2 - I*log(2)
+    assert acsc(x).as_leading_term(x, cdir=-1) == -I*log(x) - 3*pi/2 + I*log(2)
+    assert acsc(I*x + 1/2).as_leading_term(x, cdir=1) == acsc(1/2)
+    assert acsc(-I*x + 1/2).as_leading_term(x, cdir=1) == pi - acsc(1/2)
+    assert acsc(I*x - 1/2).as_leading_term(x, cdir=1) == -pi - acsc(-1/2)
+    assert acsc(-I*x - 1/2).as_leading_term(x, cdir=1) == -acsc(1/2)
+    # Tests concerning im(ndir) == 0
+    assert acsc(-I*x**2 + x - S(1)/2).as_leading_term(x, cdir=1) == -pi/2 + I*log(sqrt(3) + 2)
+    assert acsc(-I*x**2 + x - S(1)/2).as_leading_term(x, cdir=-1) == -pi/2 + I*log(sqrt(3) + 2)
+
+
+def test_acsc_series():
+    assert acsc(x).series(x, 0, 9) == \
+        -I*log(2) + pi/2 + I*log(x) + I*x**2/4 \
+        + 3*I*x**4/32 + 5*I*x**6/96 + 35*I*x**8/1024 + O(x**9)
+    t6 = acsc(x).taylor_term(6, x)
+    assert t6 == 5*I*x**6/96
+    assert acsc(x).taylor_term(8, x, t6, 0) == 35*I*x**8/1024
+
+
+def test_asin_nseries():
+    assert asin(x + 2)._eval_nseries(x, 4, None, I) == -asin(2) + pi + \
+    sqrt(3)*I*x/3 - sqrt(3)*I*x**2/9 + sqrt(3)*I*x**3/18 + O(x**4)
+    assert asin(x + 2)._eval_nseries(x, 4, None, -I) == asin(2) - \
+    sqrt(3)*I*x/3 + sqrt(3)*I*x**2/9 - sqrt(3)*I*x**3/18 + O(x**4)
+    assert asin(x - 2)._eval_nseries(x, 4, None, I) == -asin(2) - \
+    sqrt(3)*I*x/3 - sqrt(3)*I*x**2/9 - sqrt(3)*I*x**3/18 + O(x**4)
+    assert asin(x - 2)._eval_nseries(x, 4, None, -I) == asin(2) - pi + \
+    sqrt(3)*I*x/3 + sqrt(3)*I*x**2/9 + sqrt(3)*I*x**3/18 + O(x**4)
+    # testing nseries for asin at branch points
+    assert asin(1 + x)._eval_nseries(x, 3, None) == pi/2 - sqrt(2)*sqrt(-x) - \
+    sqrt(2)*(-x)**(S(3)/2)/12 - 3*sqrt(2)*(-x)**(S(5)/2)/160 + O(x**3)
+    assert asin(-1 + x)._eval_nseries(x, 3, None) == -pi/2 + sqrt(2)*sqrt(x) + \
+    sqrt(2)*x**(S(3)/2)/12 + 3*sqrt(2)*x**(S(5)/2)/160 + O(x**3)
+    assert asin(exp(x))._eval_nseries(x, 3, None) == pi/2 - sqrt(2)*sqrt(-x) + \
+    sqrt(2)*(-x)**(S(3)/2)/6 - sqrt(2)*(-x)**(S(5)/2)/120 + O(x**3)
+    assert asin(-exp(x))._eval_nseries(x, 3, None) == -pi/2 + sqrt(2)*sqrt(-x) - \
+    sqrt(2)*(-x)**(S(3)/2)/6 + sqrt(2)*(-x)**(S(5)/2)/120 + O(x**3)
+
+
+def test_acos_nseries():
+    assert acos(x + 2)._eval_nseries(x, 4, None, I) == -acos(2) - sqrt(3)*I*x/3 + \
+    sqrt(3)*I*x**2/9 - sqrt(3)*I*x**3/18 + O(x**4)
+    assert acos(x + 2)._eval_nseries(x, 4, None, -I) == acos(2) + sqrt(3)*I*x/3 - \
+    sqrt(3)*I*x**2/9 + sqrt(3)*I*x**3/18 + O(x**4)
+    assert acos(x - 2)._eval_nseries(x, 4, None, I) == acos(-2) + sqrt(3)*I*x/3 + \
+    sqrt(3)*I*x**2/9 + sqrt(3)*I*x**3/18 + O(x**4)
+    assert acos(x - 2)._eval_nseries(x, 4, None, -I) == -acos(-2) + 2*pi - \
+    sqrt(3)*I*x/3 - sqrt(3)*I*x**2/9 - sqrt(3)*I*x**3/18 + O(x**4)
+    # testing nseries for acos at branch points
+    assert acos(1 + x)._eval_nseries(x, 3, None) == sqrt(2)*sqrt(-x) + \
+    sqrt(2)*(-x)**(S(3)/2)/12 + 3*sqrt(2)*(-x)**(S(5)/2)/160 + O(x**3)
+    assert acos(-1 + x)._eval_nseries(x, 3, None) == pi - sqrt(2)*sqrt(x) - \
+    sqrt(2)*x**(S(3)/2)/12 - 3*sqrt(2)*x**(S(5)/2)/160 + O(x**3)
+    assert acos(exp(x))._eval_nseries(x, 3, None) == sqrt(2)*sqrt(-x) - \
+    sqrt(2)*(-x)**(S(3)/2)/6 + sqrt(2)*(-x)**(S(5)/2)/120 + O(x**3)
+    assert acos(-exp(x))._eval_nseries(x, 3, None) == pi - sqrt(2)*sqrt(-x) + \
+    sqrt(2)*(-x)**(S(3)/2)/6 - sqrt(2)*(-x)**(S(5)/2)/120 + O(x**3)
+
+
+def test_atan_nseries():
+    assert atan(x + 2*I)._eval_nseries(x, 4, None, 1) == I*atanh(2) - x/3 - \
+    2*I*x**2/9 + 13*x**3/81 + O(x**4)
+    assert atan(x + 2*I)._eval_nseries(x, 4, None, -1) == I*atanh(2) - pi - \
+    x/3 - 2*I*x**2/9 + 13*x**3/81 + O(x**4)
+    assert atan(x - 2*I)._eval_nseries(x, 4, None, 1) == -I*atanh(2) + pi - \
+    x/3 + 2*I*x**2/9 + 13*x**3/81 + O(x**4)
+    assert atan(x - 2*I)._eval_nseries(x, 4, None, -1) == -I*atanh(2) - x/3 + \
+    2*I*x**2/9 + 13*x**3/81 + O(x**4)
+    assert atan(1/x)._eval_nseries(x, 2, None, 1) == pi/2 - x + O(x**2)
+    assert atan(1/x)._eval_nseries(x, 2, None, -1) == -pi/2 - x + O(x**2)
+    # testing nseries for atan at branch points
+    assert atan(x + I)._eval_nseries(x, 4, None) == I*log(2)/2 + pi/4 - \
+    I*log(x)/2 + x/4 + I*x**2/16 - x**3/48 + O(x**4)
+    assert atan(x - I)._eval_nseries(x, 4, None) == -I*log(2)/2 + pi/4 + \
+    I*log(x)/2 + x/4 - I*x**2/16 - x**3/48 + O(x**4)
+
+
+def test_acot_nseries():
+    assert acot(x + S(1)/2*I)._eval_nseries(x, 4, None, 1) == -I*acoth(S(1)/2) + \
+    pi - 4*x/3 + 8*I*x**2/9 + 112*x**3/81 + O(x**4)
+    assert acot(x + S(1)/2*I)._eval_nseries(x, 4, None, -1) == -I*acoth(S(1)/2) - \
+    4*x/3 + 8*I*x**2/9 + 112*x**3/81 + O(x**4)
+    assert acot(x - S(1)/2*I)._eval_nseries(x, 4, None, 1) == I*acoth(S(1)/2) - \
+    4*x/3 - 8*I*x**2/9 + 112*x**3/81 + O(x**4)
+    assert acot(x - S(1)/2*I)._eval_nseries(x, 4, None, -1) == I*acoth(S(1)/2) - \
+    pi - 4*x/3 - 8*I*x**2/9 + 112*x**3/81 + O(x**4)
+    assert acot(x)._eval_nseries(x, 2, None, 1) == pi/2 - x + O(x**2)
+    assert acot(x)._eval_nseries(x, 2, None, -1) == -pi/2 - x + O(x**2)
+    # testing nseries for acot at branch points
+    assert acot(x + I)._eval_nseries(x, 4, None) == -I*log(2)/2 + pi/4 + \
+    I*log(x)/2 - x/4 - I*x**2/16 + x**3/48 + O(x**4)
+    assert acot(x - I)._eval_nseries(x, 4, None) == I*log(2)/2 + pi/4 - \
+    I*log(x)/2 - x/4 + I*x**2/16 + x**3/48 + O(x**4)
+
+
+def test_asec_nseries():
+    assert asec(x + S(1)/2)._eval_nseries(x, 4, None, I) == asec(S(1)/2) - \
+    4*sqrt(3)*I*x/3 + 8*sqrt(3)*I*x**2/9 - 16*sqrt(3)*I*x**3/9 + O(x**4)
+    assert asec(x + S(1)/2)._eval_nseries(x, 4, None, -I) == -asec(S(1)/2) + \
+    4*sqrt(3)*I*x/3 - 8*sqrt(3)*I*x**2/9 + 16*sqrt(3)*I*x**3/9 + O(x**4)
+    assert asec(x - S(1)/2)._eval_nseries(x, 4, None, I) == -asec(-S(1)/2) + \
+    2*pi + 4*sqrt(3)*I*x/3 + 8*sqrt(3)*I*x**2/9 + 16*sqrt(3)*I*x**3/9 + O(x**4)
+    assert asec(x - S(1)/2)._eval_nseries(x, 4, None, -I) == asec(-S(1)/2) - \
+    4*sqrt(3)*I*x/3 - 8*sqrt(3)*I*x**2/9 - 16*sqrt(3)*I*x**3/9 + O(x**4)
+    # testing nseries for asec at branch points
+    assert asec(1 + x)._eval_nseries(x, 3, None) == sqrt(2)*sqrt(x) - \
+    5*sqrt(2)*x**(S(3)/2)/12 + 43*sqrt(2)*x**(S(5)/2)/160 + O(x**3)
+    assert asec(-1 + x)._eval_nseries(x, 3, None) == pi - sqrt(2)*sqrt(-x) + \
+    5*sqrt(2)*(-x)**(S(3)/2)/12 - 43*sqrt(2)*(-x)**(S(5)/2)/160 + O(x**3)
+    assert asec(exp(x))._eval_nseries(x, 3, None) == sqrt(2)*sqrt(x) - \
+    sqrt(2)*x**(S(3)/2)/6 + sqrt(2)*x**(S(5)/2)/120 + O(x**3)
+    assert asec(-exp(x))._eval_nseries(x, 3, None) == pi - sqrt(2)*sqrt(x) + \
+    sqrt(2)*x**(S(3)/2)/6 - sqrt(2)*x**(S(5)/2)/120 + O(x**3)
+
+
+def test_acsc_nseries():
+    assert acsc(x + S(1)/2)._eval_nseries(x, 4, None, I) == acsc(S(1)/2) + \
+    4*sqrt(3)*I*x/3 - 8*sqrt(3)*I*x**2/9 + 16*sqrt(3)*I*x**3/9 + O(x**4)
+    assert acsc(x + S(1)/2)._eval_nseries(x, 4, None, -I) == -acsc(S(1)/2) + \
+    pi - 4*sqrt(3)*I*x/3 + 8*sqrt(3)*I*x**2/9 - 16*sqrt(3)*I*x**3/9 + O(x**4)
+    assert acsc(x - S(1)/2)._eval_nseries(x, 4, None, I) == acsc(S(1)/2) - pi -\
+    4*sqrt(3)*I*x/3 - 8*sqrt(3)*I*x**2/9 - 16*sqrt(3)*I*x**3/9 + O(x**4)
+    assert acsc(x - S(1)/2)._eval_nseries(x, 4, None, -I) == -acsc(S(1)/2) + \
+    4*sqrt(3)*I*x/3 + 8*sqrt(3)*I*x**2/9 + 16*sqrt(3)*I*x**3/9 + O(x**4)
+    # testing nseries for acsc at branch points
+    assert acsc(1 + x)._eval_nseries(x, 3, None) == pi/2 - sqrt(2)*sqrt(x) + \
+    5*sqrt(2)*x**(S(3)/2)/12 - 43*sqrt(2)*x**(S(5)/2)/160 + O(x**3)
+    assert acsc(-1 + x)._eval_nseries(x, 3, None) == -pi/2 + sqrt(2)*sqrt(-x) - \
+    5*sqrt(2)*(-x)**(S(3)/2)/12 + 43*sqrt(2)*(-x)**(S(5)/2)/160 + O(x**3)
+    assert acsc(exp(x))._eval_nseries(x, 3, None) == pi/2 - sqrt(2)*sqrt(x) + \
+    sqrt(2)*x**(S(3)/2)/6 - sqrt(2)*x**(S(5)/2)/120 + O(x**3)
+    assert acsc(-exp(x))._eval_nseries(x, 3, None) == -pi/2 + sqrt(2)*sqrt(x) - \
+    sqrt(2)*x**(S(3)/2)/6 + sqrt(2)*x**(S(5)/2)/120 + O(x**3)
+
+
+def test_issue_8653():
+    n = Symbol('n', integer=True)
+    assert sin(n).is_irrational is None
+    assert cos(n).is_irrational is None
+    assert tan(n).is_irrational is None
+
+
+def test_issue_9157():
+    n = Symbol('n', integer=True, positive=True)
+    assert atan(n - 1).is_nonnegative is True
+
+
+def test_trig_period():
+    x, y = symbols('x, y')
+
+    assert sin(x).period() == 2*pi
+    assert cos(x).period() == 2*pi
+    assert tan(x).period() == pi
+    assert cot(x).period() == pi
+    assert sec(x).period() == 2*pi
+    assert csc(x).period() == 2*pi
+    assert sin(2*x).period() == pi
+    assert cot(4*x - 6).period() == pi/4
+    assert cos((-3)*x).period() == pi*Rational(2, 3)
+    assert cos(x*y).period(x) == 2*pi/abs(y)
+    assert sin(3*x*y + 2*pi).period(y) == 2*pi/abs(3*x)
+    assert tan(3*x).period(y) is S.Zero
+    raises(NotImplementedError, lambda: sin(x**2).period(x))
+
+
+def test_issue_7171():
+    assert sin(x).rewrite(sqrt) == sin(x)
+    assert sin(x).rewrite(pow) == sin(x)
+
+
+def test_issue_11864():
+    w, k = symbols('w, k', real=True)
+    F = Piecewise((1, Eq(2*pi*k, 0)), (sin(pi*k)/(pi*k), True))
+    soln = Piecewise((1, Eq(2*pi*k, 0)), (sinc(pi*k), True))
+    assert F.rewrite(sinc) == soln
+
+def test_real_assumptions():
+    z = Symbol('z', real=False, finite=True)
+    assert sin(z).is_real is None
+    assert cos(z).is_real is None
+    assert tan(z).is_real is False
+    assert sec(z).is_real is None
+    assert csc(z).is_real is None
+    assert cot(z).is_real is False
+    assert asin(p).is_real is None
+    assert asin(n).is_real is None
+    assert asec(p).is_real is None
+    assert asec(n).is_real is None
+    assert acos(p).is_real is None
+    assert acos(n).is_real is None
+    assert acsc(p).is_real is None
+    assert acsc(n).is_real is None
+    assert atan(p).is_positive is True
+    assert atan(n).is_negative is True
+    assert acot(p).is_positive is True
+    assert acot(n).is_negative is True
+
+def test_issue_14320():
+    assert asin(sin(2)) == -2 + pi and (-pi/2 <= -2 + pi <= pi/2) and sin(2) == sin(-2 + pi)
+    assert asin(cos(2)) == -2 + pi/2 and (-pi/2 <= -2 + pi/2 <= pi/2) and cos(2) == sin(-2 + pi/2)
+    assert acos(sin(2)) == -pi/2 + 2 and (0 <= -pi/2 + 2 <= pi) and sin(2) == cos(-pi/2 + 2)
+    assert acos(cos(20)) == -6*pi + 20 and (0 <= -6*pi + 20 <= pi) and cos(20) == cos(-6*pi + 20)
+    assert acos(cos(30)) == -30 + 10*pi and (0 <= -30 + 10*pi <= pi) and cos(30) == cos(-30 + 10*pi)
+
+    assert atan(tan(17)) == -5*pi + 17 and (-pi/2 < -5*pi + 17 < pi/2) and tan(17) == tan(-5*pi + 17)
+    assert atan(tan(15)) == -5*pi + 15 and (-pi/2 < -5*pi + 15 < pi/2) and tan(15) == tan(-5*pi + 15)
+    assert atan(cot(12)) == -12 + pi*Rational(7, 2) and (-pi/2 < -12 + pi*Rational(7, 2) < pi/2) and cot(12) == tan(-12 + pi*Rational(7, 2))
+    assert acot(cot(15)) == -5*pi + 15 and (-pi/2 < -5*pi + 15 <= pi/2) and cot(15) == cot(-5*pi + 15)
+    assert acot(tan(19)) == -19 + pi*Rational(13, 2) and (-pi/2 < -19 + pi*Rational(13, 2) <= pi/2) and tan(19) == cot(-19 + pi*Rational(13, 2))
+
+    assert asec(sec(11)) == -11 + 4*pi and (0 <= -11 + 4*pi <= pi) and cos(11) == cos(-11 + 4*pi)
+    assert asec(csc(13)) == -13 + pi*Rational(9, 2) and (0 <= -13 + pi*Rational(9, 2) <= pi) and sin(13) == cos(-13 + pi*Rational(9, 2))
+    assert acsc(csc(14)) == -4*pi + 14 and (-pi/2 <= -4*pi + 14 <= pi/2) and sin(14) == sin(-4*pi + 14)
+    assert acsc(sec(10)) == pi*Rational(-7, 2) + 10 and (-pi/2 <= pi*Rational(-7, 2) + 10 <= pi/2) and cos(10) == sin(pi*Rational(-7, 2) + 10)
+
+def test_issue_14543():
+    assert sec(2*pi + 11) == sec(11)
+    assert sec(2*pi - 11) == sec(11)
+    assert sec(pi + 11) == -sec(11)
+    assert sec(pi - 11) == -sec(11)
+
+    assert csc(2*pi + 17) == csc(17)
+    assert csc(2*pi - 17) == -csc(17)
+    assert csc(pi + 17) == -csc(17)
+    assert csc(pi - 17) == csc(17)
+
+    x = Symbol('x')
+    assert csc(pi/2 + x) == sec(x)
+    assert csc(pi/2 - x) == sec(x)
+    assert csc(pi*Rational(3, 2) + x) == -sec(x)
+    assert csc(pi*Rational(3, 2) - x) == -sec(x)
+
+    assert sec(pi/2 - x) == csc(x)
+    assert sec(pi/2 + x) == -csc(x)
+    assert sec(pi*Rational(3, 2) + x) == csc(x)
+    assert sec(pi*Rational(3, 2) - x) == -csc(x)
+
+
+def test_as_real_imag():
+    # This is for https://github.com/sympy/sympy/issues/17142
+    # If it start failing again in irrelevant builds or in the master
+    # please open up the issue again.
+    expr = atan(I/(I + I*tan(1)))
+    assert expr.as_real_imag() == (expr, 0)
+
+
+def test_issue_18746():
+    e3 = cos(S.Pi*(x/4 + 1/4))
+    assert e3.period() == 8

llmeval-env/lib/python3.10/site-packages/sympy/functions/special/__init__.py

@@ -0,0 +1 @@
+# Stub __init__.py for the sympy.functions.special package

llmeval-env/lib/python3.10/site-packages/sympy/functions/special/bessel.py

@@ -0,0 +1,2089 @@
+from functools import wraps
+
+from sympy.core import S
+from sympy.core.add import Add
+from sympy.core.cache import cacheit
+from sympy.core.expr import Expr
+from sympy.core.function import Function, ArgumentIndexError, _mexpand
+from sympy.core.logic import fuzzy_or, fuzzy_not
+from sympy.core.numbers import Rational, pi, I
+from sympy.core.power import Pow
+from sympy.core.symbol import Dummy, Wild
+from sympy.core.sympify import sympify
+from sympy.functions.combinatorial.factorials import factorial
+from sympy.functions.elementary.trigonometric import sin, cos, csc, cot
+from sympy.functions.elementary.integers import ceiling
+from sympy.functions.elementary.exponential import exp, log
+from sympy.functions.elementary.miscellaneous import cbrt, sqrt, root
+from sympy.functions.elementary.complexes import (Abs, re, im, polar_lift, unpolarify)
+from sympy.functions.special.gamma_functions import gamma, digamma, uppergamma
+from sympy.functions.special.hyper import hyper
+from sympy.polys.orthopolys import spherical_bessel_fn
+
+from mpmath import mp, workprec
+
+# TODO
+# o Scorer functions G1 and G2
+# o Asymptotic expansions
+#   These are possible, e.g. for fixed order, but since the bessel type
+#   functions are oscillatory they are not actually tractable at
+#   infinity, so this is not particularly useful right now.
+# o Nicer series expansions.
+# o More rewriting.
+# o Add solvers to ode.py (or rather add solvers for the hypergeometric equation).
+
+
+class BesselBase(Function):
+    """
+    Abstract base class for Bessel-type functions.
+
+    This class is meant to reduce code duplication.
+    All Bessel-type functions can 1) be differentiated, with the derivatives
+    expressed in terms of similar functions, and 2) be rewritten in terms
+    of other Bessel-type functions.
+
+    Here, Bessel-type functions are assumed to have one complex parameter.
+
+    To use this base class, define class attributes ``_a`` and ``_b`` such that
+    ``2*F_n' = -_a*F_{n+1} + b*F_{n-1}``.
+
+    """
+
+    @property
+    def order(self):
+        """ The order of the Bessel-type function. """
+        return self.args[0]
+
+    @property
+    def argument(self):
+        """ The argument of the Bessel-type function. """
+        return self.args[1]
+
+    @classmethod
+    def eval(cls, nu, z):
+        return
+
+    def fdiff(self, argindex=2):
+        if argindex != 2:
+            raise ArgumentIndexError(self, argindex)
+        return (self._b/2 * self.__class__(self.order - 1, self.argument) -
+                self._a/2 * self.__class__(self.order + 1, self.argument))
+
+    def _eval_conjugate(self):
+        z = self.argument
+        if z.is_extended_negative is False:
+            return self.__class__(self.order.conjugate(), z.conjugate())
+
+    def _eval_is_meromorphic(self, x, a):
+        nu, z = self.order, self.argument
+
+        if nu.has(x):
+            return False
+        if not z._eval_is_meromorphic(x, a):
+            return None
+        z0 = z.subs(x, a)
+        if nu.is_integer:
+            if isinstance(self, (besselj, besseli, hn1, hn2, jn, yn)) or not nu.is_zero:
+                return fuzzy_not(z0.is_infinite)
+        return fuzzy_not(fuzzy_or([z0.is_zero, z0.is_infinite]))
+
+    def _eval_expand_func(self, **hints):
+        nu, z, f = self.order, self.argument, self.__class__
+        if nu.is_real:
+            if (nu - 1).is_positive:
+                return (-self._a*self._b*f(nu - 2, z)._eval_expand_func() +
+                        2*self._a*(nu - 1)*f(nu - 1, z)._eval_expand_func()/z)
+            elif (nu + 1).is_negative:
+                return (2*self._b*(nu + 1)*f(nu + 1, z)._eval_expand_func()/z -
+                        self._a*self._b*f(nu + 2, z)._eval_expand_func())
+        return self
+
+    def _eval_simplify(self, **kwargs):
+        from sympy.simplify.simplify import besselsimp
+        return besselsimp(self)
+
+
+class besselj(BesselBase):
+    r"""
+    Bessel function of the first kind.
+
+    Explanation
+    ===========
+
+    The Bessel $J$ function of order $\nu$ is defined to be the function
+    satisfying Bessel's differential equation
+
+    .. math ::
+        z^2 \frac{\mathrm{d}^2 w}{\mathrm{d}z^2}
+        + z \frac{\mathrm{d}w}{\mathrm{d}z} + (z^2 - \nu^2) w = 0,
+
+    with Laurent expansion
+
+    .. math ::
+        J_\nu(z) = z^\nu \left(\frac{1}{\Gamma(\nu + 1) 2^\nu} + O(z^2) \right),
+
+    if $\nu$ is not a negative integer. If $\nu=-n \in \mathbb{Z}_{<0}$
+    *is* a negative integer, then the definition is
+
+    .. math ::
+        J_{-n}(z) = (-1)^n J_n(z).
+
+    Examples
+    ========
+
+    Create a Bessel function object:
+
+    >>> from sympy import besselj, jn
+    >>> from sympy.abc import z, n
+    >>> b = besselj(n, z)
+
+    Differentiate it:
+
+    >>> b.diff(z)
+    besselj(n - 1, z)/2 - besselj(n + 1, z)/2
+
+    Rewrite in terms of spherical Bessel functions:
+
+    >>> b.rewrite(jn)
+    sqrt(2)*sqrt(z)*jn(n - 1/2, z)/sqrt(pi)
+
+    Access the parameter and argument:
+
+    >>> b.order
+    n
+    >>> b.argument
+    z
+
+    See Also
+    ========
+
+    bessely, besseli, besselk
+
+    References
+    ==========
+
+    .. [1] Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 9",
+           Handbook of Mathematical Functions with Formulas, Graphs, and
+           Mathematical Tables
+    .. [2] Luke, Y. L. (1969), The Special Functions and Their
+           Approximations, Volume 1
+    .. [3] https://en.wikipedia.org/wiki/Bessel_function
+    .. [4] https://functions.wolfram.com/Bessel-TypeFunctions/BesselJ/
+
+    """
+
+    _a = S.One
+    _b = S.One
+
+    @classmethod
+    def eval(cls, nu, z):
+        if z.is_zero:
+            if nu.is_zero:
+                return S.One
+            elif (nu.is_integer and nu.is_zero is False) or re(nu).is_positive:
+                return S.Zero
+            elif re(nu).is_negative and not (nu.is_integer is True):
+                return S.ComplexInfinity
+            elif nu.is_imaginary:
+                return S.NaN
+        if z in (S.Infinity, S.NegativeInfinity):
+            return S.Zero
+
+        if z.could_extract_minus_sign():
+            return (z)**nu*(-z)**(-nu)*besselj(nu, -z)
+        if nu.is_integer:
+            if nu.could_extract_minus_sign():
+                return S.NegativeOne**(-nu)*besselj(-nu, z)
+            newz = z.extract_multiplicatively(I)
+            if newz:  # NOTE we don't want to change the function if z==0
+                return I**(nu)*besseli(nu, newz)
+
+        # branch handling:
+        if nu.is_integer:
+            newz = unpolarify(z)
+            if newz != z:
+                return besselj(nu, newz)
+        else:
+            newz, n = z.extract_branch_factor()
+            if n != 0:
+                return exp(2*n*pi*nu*I)*besselj(nu, newz)
+        nnu = unpolarify(nu)
+        if nu != nnu:
+            return besselj(nnu, z)
+
+    def _eval_rewrite_as_besseli(self, nu, z, **kwargs):
+        return exp(I*pi*nu/2)*besseli(nu, polar_lift(-I)*z)
+
+    def _eval_rewrite_as_bessely(self, nu, z, **kwargs):
+        if nu.is_integer is False:
+            return csc(pi*nu)*bessely(-nu, z) - cot(pi*nu)*bessely(nu, z)
+
+    def _eval_rewrite_as_jn(self, nu, z, **kwargs):
+        return sqrt(2*z/pi)*jn(nu - S.Half, self.argument)
+
+    def _eval_as_leading_term(self, x, logx=None, cdir=0):
+        nu, z = self.args
+        try:
+            arg = z.as_leading_term(x)
+        except NotImplementedError:
+            return self
+        c, e = arg.as_coeff_exponent(x)
+
+        if e.is_positive:
+            return arg**nu/(2**nu*gamma(nu + 1))
+        elif e.is_negative:
+            cdir = 1 if cdir == 0 else cdir
+            sign = c*cdir**e
+            if not sign.is_negative:
+                # Refer Abramowitz and Stegun 1965, p. 364 for more information on
+                # asymptotic approximation of besselj function.
+                return sqrt(2)*cos(z - pi*(2*nu + 1)/4)/sqrt(pi*z)
+            return self
+
+        return super(besselj, self)._eval_as_leading_term(x, logx, cdir)
+
+    def _eval_is_extended_real(self):
+        nu, z = self.args
+        if nu.is_integer and z.is_extended_real:
+            return True
+
+    def _eval_nseries(self, x, n, logx, cdir=0):
+        # Refer https://functions.wolfram.com/Bessel-TypeFunctions/BesselJ/06/01/04/01/01/0003/
+        # for more information on nseries expansion of besselj function.
+        from sympy.series.order import Order
+        nu, z = self.args
+
+        # In case of powers less than 1, number of terms need to be computed
+        # separately to avoid repeated callings of _eval_nseries with wrong n
+        try:
+            _, exp = z.leadterm(x)
+        except (ValueError, NotImplementedError):
+            return self
+
+        if exp.is_positive:
+            newn = ceiling(n/exp)
+            o = Order(x**n, x)
+            r = (z/2)._eval_nseries(x, n, logx, cdir).removeO()
+            if r is S.Zero:
+                return o
+            t = (_mexpand(r**2) + o).removeO()
+
+            term = r**nu/gamma(nu + 1)
+            s = [term]
+            for k in range(1, (newn + 1)//2):
+                term *= -t/(k*(nu + k))
+                term = (_mexpand(term) + o).removeO()
+                s.append(term)
+            return Add(*s) + o
+
+        return super(besselj, self)._eval_nseries(x, n, logx, cdir)
+
+
+class bessely(BesselBase):
+    r"""
+    Bessel function of the second kind.
+
+    Explanation
+    ===========
+
+    The Bessel $Y$ function of order $\nu$ is defined as
+
+    .. math ::
+        Y_\nu(z) = \lim_{\mu \to \nu} \frac{J_\mu(z) \cos(\pi \mu)
+                                            - J_{-\mu}(z)}{\sin(\pi \mu)},
+
+    where $J_\mu(z)$ is the Bessel function of the first kind.
+
+    It is a solution to Bessel's equation, and linearly independent from
+    $J_\nu$.
+
+    Examples
+    ========
+
+    >>> from sympy import bessely, yn
+    >>> from sympy.abc import z, n
+    >>> b = bessely(n, z)
+    >>> b.diff(z)
+    bessely(n - 1, z)/2 - bessely(n + 1, z)/2
+    >>> b.rewrite(yn)
+    sqrt(2)*sqrt(z)*yn(n - 1/2, z)/sqrt(pi)
+
+    See Also
+    ========
+
+    besselj, besseli, besselk
+
+    References
+    ==========
+
+    .. [1] https://functions.wolfram.com/Bessel-TypeFunctions/BesselY/
+
+    """
+
+    _a = S.One
+    _b = S.One
+
+    @classmethod
+    def eval(cls, nu, z):
+        if z.is_zero:
+            if nu.is_zero:
+                return S.NegativeInfinity
+            elif re(nu).is_zero is False:
+                return S.ComplexInfinity
+            elif re(nu).is_zero:
+                return S.NaN
+        if z in (S.Infinity, S.NegativeInfinity):
+            return S.Zero
+        if z == I*S.Infinity:
+            return exp(I*pi*(nu + 1)/2) * S.Infinity
+        if z == I*S.NegativeInfinity:
+            return exp(-I*pi*(nu + 1)/2) * S.Infinity
+
+        if nu.is_integer:
+            if nu.could_extract_minus_sign():
+                return S.NegativeOne**(-nu)*bessely(-nu, z)
+
+    def _eval_rewrite_as_besselj(self, nu, z, **kwargs):
+        if nu.is_integer is False:
+            return csc(pi*nu)*(cos(pi*nu)*besselj(nu, z) - besselj(-nu, z))
+
+    def _eval_rewrite_as_besseli(self, nu, z, **kwargs):
+        aj = self._eval_rewrite_as_besselj(*self.args)
+        if aj:
+            return aj.rewrite(besseli)
+
+    def _eval_rewrite_as_yn(self, nu, z, **kwargs):
+        return sqrt(2*z/pi) * yn(nu - S.Half, self.argument)
+
+    def _eval_as_leading_term(self, x, logx=None, cdir=0):
+        nu, z = self.args
+        try:
+            arg = z.as_leading_term(x)
+        except NotImplementedError:
+            return self
+        c, e = arg.as_coeff_exponent(x)
+
+        if e.is_positive:
+            term_one = ((2/pi)*log(z/2)*besselj(nu, z))
+            term_two = -(z/2)**(-nu)*factorial(nu - 1)/pi if (nu).is_positive else S.Zero
+            term_three = -(z/2)**nu/(pi*factorial(nu))*(digamma(nu + 1) - S.EulerGamma)
+            arg = Add(*[term_one, term_two, term_three]).as_leading_term(x, logx=logx)
+            return arg
+        elif e.is_negative:
+            cdir = 1 if cdir == 0 else cdir
+            sign = c*cdir**e
+            if not sign.is_negative:
+                # Refer Abramowitz and Stegun 1965, p. 364 for more information on
+                # asymptotic approximation of bessely function.
+                return sqrt(2)*(-sin(pi*nu/2 - z + pi/4) + 3*cos(pi*nu/2 - z + pi/4)/(8*z))*sqrt(1/z)/sqrt(pi)
+            return self
+
+        return super(bessely, self)._eval_as_leading_term(x, logx, cdir)
+
+    def _eval_is_extended_real(self):
+        nu, z = self.args
+        if nu.is_integer and z.is_positive:
+            return True
+
+    def _eval_nseries(self, x, n, logx, cdir=0):
+        # Refer https://functions.wolfram.com/Bessel-TypeFunctions/BesselY/06/01/04/01/02/0008/
+        # for more information on nseries expansion of bessely function.
+        from sympy.series.order import Order
+        nu, z = self.args
+
+        # In case of powers less than 1, number of terms need to be computed
+        # separately to avoid repeated callings of _eval_nseries with wrong n
+        try:
+            _, exp = z.leadterm(x)
+        except (ValueError, NotImplementedError):
+            return self
+
+        if exp.is_positive and nu.is_integer:
+            newn = ceiling(n/exp)
+            bn = besselj(nu, z)
+            a = ((2/pi)*log(z/2)*bn)._eval_nseries(x, n, logx, cdir)
+
+            b, c = [], []
+            o = Order(x**n, x)
+            r = (z/2)._eval_nseries(x, n, logx, cdir).removeO()
+            if r is S.Zero:
+                return o
+            t = (_mexpand(r**2) + o).removeO()
+
+            if nu > S.Zero:
+                term = r**(-nu)*factorial(nu - 1)/pi
+                b.append(term)
+                for k in range(1, nu):
+                    denom = (nu - k)*k
+                    if denom == S.Zero:
+                        term *= t/k
+                    else:
+                        term *= t/denom
+                    term = (_mexpand(term) + o).removeO()
+                    b.append(term)
+
+            p = r**nu/(pi*factorial(nu))
+            term = p*(digamma(nu + 1) - S.EulerGamma)
+            c.append(term)
+            for k in range(1, (newn + 1)//2):
+                p *= -t/(k*(k + nu))
+                p = (_mexpand(p) + o).removeO()
+                term = p*(digamma(k + nu + 1) + digamma(k + 1))
+                c.append(term)
+            return a - Add(*b) - Add(*c) # Order term comes from a
+
+        return super(bessely, self)._eval_nseries(x, n, logx, cdir)
+
+
+class besseli(BesselBase):
+    r"""
+    Modified Bessel function of the first kind.
+
+    Explanation
+    ===========
+
+    The Bessel $I$ function is a solution to the modified Bessel equation
+
+    .. math ::
+        z^2 \frac{\mathrm{d}^2 w}{\mathrm{d}z^2}
+        + z \frac{\mathrm{d}w}{\mathrm{d}z} + (z^2 + \nu^2)^2 w = 0.
+
+    It can be defined as
+
+    .. math ::
+        I_\nu(z) = i^{-\nu} J_\nu(iz),
+
+    where $J_\nu(z)$ is the Bessel function of the first kind.
+
+    Examples
+    ========
+
+    >>> from sympy import besseli
+    >>> from sympy.abc import z, n
+    >>> besseli(n, z).diff(z)
+    besseli(n - 1, z)/2 + besseli(n + 1, z)/2
+
+    See Also
+    ========
+
+    besselj, bessely, besselk
+
+    References
+    ==========
+
+    .. [1] https://functions.wolfram.com/Bessel-TypeFunctions/BesselI/
+
+    """
+
+    _a = -S.One
+    _b = S.One
+
+    @classmethod
+    def eval(cls, nu, z):
+        if z.is_zero:
+            if nu.is_zero:
+                return S.One
+            elif (nu.is_integer and nu.is_zero is False) or re(nu).is_positive:
+                return S.Zero
+            elif re(nu).is_negative and not (nu.is_integer is True):
+                return S.ComplexInfinity
+            elif nu.is_imaginary:
+                return S.NaN
+        if im(z) in (S.Infinity, S.NegativeInfinity):
+            return S.Zero
+        if z is S.Infinity:
+            return S.Infinity
+        if z is S.NegativeInfinity:
+            return (-1)**nu*S.Infinity
+
+        if z.could_extract_minus_sign():
+            return (z)**nu*(-z)**(-nu)*besseli(nu, -z)
+        if nu.is_integer:
+            if nu.could_extract_minus_sign():
+                return besseli(-nu, z)
+            newz = z.extract_multiplicatively(I)
+            if newz:  # NOTE we don't want to change the function if z==0
+                return I**(-nu)*besselj(nu, -newz)
+
+        # branch handling:
+        if nu.is_integer:
+            newz = unpolarify(z)
+            if newz != z:
+                return besseli(nu, newz)
+        else:
+            newz, n = z.extract_branch_factor()
+            if n != 0:
+                return exp(2*n*pi*nu*I)*besseli(nu, newz)
+        nnu = unpolarify(nu)
+        if nu != nnu:
+            return besseli(nnu, z)
+
+    def _eval_rewrite_as_besselj(self, nu, z, **kwargs):
+        return exp(-I*pi*nu/2)*besselj(nu, polar_lift(I)*z)
+
+    def _eval_rewrite_as_bessely(self, nu, z, **kwargs):
+        aj = self._eval_rewrite_as_besselj(*self.args)
+        if aj:
+            return aj.rewrite(bessely)
+
+    def _eval_rewrite_as_jn(self, nu, z, **kwargs):
+        return self._eval_rewrite_as_besselj(*self.args).rewrite(jn)
+
+    def _eval_is_extended_real(self):
+        nu, z = self.args
+        if nu.is_integer and z.is_extended_real:
+            return True
+
+    def _eval_as_leading_term(self, x, logx=None, cdir=0):
+        nu, z = self.args
+        try:
+            arg = z.as_leading_term(x)
+        except NotImplementedError:
+            return self
+        c, e = arg.as_coeff_exponent(x)
+
+        if e.is_positive:
+            return arg**nu/(2**nu*gamma(nu + 1))
+        elif e.is_negative:
+            cdir = 1 if cdir == 0 else cdir
+            sign = c*cdir**e
+            if not sign.is_negative:
+                # Refer Abramowitz and Stegun 1965, p. 377 for more information on
+                # asymptotic approximation of besseli function.
+                return exp(z)/sqrt(2*pi*z)
+            return self
+
+        return super(besseli, self)._eval_as_leading_term(x, logx, cdir)
+
+    def _eval_nseries(self, x, n, logx, cdir=0):
+        # Refer https://functions.wolfram.com/Bessel-TypeFunctions/BesselI/06/01/04/01/01/0003/
+        # for more information on nseries expansion of besseli function.
+        from sympy.series.order import Order
+        nu, z = self.args
+
+        # In case of powers less than 1, number of terms need to be computed
+        # separately to avoid repeated callings of _eval_nseries with wrong n
+        try:
+            _, exp = z.leadterm(x)
+        except (ValueError, NotImplementedError):
+            return self
+
+        if exp.is_positive:
+            newn = ceiling(n/exp)
+            o = Order(x**n, x)
+            r = (z/2)._eval_nseries(x, n, logx, cdir).removeO()
+            if r is S.Zero:
+                return o
+            t = (_mexpand(r**2) + o).removeO()
+
+            term = r**nu/gamma(nu + 1)
+            s = [term]
+            for k in range(1, (newn + 1)//2):
+                term *= t/(k*(nu + k))
+                term = (_mexpand(term) + o).removeO()
+                s.append(term)
+            return Add(*s) + o
+
+        return super(besseli, self)._eval_nseries(x, n, logx, cdir)
+
+
+class besselk(BesselBase):
+    r"""
+    Modified Bessel function of the second kind.
+
+    Explanation
+    ===========
+
+    The Bessel $K$ function of order $\nu$ is defined as
+
+    .. math ::
+        K_\nu(z) = \lim_{\mu \to \nu} \frac{\pi}{2}
+                   \frac{I_{-\mu}(z) -I_\mu(z)}{\sin(\pi \mu)},
+
+    where $I_\mu(z)$ is the modified Bessel function of the first kind.
+
+    It is a solution of the modified Bessel equation, and linearly independent
+    from $Y_\nu$.
+
+    Examples
+    ========
+
+    >>> from sympy import besselk
+    >>> from sympy.abc import z, n
+    >>> besselk(n, z).diff(z)
+    -besselk(n - 1, z)/2 - besselk(n + 1, z)/2
+
+    See Also
+    ========
+
+    besselj, besseli, bessely
+
+    References
+    ==========
+
+    .. [1] https://functions.wolfram.com/Bessel-TypeFunctions/BesselK/
+
+    """
+
+    _a = S.One
+    _b = -S.One
+
+    @classmethod
+    def eval(cls, nu, z):
+        if z.is_zero:
+            if nu.is_zero:
+                return S.Infinity
+            elif re(nu).is_zero is False:
+                return S.ComplexInfinity
+            elif re(nu).is_zero:
+                return S.NaN
+        if z in (S.Infinity, I*S.Infinity, I*S.NegativeInfinity):
+            return S.Zero
+
+        if nu.is_integer:
+            if nu.could_extract_minus_sign():
+                return besselk(-nu, z)
+
+    def _eval_rewrite_as_besseli(self, nu, z, **kwargs):
+        if nu.is_integer is False:
+            return pi*csc(pi*nu)*(besseli(-nu, z) - besseli(nu, z))/2
+
+    def _eval_rewrite_as_besselj(self, nu, z, **kwargs):
+        ai = self._eval_rewrite_as_besseli(*self.args)
+        if ai:
+            return ai.rewrite(besselj)
+
+    def _eval_rewrite_as_bessely(self, nu, z, **kwargs):
+        aj = self._eval_rewrite_as_besselj(*self.args)
+        if aj:
+            return aj.rewrite(bessely)
+
+    def _eval_rewrite_as_yn(self, nu, z, **kwargs):
+        ay = self._eval_rewrite_as_bessely(*self.args)
+        if ay:
+            return ay.rewrite(yn)
+
+    def _eval_is_extended_real(self):
+        nu, z = self.args
+        if nu.is_integer and z.is_positive:
+            return True
+
+    def _eval_as_leading_term(self, x, logx=None, cdir=0):
+        nu, z = self.args
+        try:
+            arg = z.as_leading_term(x)
+        except NotImplementedError:
+            return self
+        _, e = arg.as_coeff_exponent(x)
+
+        if e.is_positive:
+            term_one = ((-1)**(nu -1)*log(z/2)*besseli(nu, z))
+            term_two = (z/2)**(-nu)*factorial(nu - 1)/2 if (nu).is_positive else S.Zero
+            term_three = (-1)**nu*(z/2)**nu/(2*factorial(nu))*(digamma(nu + 1) - S.EulerGamma)
+            arg = Add(*[term_one, term_two, term_three]).as_leading_term(x, logx=logx)
+            return arg
+        elif e.is_negative:
+            # Refer Abramowitz and Stegun 1965, p. 378 for more information on
+            # asymptotic approximation of besselk function.
+            return sqrt(pi)*exp(-z)/sqrt(2*z)
+
+        return super(besselk, self)._eval_as_leading_term(x, logx, cdir)
+
+    def _eval_nseries(self, x, n, logx, cdir=0):
+        # Refer https://functions.wolfram.com/Bessel-TypeFunctions/BesselK/06/01/04/01/02/0008/
+        # for more information on nseries expansion of besselk function.
+        from sympy.series.order import Order
+        nu, z = self.args
+
+        # In case of powers less than 1, number of terms need to be computed
+        # separately to avoid repeated callings of _eval_nseries with wrong n
+        try:
+            _, exp = z.leadterm(x)
+        except (ValueError, NotImplementedError):
+            return self
+
+        if exp.is_positive and nu.is_integer:
+            newn = ceiling(n/exp)
+            bn = besseli(nu, z)
+            a = ((-1)**(nu - 1)*log(z/2)*bn)._eval_nseries(x, n, logx, cdir)
+
+            b, c = [], []
+            o = Order(x**n, x)
+            r = (z/2)._eval_nseries(x, n, logx, cdir).removeO()
+            if r is S.Zero:
+                return o
+            t = (_mexpand(r**2) + o).removeO()
+
+            if nu > S.Zero:
+                term = r**(-nu)*factorial(nu - 1)/2
+                b.append(term)
+                for k in range(1, nu):
+                    denom = (k - nu)*k
+                    if denom == S.Zero:
+                        term *= t/k
+                    else:
+                        term *= t/denom
+                    term = (_mexpand(term) + o).removeO()
+                    b.append(term)
+
+            p = r**nu*(-1)**nu/(2*factorial(nu))
+            term = p*(digamma(nu + 1) - S.EulerGamma)
+            c.append(term)
+            for k in range(1, (newn + 1)//2):
+                p *= t/(k*(k + nu))
+                p = (_mexpand(p) + o).removeO()
+                term = p*(digamma(k + nu + 1) + digamma(k + 1))
+                c.append(term)
+            return a + Add(*b) + Add(*c) # Order term comes from a
+
+        return super(besselk, self)._eval_nseries(x, n, logx, cdir)
+
+
+class hankel1(BesselBase):
+    r"""
+    Hankel function of the first kind.
+
+    Explanation
+    ===========
+
+    This function is defined as
+
+    .. math ::
+        H_\nu^{(1)} = J_\nu(z) + iY_\nu(z),
+
+    where $J_\nu(z)$ is the Bessel function of the first kind, and
+    $Y_\nu(z)$ is the Bessel function of the second kind.
+
+    It is a solution to Bessel's equation.
+
+    Examples
+    ========
+
+    >>> from sympy import hankel1
+    >>> from sympy.abc import z, n
+    >>> hankel1(n, z).diff(z)
+    hankel1(n - 1, z)/2 - hankel1(n + 1, z)/2
+
+    See Also
+    ========
+
+    hankel2, besselj, bessely
+
+    References
+    ==========
+
+    .. [1] https://functions.wolfram.com/Bessel-TypeFunctions/HankelH1/
+
+    """
+
+    _a = S.One
+    _b = S.One
+
+    def _eval_conjugate(self):
+        z = self.argument
+        if z.is_extended_negative is False:
+            return hankel2(self.order.conjugate(), z.conjugate())
+
+
+class hankel2(BesselBase):
+    r"""
+    Hankel function of the second kind.
+
+    Explanation
+    ===========
+
+    This function is defined as
+
+    .. math ::
+        H_\nu^{(2)} = J_\nu(z) - iY_\nu(z),
+
+    where $J_\nu(z)$ is the Bessel function of the first kind, and
+    $Y_\nu(z)$ is the Bessel function of the second kind.
+
+    It is a solution to Bessel's equation, and linearly independent from
+    $H_\nu^{(1)}$.
+
+    Examples
+    ========
+
+    >>> from sympy import hankel2
+    >>> from sympy.abc import z, n
+    >>> hankel2(n, z).diff(z)
+    hankel2(n - 1, z)/2 - hankel2(n + 1, z)/2
+
+    See Also
+    ========
+
+    hankel1, besselj, bessely
+
+    References
+    ==========
+
+    .. [1] https://functions.wolfram.com/Bessel-TypeFunctions/HankelH2/
+
+    """
+
+    _a = S.One
+    _b = S.One
+
+    def _eval_conjugate(self):
+        z = self.argument
+        if z.is_extended_negative is False:
+            return hankel1(self.order.conjugate(), z.conjugate())
+
+
+def assume_integer_order(fn):
+    @wraps(fn)
+    def g(self, nu, z):
+        if nu.is_integer:
+            return fn(self, nu, z)
+    return g
+
+
+class SphericalBesselBase(BesselBase):
+    """
+    Base class for spherical Bessel functions.
+
+    These are thin wrappers around ordinary Bessel functions,
+    since spherical Bessel functions differ from the ordinary
+    ones just by a slight change in order.
+
+    To use this class, define the ``_eval_evalf()`` and ``_expand()`` methods.
+
+    """
+
+    def _expand(self, **hints):
+        """ Expand self into a polynomial. Nu is guaranteed to be Integer. """
+        raise NotImplementedError('expansion')
+
+    def _eval_expand_func(self, **hints):
+        if self.order.is_Integer:
+            return self._expand(**hints)
+        return self
+
+    def fdiff(self, argindex=2):
+        if argindex != 2:
+            raise ArgumentIndexError(self, argindex)
+        return self.__class__(self.order - 1, self.argument) - \
+            self * (self.order + 1)/self.argument
+
+
+def _jn(n, z):
+    return (spherical_bessel_fn(n, z)*sin(z) +
+            S.NegativeOne**(n + 1)*spherical_bessel_fn(-n - 1, z)*cos(z))
+
+
+def _yn(n, z):
+    # (-1)**(n + 1) * _jn(-n - 1, z)
+    return (S.NegativeOne**(n + 1) * spherical_bessel_fn(-n - 1, z)*sin(z) -
+            spherical_bessel_fn(n, z)*cos(z))
+
+
+class jn(SphericalBesselBase):
+    r"""
+    Spherical Bessel function of the first kind.
+
+    Explanation
+    ===========
+
+    This function is a solution to the spherical Bessel equation
+
+    .. math ::
+        z^2 \frac{\mathrm{d}^2 w}{\mathrm{d}z^2}
+          + 2z \frac{\mathrm{d}w}{\mathrm{d}z} + (z^2 - \nu(\nu + 1)) w = 0.
+
+    It can be defined as
+
+    .. math ::
+        j_\nu(z) = \sqrt{\frac{\pi}{2z}} J_{\nu + \frac{1}{2}}(z),
+
+    where $J_\nu(z)$ is the Bessel function of the first kind.
+
+    The spherical Bessel functions of integral order are
+    calculated using the formula:
+
+    .. math:: j_n(z) = f_n(z) \sin{z} + (-1)^{n+1} f_{-n-1}(z) \cos{z},
+
+    where the coefficients $f_n(z)$ are available as
+    :func:`sympy.polys.orthopolys.spherical_bessel_fn`.
+
+    Examples
+    ========
+
+    >>> from sympy import Symbol, jn, sin, cos, expand_func, besselj, bessely
+    >>> z = Symbol("z")
+    >>> nu = Symbol("nu", integer=True)
+    >>> print(expand_func(jn(0, z)))
+    sin(z)/z
+    >>> expand_func(jn(1, z)) == sin(z)/z**2 - cos(z)/z
+    True
+    >>> expand_func(jn(3, z))
+    (-6/z**2 + 15/z**4)*sin(z) + (1/z - 15/z**3)*cos(z)
+    >>> jn(nu, z).rewrite(besselj)
+    sqrt(2)*sqrt(pi)*sqrt(1/z)*besselj(nu + 1/2, z)/2
+    >>> jn(nu, z).rewrite(bessely)
+    (-1)**nu*sqrt(2)*sqrt(pi)*sqrt(1/z)*bessely(-nu - 1/2, z)/2
+    >>> jn(2, 5.2+0.3j).evalf(20)
+    0.099419756723640344491 - 0.054525080242173562897*I
+
+    See Also
+    ========
+
+    besselj, bessely, besselk, yn
+
+    References
+    ==========
+
+    .. [1] https://dlmf.nist.gov/10.47
+
+    """
+    @classmethod
+    def eval(cls, nu, z):
+        if z.is_zero:
+            if nu.is_zero:
+                return S.One
+            elif nu.is_integer:
+                if nu.is_positive:
+                    return S.Zero
+                else:
+                    return S.ComplexInfinity
+        if z in (S.NegativeInfinity, S.Infinity):
+            return S.Zero
+
+    def _eval_rewrite_as_besselj(self, nu, z, **kwargs):
+        return sqrt(pi/(2*z)) * besselj(nu + S.Half, z)
+
+    def _eval_rewrite_as_bessely(self, nu, z, **kwargs):
+        return S.NegativeOne**nu * sqrt(pi/(2*z)) * bessely(-nu - S.Half, z)
+
+    def _eval_rewrite_as_yn(self, nu, z, **kwargs):
+        return S.NegativeOne**(nu) * yn(-nu - 1, z)
+
+    def _expand(self, **hints):
+        return _jn(self.order, self.argument)
+
+    def _eval_evalf(self, prec):
+        if self.order.is_Integer:
+            return self.rewrite(besselj)._eval_evalf(prec)
+
+
+class yn(SphericalBesselBase):
+    r"""
+    Spherical Bessel function of the second kind.
+
+    Explanation
+    ===========
+
+    This function is another solution to the spherical Bessel equation, and
+    linearly independent from $j_n$. It can be defined as
+
+    .. math ::
+        y_\nu(z) = \sqrt{\frac{\pi}{2z}} Y_{\nu + \frac{1}{2}}(z),
+
+    where $Y_\nu(z)$ is the Bessel function of the second kind.
+
+    For integral orders $n$, $y_n$ is calculated using the formula:
+
+    .. math:: y_n(z) = (-1)^{n+1} j_{-n-1}(z)
+
+    Examples
+    ========
+
+    >>> from sympy import Symbol, yn, sin, cos, expand_func, besselj, bessely
+    >>> z = Symbol("z")
+    >>> nu = Symbol("nu", integer=True)
+    >>> print(expand_func(yn(0, z)))
+    -cos(z)/z
+    >>> expand_func(yn(1, z)) == -cos(z)/z**2-sin(z)/z
+    True
+    >>> yn(nu, z).rewrite(besselj)
+    (-1)**(nu + 1)*sqrt(2)*sqrt(pi)*sqrt(1/z)*besselj(-nu - 1/2, z)/2
+    >>> yn(nu, z).rewrite(bessely)
+    sqrt(2)*sqrt(pi)*sqrt(1/z)*bessely(nu + 1/2, z)/2
+    >>> yn(2, 5.2+0.3j).evalf(20)
+    0.18525034196069722536 + 0.014895573969924817587*I
+
+    See Also
+    ========
+
+    besselj, bessely, besselk, jn
+
+    References
+    ==========
+
+    .. [1] https://dlmf.nist.gov/10.47
+
+    """
+    @assume_integer_order
+    def _eval_rewrite_as_besselj(self, nu, z, **kwargs):
+        return S.NegativeOne**(nu+1) * sqrt(pi/(2*z)) * besselj(-nu - S.Half, z)
+
+    @assume_integer_order
+    def _eval_rewrite_as_bessely(self, nu, z, **kwargs):
+        return sqrt(pi/(2*z)) * bessely(nu + S.Half, z)
+
+    def _eval_rewrite_as_jn(self, nu, z, **kwargs):
+        return S.NegativeOne**(nu + 1) * jn(-nu - 1, z)
+
+    def _expand(self, **hints):
+        return _yn(self.order, self.argument)
+
+    def _eval_evalf(self, prec):
+        if self.order.is_Integer:
+            return self.rewrite(bessely)._eval_evalf(prec)
+
+
+class SphericalHankelBase(SphericalBesselBase):
+
+    @assume_integer_order
+    def _eval_rewrite_as_besselj(self, nu, z, **kwargs):
+        # jn +- I*yn
+        # jn as beeselj: sqrt(pi/(2*z)) * besselj(nu + S.Half, z)
+        # yn as besselj: (-1)**(nu+1) * sqrt(pi/(2*z)) * besselj(-nu - S.Half, z)
+        hks = self._hankel_kind_sign
+        return sqrt(pi/(2*z))*(besselj(nu + S.Half, z) +
+                               hks*I*S.NegativeOne**(nu+1)*besselj(-nu - S.Half, z))
+
+    @assume_integer_order
+    def _eval_rewrite_as_bessely(self, nu, z, **kwargs):
+        # jn +- I*yn
+        # jn as bessely: (-1)**nu * sqrt(pi/(2*z)) * bessely(-nu - S.Half, z)
+        # yn as bessely: sqrt(pi/(2*z)) * bessely(nu + S.Half, z)
+        hks = self._hankel_kind_sign
+        return sqrt(pi/(2*z))*(S.NegativeOne**nu*bessely(-nu - S.Half, z) +
+                               hks*I*bessely(nu + S.Half, z))
+
+    def _eval_rewrite_as_yn(self, nu, z, **kwargs):
+        hks = self._hankel_kind_sign
+        return jn(nu, z).rewrite(yn) + hks*I*yn(nu, z)
+
+    def _eval_rewrite_as_jn(self, nu, z, **kwargs):
+        hks = self._hankel_kind_sign
+        return jn(nu, z) + hks*I*yn(nu, z).rewrite(jn)
+
+    def _eval_expand_func(self, **hints):
+        if self.order.is_Integer:
+            return self._expand(**hints)
+        else:
+            nu = self.order
+            z = self.argument
+            hks = self._hankel_kind_sign
+            return jn(nu, z) + hks*I*yn(nu, z)
+
+    def _expand(self, **hints):
+        n = self.order
+        z = self.argument
+        hks = self._hankel_kind_sign
+
+        # fully expanded version
+        # return ((fn(n, z) * sin(z) +
+        #          (-1)**(n + 1) * fn(-n - 1, z) * cos(z)) +  # jn
+        #         (hks * I * (-1)**(n + 1) *
+        #          (fn(-n - 1, z) * hk * I * sin(z) +
+        #           (-1)**(-n) * fn(n, z) * I * cos(z)))  # +-I*yn
+        #         )
+
+        return (_jn(n, z) + hks*I*_yn(n, z)).expand()
+
+    def _eval_evalf(self, prec):
+        if self.order.is_Integer:
+            return self.rewrite(besselj)._eval_evalf(prec)
+
+
+class hn1(SphericalHankelBase):
+    r"""
+    Spherical Hankel function of the first kind.
+
+    Explanation
+    ===========
+
+    This function is defined as
+
+    .. math:: h_\nu^(1)(z) = j_\nu(z) + i y_\nu(z),
+
+    where $j_\nu(z)$ and $y_\nu(z)$ are the spherical
+    Bessel function of the first and second kinds.
+
+    For integral orders $n$, $h_n^(1)$ is calculated using the formula:
+
+    .. math:: h_n^(1)(z) = j_{n}(z) + i (-1)^{n+1} j_{-n-1}(z)
+
+    Examples
+    ========
+
+    >>> from sympy import Symbol, hn1, hankel1, expand_func, yn, jn
+    >>> z = Symbol("z")
+    >>> nu = Symbol("nu", integer=True)
+    >>> print(expand_func(hn1(nu, z)))
+    jn(nu, z) + I*yn(nu, z)
+    >>> print(expand_func(hn1(0, z)))
+    sin(z)/z - I*cos(z)/z
+    >>> print(expand_func(hn1(1, z)))
+    -I*sin(z)/z - cos(z)/z + sin(z)/z**2 - I*cos(z)/z**2
+    >>> hn1(nu, z).rewrite(jn)
+    (-1)**(nu + 1)*I*jn(-nu - 1, z) + jn(nu, z)
+    >>> hn1(nu, z).rewrite(yn)
+    (-1)**nu*yn(-nu - 1, z) + I*yn(nu, z)
+    >>> hn1(nu, z).rewrite(hankel1)
+    sqrt(2)*sqrt(pi)*sqrt(1/z)*hankel1(nu, z)/2
+
+    See Also
+    ========
+
+    hn2, jn, yn, hankel1, hankel2
+
+    References
+    ==========
+
+    .. [1] https://dlmf.nist.gov/10.47
+
+    """
+
+    _hankel_kind_sign = S.One
+
+    @assume_integer_order
+    def _eval_rewrite_as_hankel1(self, nu, z, **kwargs):
+        return sqrt(pi/(2*z))*hankel1(nu, z)
+
+
+class hn2(SphericalHankelBase):
+    r"""
+    Spherical Hankel function of the second kind.
+
+    Explanation
+    ===========
+
+    This function is defined as
+
+    .. math:: h_\nu^(2)(z) = j_\nu(z) - i y_\nu(z),
+
+    where $j_\nu(z)$ and $y_\nu(z)$ are the spherical
+    Bessel function of the first and second kinds.
+
+    For integral orders $n$, $h_n^(2)$ is calculated using the formula:
+
+    .. math:: h_n^(2)(z) = j_{n} - i (-1)^{n+1} j_{-n-1}(z)
+
+    Examples
+    ========
+
+    >>> from sympy import Symbol, hn2, hankel2, expand_func, jn, yn
+    >>> z = Symbol("z")
+    >>> nu = Symbol("nu", integer=True)
+    >>> print(expand_func(hn2(nu, z)))
+    jn(nu, z) - I*yn(nu, z)
+    >>> print(expand_func(hn2(0, z)))
+    sin(z)/z + I*cos(z)/z
+    >>> print(expand_func(hn2(1, z)))
+    I*sin(z)/z - cos(z)/z + sin(z)/z**2 + I*cos(z)/z**2
+    >>> hn2(nu, z).rewrite(hankel2)
+    sqrt(2)*sqrt(pi)*sqrt(1/z)*hankel2(nu, z)/2
+    >>> hn2(nu, z).rewrite(jn)
+    -(-1)**(nu + 1)*I*jn(-nu - 1, z) + jn(nu, z)
+    >>> hn2(nu, z).rewrite(yn)
+    (-1)**nu*yn(-nu - 1, z) - I*yn(nu, z)
+
+    See Also
+    ========
+
+    hn1, jn, yn, hankel1, hankel2
+
+    References
+    ==========
+
+    .. [1] https://dlmf.nist.gov/10.47
+
+    """
+
+    _hankel_kind_sign = -S.One
+
+    @assume_integer_order
+    def _eval_rewrite_as_hankel2(self, nu, z, **kwargs):
+        return sqrt(pi/(2*z))*hankel2(nu, z)
+
+
+def jn_zeros(n, k, method="sympy", dps=15):
+    """
+    Zeros of the spherical Bessel function of the first kind.
+
+    Explanation
+    ===========
+
+    This returns an array of zeros of $jn$ up to the $k$-th zero.
+
+    * method = "sympy": uses `mpmath.besseljzero
+      <https://mpmath.org/doc/current/functions/bessel.html#mpmath.besseljzero>`_
+    * method = "scipy": uses the
+      `SciPy's sph_jn <https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.jn_zeros.html>`_
+      and
+      `newton <https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.newton.html>`_
+      to find all
+      roots, which is faster than computing the zeros using a general
+      numerical solver, but it requires SciPy and only works with low
+      precision floating point numbers. (The function used with
+      method="sympy" is a recent addition to mpmath; before that a general
+      solver was used.)
+
+    Examples
+    ========
+
+    >>> from sympy import jn_zeros
+    >>> jn_zeros(2, 4, dps=5)
+    [5.7635, 9.095, 12.323, 15.515]
+
+    See Also
+    ========
+
+    jn, yn, besselj, besselk, bessely
+
+    Parameters
+    ==========
+
+    n : integer
+        order of Bessel function
+
+    k : integer
+        number of zeros to return
+
+
+    """
+    from math import pi as math_pi
+
+    if method == "sympy":
+        from mpmath import besseljzero
+        from mpmath.libmp.libmpf import dps_to_prec
+        prec = dps_to_prec(dps)
+        return [Expr._from_mpmath(besseljzero(S(n + 0.5)._to_mpmath(prec),
+                                              int(l)), prec)
+                for l in range(1, k + 1)]
+    elif method == "scipy":
+        from scipy.optimize import newton
+        try:
+            from scipy.special import spherical_jn
+            f = lambda x: spherical_jn(n, x)
+        except ImportError:
+            from scipy.special import sph_jn
+            f = lambda x: sph_jn(n, x)[0][-1]
+    else:
+        raise NotImplementedError("Unknown method.")
+
+    def solver(f, x):
+        if method == "scipy":
+            root = newton(f, x)
+        else:
+            raise NotImplementedError("Unknown method.")
+        return root
+
+    # we need to approximate the position of the first root:
+    root = n + math_pi
+    # determine the first root exactly:
+    root = solver(f, root)
+    roots = [root]
+    for i in range(k - 1):
+        # estimate the position of the next root using the last root + pi:
+        root = solver(f, root + math_pi)
+        roots.append(root)
+    return roots
+
+
+class AiryBase(Function):
+    """
+    Abstract base class for Airy functions.
+
+    This class is meant to reduce code duplication.
+
+    """
+
+    def _eval_conjugate(self):
+        return self.func(self.args[0].conjugate())
+
+    def _eval_is_extended_real(self):
+        return self.args[0].is_extended_real
+
+    def as_real_imag(self, deep=True, **hints):
+        z = self.args[0]
+        zc = z.conjugate()
+        f = self.func
+        u = (f(z)+f(zc))/2
+        v = I*(f(zc)-f(z))/2
+        return u, v
+
+    def _eval_expand_complex(self, deep=True, **hints):
+        re_part, im_part = self.as_real_imag(deep=deep, **hints)
+        return re_part + im_part*I
+
+
+class airyai(AiryBase):
+    r"""
+    The Airy function $\operatorname{Ai}$ of the first kind.
+
+    Explanation
+    ===========
+
+    The Airy function $\operatorname{Ai}(z)$ is defined to be the function
+    satisfying Airy's differential equation
+
+    .. math::
+        \frac{\mathrm{d}^2 w(z)}{\mathrm{d}z^2} - z w(z) = 0.
+
+    Equivalently, for real $z$
+
+    .. math::
+        \operatorname{Ai}(z) := \frac{1}{\pi}
+        \int_0^\infty \cos\left(\frac{t^3}{3} + z t\right) \mathrm{d}t.
+
+    Examples
+    ========
+
+    Create an Airy function object:
+
+    >>> from sympy import airyai
+    >>> from sympy.abc import z
+
+    >>> airyai(z)
+    airyai(z)
+
+    Several special values are known:
+
+    >>> airyai(0)
+    3**(1/3)/(3*gamma(2/3))
+    >>> from sympy import oo
+    >>> airyai(oo)
+    0
+    >>> airyai(-oo)
+    0
+
+    The Airy function obeys the mirror symmetry:
+
+    >>> from sympy import conjugate
+    >>> conjugate(airyai(z))
+    airyai(conjugate(z))
+
+    Differentiation with respect to $z$ is supported:
+
+    >>> from sympy import diff
+    >>> diff(airyai(z), z)
+    airyaiprime(z)
+    >>> diff(airyai(z), z, 2)
+    z*airyai(z)
+
+    Series expansion is also supported:
+
+    >>> from sympy import series
+    >>> series(airyai(z), z, 0, 3)
+    3**(5/6)*gamma(1/3)/(6*pi) - 3**(1/6)*z*gamma(2/3)/(2*pi) + O(z**3)
+
+    We can numerically evaluate the Airy function to arbitrary precision
+    on the whole complex plane:
+
+    >>> airyai(-2).evalf(50)
+    0.22740742820168557599192443603787379946077222541710
+
+    Rewrite $\operatorname{Ai}(z)$ in terms of hypergeometric functions:
+
+    >>> from sympy import hyper
+    >>> airyai(z).rewrite(hyper)
+    -3**(2/3)*z*hyper((), (4/3,), z**3/9)/(3*gamma(1/3)) + 3**(1/3)*hyper((), (2/3,), z**3/9)/(3*gamma(2/3))
+
+    See Also
+    ========
+
+    airybi: Airy function of the second kind.
+    airyaiprime: Derivative of the Airy function of the first kind.
+    airybiprime: Derivative of the Airy function of the second kind.
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Airy_function
+    .. [2] https://dlmf.nist.gov/9
+    .. [3] https://encyclopediaofmath.org/wiki/Airy_functions
+    .. [4] https://mathworld.wolfram.com/AiryFunctions.html
+
+    """
+
+    nargs = 1
+    unbranched = True
+
+    @classmethod
+    def eval(cls, arg):
+        if arg.is_Number:
+            if arg is S.NaN:
+                return S.NaN
+            elif arg is S.Infinity:
+                return S.Zero
+            elif arg is S.NegativeInfinity:
+                return S.Zero
+            elif arg.is_zero:
+                return S.One / (3**Rational(2, 3) * gamma(Rational(2, 3)))
+        if arg.is_zero:
+            return S.One / (3**Rational(2, 3) * gamma(Rational(2, 3)))
+
+    def fdiff(self, argindex=1):
+        if argindex == 1:
+            return airyaiprime(self.args[0])
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    @staticmethod
+    @cacheit
+    def taylor_term(n, x, *previous_terms):
+        if n < 0:
+            return S.Zero
+        else:
+            x = sympify(x)
+            if len(previous_terms) > 1:
+                p = previous_terms[-1]
+                return ((cbrt(3)*x)**(-n)*(cbrt(3)*x)**(n + 1)*sin(pi*(n*Rational(2, 3) + Rational(4, 3)))*factorial(n) *
+                        gamma(n/3 + Rational(2, 3))/(sin(pi*(n*Rational(2, 3) + Rational(2, 3)))*factorial(n + 1)*gamma(n/3 + Rational(1, 3))) * p)
+            else:
+                return (S.One/(3**Rational(2, 3)*pi) * gamma((n+S.One)/S(3)) * sin(Rational(2, 3)*pi*(n+S.One)) /
+                        factorial(n) * (cbrt(3)*x)**n)
+
+    def _eval_rewrite_as_besselj(self, z, **kwargs):
+        ot = Rational(1, 3)
+        tt = Rational(2, 3)
+        a = Pow(-z, Rational(3, 2))
+        if re(z).is_negative:
+            return ot*sqrt(-z) * (besselj(-ot, tt*a) + besselj(ot, tt*a))
+
+    def _eval_rewrite_as_besseli(self, z, **kwargs):
+        ot = Rational(1, 3)
+        tt = Rational(2, 3)
+        a = Pow(z, Rational(3, 2))
+        if re(z).is_positive:
+            return ot*sqrt(z) * (besseli(-ot, tt*a) - besseli(ot, tt*a))
+        else:
+            return ot*(Pow(a, ot)*besseli(-ot, tt*a) - z*Pow(a, -ot)*besseli(ot, tt*a))
+
+    def _eval_rewrite_as_hyper(self, z, **kwargs):
+        pf1 = S.One / (3**Rational(2, 3)*gamma(Rational(2, 3)))
+        pf2 = z / (root(3, 3)*gamma(Rational(1, 3)))
+        return pf1 * hyper([], [Rational(2, 3)], z**3/9) - pf2 * hyper([], [Rational(4, 3)], z**3/9)
+
+    def _eval_expand_func(self, **hints):
+        arg = self.args[0]
+        symbs = arg.free_symbols
+
+        if len(symbs) == 1:
+            z = symbs.pop()
+            c = Wild("c", exclude=[z])
+            d = Wild("d", exclude=[z])
+            m = Wild("m", exclude=[z])
+            n = Wild("n", exclude=[z])
+            M = arg.match(c*(d*z**n)**m)
+            if M is not None:
+                m = M[m]
+                # The transformation is given by 03.05.16.0001.01
+                # https://functions.wolfram.com/Bessel-TypeFunctions/AiryAi/16/01/01/0001/
+                if (3*m).is_integer:
+                    c = M[c]
+                    d = M[d]
+                    n = M[n]
+                    pf = (d * z**n)**m / (d**m * z**(m*n))
+                    newarg = c * d**m * z**(m*n)
+                    return S.Half * ((pf + S.One)*airyai(newarg) - (pf - S.One)/sqrt(3)*airybi(newarg))
+
+
+class airybi(AiryBase):
+    r"""
+    The Airy function $\operatorname{Bi}$ of the second kind.
+
+    Explanation
+    ===========
+
+    The Airy function $\operatorname{Bi}(z)$ is defined to be the function
+    satisfying Airy's differential equation
+
+    .. math::
+        \frac{\mathrm{d}^2 w(z)}{\mathrm{d}z^2} - z w(z) = 0.
+
+    Equivalently, for real $z$
+
+    .. math::
+        \operatorname{Bi}(z) := \frac{1}{\pi}
+                 \int_0^\infty
+                   \exp\left(-\frac{t^3}{3} + z t\right)
+                   + \sin\left(\frac{t^3}{3} + z t\right) \mathrm{d}t.
+
+    Examples
+    ========
+
+    Create an Airy function object:
+
+    >>> from sympy import airybi
+    >>> from sympy.abc import z
+
+    >>> airybi(z)
+    airybi(z)
+
+    Several special values are known:
+
+    >>> airybi(0)
+    3**(5/6)/(3*gamma(2/3))
+    >>> from sympy import oo
+    >>> airybi(oo)
+    oo
+    >>> airybi(-oo)
+    0
+
+    The Airy function obeys the mirror symmetry:
+
+    >>> from sympy import conjugate
+    >>> conjugate(airybi(z))
+    airybi(conjugate(z))
+
+    Differentiation with respect to $z$ is supported:
+
+    >>> from sympy import diff
+    >>> diff(airybi(z), z)
+    airybiprime(z)
+    >>> diff(airybi(z), z, 2)
+    z*airybi(z)
+
+    Series expansion is also supported:
+
+    >>> from sympy import series
+    >>> series(airybi(z), z, 0, 3)
+    3**(1/3)*gamma(1/3)/(2*pi) + 3**(2/3)*z*gamma(2/3)/(2*pi) + O(z**3)
+
+    We can numerically evaluate the Airy function to arbitrary precision
+    on the whole complex plane:
+
+    >>> airybi(-2).evalf(50)
+    -0.41230258795639848808323405461146104203453483447240
+
+    Rewrite $\operatorname{Bi}(z)$ in terms of hypergeometric functions:
+
+    >>> from sympy import hyper
+    >>> airybi(z).rewrite(hyper)
+    3**(1/6)*z*hyper((), (4/3,), z**3/9)/gamma(1/3) + 3**(5/6)*hyper((), (2/3,), z**3/9)/(3*gamma(2/3))
+
+    See Also
+    ========
+
+    airyai: Airy function of the first kind.
+    airyaiprime: Derivative of the Airy function of the first kind.
+    airybiprime: Derivative of the Airy function of the second kind.
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Airy_function
+    .. [2] https://dlmf.nist.gov/9
+    .. [3] https://encyclopediaofmath.org/wiki/Airy_functions
+    .. [4] https://mathworld.wolfram.com/AiryFunctions.html
+
+    """
+
+    nargs = 1
+    unbranched = True
+
+    @classmethod
+    def eval(cls, arg):
+        if arg.is_Number:
+            if arg is S.NaN:
+                return S.NaN
+            elif arg is S.Infinity:
+                return S.Infinity
+            elif arg is S.NegativeInfinity:
+                return S.Zero
+            elif arg.is_zero:
+                return S.One / (3**Rational(1, 6) * gamma(Rational(2, 3)))
+
+        if arg.is_zero:
+            return S.One / (3**Rational(1, 6) * gamma(Rational(2, 3)))
+
+    def fdiff(self, argindex=1):
+        if argindex == 1:
+            return airybiprime(self.args[0])
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    @staticmethod
+    @cacheit
+    def taylor_term(n, x, *previous_terms):
+        if n < 0:
+            return S.Zero
+        else:
+            x = sympify(x)
+            if len(previous_terms) > 1:
+                p = previous_terms[-1]
+                return (cbrt(3)*x * Abs(sin(Rational(2, 3)*pi*(n + S.One))) * factorial((n - S.One)/S(3)) /
+                        ((n + S.One) * Abs(cos(Rational(2, 3)*pi*(n + S.Half))) * factorial((n - 2)/S(3))) * p)
+            else:
+                return (S.One/(root(3, 6)*pi) * gamma((n + S.One)/S(3)) * Abs(sin(Rational(2, 3)*pi*(n + S.One))) /
+                        factorial(n) * (cbrt(3)*x)**n)
+
+    def _eval_rewrite_as_besselj(self, z, **kwargs):
+        ot = Rational(1, 3)
+        tt = Rational(2, 3)
+        a = Pow(-z, Rational(3, 2))
+        if re(z).is_negative:
+            return sqrt(-z/3) * (besselj(-ot, tt*a) - besselj(ot, tt*a))
+
+    def _eval_rewrite_as_besseli(self, z, **kwargs):
+        ot = Rational(1, 3)
+        tt = Rational(2, 3)
+        a = Pow(z, Rational(3, 2))
+        if re(z).is_positive:
+            return sqrt(z)/sqrt(3) * (besseli(-ot, tt*a) + besseli(ot, tt*a))
+        else:
+            b = Pow(a, ot)
+            c = Pow(a, -ot)
+            return sqrt(ot)*(b*besseli(-ot, tt*a) + z*c*besseli(ot, tt*a))
+
+    def _eval_rewrite_as_hyper(self, z, **kwargs):
+        pf1 = S.One / (root(3, 6)*gamma(Rational(2, 3)))
+        pf2 = z*root(3, 6) / gamma(Rational(1, 3))
+        return pf1 * hyper([], [Rational(2, 3)], z**3/9) + pf2 * hyper([], [Rational(4, 3)], z**3/9)
+
+    def _eval_expand_func(self, **hints):
+        arg = self.args[0]
+        symbs = arg.free_symbols
+
+        if len(symbs) == 1:
+            z = symbs.pop()
+            c = Wild("c", exclude=[z])
+            d = Wild("d", exclude=[z])
+            m = Wild("m", exclude=[z])
+            n = Wild("n", exclude=[z])
+            M = arg.match(c*(d*z**n)**m)
+            if M is not None:
+                m = M[m]
+                # The transformation is given by 03.06.16.0001.01
+                # https://functions.wolfram.com/Bessel-TypeFunctions/AiryBi/16/01/01/0001/
+                if (3*m).is_integer:
+                    c = M[c]
+                    d = M[d]
+                    n = M[n]
+                    pf = (d * z**n)**m / (d**m * z**(m*n))
+                    newarg = c * d**m * z**(m*n)
+                    return S.Half * (sqrt(3)*(S.One - pf)*airyai(newarg) + (S.One + pf)*airybi(newarg))
+
+
+class airyaiprime(AiryBase):
+    r"""
+    The derivative $\operatorname{Ai}^\prime$ of the Airy function of the first
+    kind.
+
+    Explanation
+    ===========
+
+    The Airy function $\operatorname{Ai}^\prime(z)$ is defined to be the
+    function
+
+    .. math::
+        \operatorname{Ai}^\prime(z) := \frac{\mathrm{d} \operatorname{Ai}(z)}{\mathrm{d} z}.
+
+    Examples
+    ========
+
+    Create an Airy function object:
+
+    >>> from sympy import airyaiprime
+    >>> from sympy.abc import z
+
+    >>> airyaiprime(z)
+    airyaiprime(z)
+
+    Several special values are known:
+
+    >>> airyaiprime(0)
+    -3**(2/3)/(3*gamma(1/3))
+    >>> from sympy import oo
+    >>> airyaiprime(oo)
+    0
+
+    The Airy function obeys the mirror symmetry:
+
+    >>> from sympy import conjugate
+    >>> conjugate(airyaiprime(z))
+    airyaiprime(conjugate(z))
+
+    Differentiation with respect to $z$ is supported:
+
+    >>> from sympy import diff
+    >>> diff(airyaiprime(z), z)
+    z*airyai(z)
+    >>> diff(airyaiprime(z), z, 2)
+    z*airyaiprime(z) + airyai(z)
+
+    Series expansion is also supported:
+
+    >>> from sympy import series
+    >>> series(airyaiprime(z), z, 0, 3)
+    -3**(2/3)/(3*gamma(1/3)) + 3**(1/3)*z**2/(6*gamma(2/3)) + O(z**3)
+
+    We can numerically evaluate the Airy function to arbitrary precision
+    on the whole complex plane:
+
+    >>> airyaiprime(-2).evalf(50)
+    0.61825902074169104140626429133247528291577794512415
+
+    Rewrite $\operatorname{Ai}^\prime(z)$ in terms of hypergeometric functions:
+
+    >>> from sympy import hyper
+    >>> airyaiprime(z).rewrite(hyper)
+    3**(1/3)*z**2*hyper((), (5/3,), z**3/9)/(6*gamma(2/3)) - 3**(2/3)*hyper((), (1/3,), z**3/9)/(3*gamma(1/3))
+
+    See Also
+    ========
+
+    airyai: Airy function of the first kind.
+    airybi: Airy function of the second kind.
+    airybiprime: Derivative of the Airy function of the second kind.
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Airy_function
+    .. [2] https://dlmf.nist.gov/9
+    .. [3] https://encyclopediaofmath.org/wiki/Airy_functions
+    .. [4] https://mathworld.wolfram.com/AiryFunctions.html
+
+    """
+
+    nargs = 1
+    unbranched = True
+
+    @classmethod
+    def eval(cls, arg):
+        if arg.is_Number:
+            if arg is S.NaN:
+                return S.NaN
+            elif arg is S.Infinity:
+                return S.Zero
+
+        if arg.is_zero:
+            return S.NegativeOne / (3**Rational(1, 3) * gamma(Rational(1, 3)))
+
+    def fdiff(self, argindex=1):
+        if argindex == 1:
+            return self.args[0]*airyai(self.args[0])
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    def _eval_evalf(self, prec):
+        z = self.args[0]._to_mpmath(prec)
+        with workprec(prec):
+            res = mp.airyai(z, derivative=1)
+        return Expr._from_mpmath(res, prec)
+
+    def _eval_rewrite_as_besselj(self, z, **kwargs):
+        tt = Rational(2, 3)
+        a = Pow(-z, Rational(3, 2))
+        if re(z).is_negative:
+            return z/3 * (besselj(-tt, tt*a) - besselj(tt, tt*a))
+
+    def _eval_rewrite_as_besseli(self, z, **kwargs):
+        ot = Rational(1, 3)
+        tt = Rational(2, 3)
+        a = tt * Pow(z, Rational(3, 2))
+        if re(z).is_positive:
+            return z/3 * (besseli(tt, a) - besseli(-tt, a))
+        else:
+            a = Pow(z, Rational(3, 2))
+            b = Pow(a, tt)
+            c = Pow(a, -tt)
+            return ot * (z**2*c*besseli(tt, tt*a) - b*besseli(-ot, tt*a))
+
+    def _eval_rewrite_as_hyper(self, z, **kwargs):
+        pf1 = z**2 / (2*3**Rational(2, 3)*gamma(Rational(2, 3)))
+        pf2 = 1 / (root(3, 3)*gamma(Rational(1, 3)))
+        return pf1 * hyper([], [Rational(5, 3)], z**3/9) - pf2 * hyper([], [Rational(1, 3)], z**3/9)
+
+    def _eval_expand_func(self, **hints):
+        arg = self.args[0]
+        symbs = arg.free_symbols
+
+        if len(symbs) == 1:
+            z = symbs.pop()
+            c = Wild("c", exclude=[z])
+            d = Wild("d", exclude=[z])
+            m = Wild("m", exclude=[z])
+            n = Wild("n", exclude=[z])
+            M = arg.match(c*(d*z**n)**m)
+            if M is not None:
+                m = M[m]
+                # The transformation is in principle
+                # given by 03.07.16.0001.01 but note
+                # that there is an error in this formula.
+                # https://functions.wolfram.com/Bessel-TypeFunctions/AiryAiPrime/16/01/01/0001/
+                if (3*m).is_integer:
+                    c = M[c]
+                    d = M[d]
+                    n = M[n]
+                    pf = (d**m * z**(n*m)) / (d * z**n)**m
+                    newarg = c * d**m * z**(n*m)
+                    return S.Half * ((pf + S.One)*airyaiprime(newarg) + (pf - S.One)/sqrt(3)*airybiprime(newarg))
+
+
+class airybiprime(AiryBase):
+    r"""
+    The derivative $\operatorname{Bi}^\prime$ of the Airy function of the first
+    kind.
+
+    Explanation
+    ===========
+
+    The Airy function $\operatorname{Bi}^\prime(z)$ is defined to be the
+    function
+
+    .. math::
+        \operatorname{Bi}^\prime(z) := \frac{\mathrm{d} \operatorname{Bi}(z)}{\mathrm{d} z}.
+
+    Examples
+    ========
+
+    Create an Airy function object:
+
+    >>> from sympy import airybiprime
+    >>> from sympy.abc import z
+
+    >>> airybiprime(z)
+    airybiprime(z)
+
+    Several special values are known:
+
+    >>> airybiprime(0)
+    3**(1/6)/gamma(1/3)
+    >>> from sympy import oo
+    >>> airybiprime(oo)
+    oo
+    >>> airybiprime(-oo)
+    0
+
+    The Airy function obeys the mirror symmetry:
+
+    >>> from sympy import conjugate
+    >>> conjugate(airybiprime(z))
+    airybiprime(conjugate(z))
+
+    Differentiation with respect to $z$ is supported:
+
+    >>> from sympy import diff
+    >>> diff(airybiprime(z), z)
+    z*airybi(z)
+    >>> diff(airybiprime(z), z, 2)
+    z*airybiprime(z) + airybi(z)
+
+    Series expansion is also supported:
+
+    >>> from sympy import series
+    >>> series(airybiprime(z), z, 0, 3)
+    3**(1/6)/gamma(1/3) + 3**(5/6)*z**2/(6*gamma(2/3)) + O(z**3)
+
+    We can numerically evaluate the Airy function to arbitrary precision
+    on the whole complex plane:
+
+    >>> airybiprime(-2).evalf(50)
+    0.27879516692116952268509756941098324140300059345163
+
+    Rewrite $\operatorname{Bi}^\prime(z)$ in terms of hypergeometric functions:
+
+    >>> from sympy import hyper
+    >>> airybiprime(z).rewrite(hyper)
+    3**(5/6)*z**2*hyper((), (5/3,), z**3/9)/(6*gamma(2/3)) + 3**(1/6)*hyper((), (1/3,), z**3/9)/gamma(1/3)
+
+    See Also
+    ========
+
+    airyai: Airy function of the first kind.
+    airybi: Airy function of the second kind.
+    airyaiprime: Derivative of the Airy function of the first kind.
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Airy_function
+    .. [2] https://dlmf.nist.gov/9
+    .. [3] https://encyclopediaofmath.org/wiki/Airy_functions
+    .. [4] https://mathworld.wolfram.com/AiryFunctions.html
+
+    """
+
+    nargs = 1
+    unbranched = True
+
+    @classmethod
+    def eval(cls, arg):
+        if arg.is_Number:
+            if arg is S.NaN:
+                return S.NaN
+            elif arg is S.Infinity:
+                return S.Infinity
+            elif arg is S.NegativeInfinity:
+                return S.Zero
+            elif arg.is_zero:
+                return 3**Rational(1, 6) / gamma(Rational(1, 3))
+
+        if arg.is_zero:
+            return 3**Rational(1, 6) / gamma(Rational(1, 3))
+
+
+    def fdiff(self, argindex=1):
+        if argindex == 1:
+            return self.args[0]*airybi(self.args[0])
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    def _eval_evalf(self, prec):
+        z = self.args[0]._to_mpmath(prec)
+        with workprec(prec):
+            res = mp.airybi(z, derivative=1)
+        return Expr._from_mpmath(res, prec)
+
+    def _eval_rewrite_as_besselj(self, z, **kwargs):
+        tt = Rational(2, 3)
+        a = tt * Pow(-z, Rational(3, 2))
+        if re(z).is_negative:
+            return -z/sqrt(3) * (besselj(-tt, a) + besselj(tt, a))
+
+    def _eval_rewrite_as_besseli(self, z, **kwargs):
+        ot = Rational(1, 3)
+        tt = Rational(2, 3)
+        a = tt * Pow(z, Rational(3, 2))
+        if re(z).is_positive:
+            return z/sqrt(3) * (besseli(-tt, a) + besseli(tt, a))
+        else:
+            a = Pow(z, Rational(3, 2))
+            b = Pow(a, tt)
+            c = Pow(a, -tt)
+            return sqrt(ot) * (b*besseli(-tt, tt*a) + z**2*c*besseli(tt, tt*a))
+
+    def _eval_rewrite_as_hyper(self, z, **kwargs):
+        pf1 = z**2 / (2*root(3, 6)*gamma(Rational(2, 3)))
+        pf2 = root(3, 6) / gamma(Rational(1, 3))
+        return pf1 * hyper([], [Rational(5, 3)], z**3/9) + pf2 * hyper([], [Rational(1, 3)], z**3/9)
+
+    def _eval_expand_func(self, **hints):
+        arg = self.args[0]
+        symbs = arg.free_symbols
+
+        if len(symbs) == 1:
+            z = symbs.pop()
+            c = Wild("c", exclude=[z])
+            d = Wild("d", exclude=[z])
+            m = Wild("m", exclude=[z])
+            n = Wild("n", exclude=[z])
+            M = arg.match(c*(d*z**n)**m)
+            if M is not None:
+                m = M[m]
+                # The transformation is in principle
+                # given by 03.08.16.0001.01 but note
+                # that there is an error in this formula.
+                # https://functions.wolfram.com/Bessel-TypeFunctions/AiryBiPrime/16/01/01/0001/
+                if (3*m).is_integer:
+                    c = M[c]
+                    d = M[d]
+                    n = M[n]
+                    pf = (d**m * z**(n*m)) / (d * z**n)**m
+                    newarg = c * d**m * z**(n*m)
+                    return S.Half * (sqrt(3)*(pf - S.One)*airyaiprime(newarg) + (pf + S.One)*airybiprime(newarg))
+
+
+class marcumq(Function):
+    r"""
+    The Marcum Q-function.
+
+    Explanation
+    ===========
+
+    The Marcum Q-function is defined by the meromorphic continuation of
+
+    .. math::
+        Q_m(a, b) = a^{- m + 1} \int_{b}^{\infty} x^{m} e^{- \frac{a^{2}}{2} - \frac{x^{2}}{2}} I_{m - 1}\left(a x\right)\, dx
+
+    Examples
+    ========
+
+    >>> from sympy import marcumq
+    >>> from sympy.abc import m, a, b
+    >>> marcumq(m, a, b)
+    marcumq(m, a, b)
+
+    Special values:
+
+    >>> marcumq(m, 0, b)
+    uppergamma(m, b**2/2)/gamma(m)
+    >>> marcumq(0, 0, 0)
+    0
+    >>> marcumq(0, a, 0)
+    1 - exp(-a**2/2)
+    >>> marcumq(1, a, a)
+    1/2 + exp(-a**2)*besseli(0, a**2)/2
+    >>> marcumq(2, a, a)
+    1/2 + exp(-a**2)*besseli(0, a**2)/2 + exp(-a**2)*besseli(1, a**2)
+
+    Differentiation with respect to $a$ and $b$ is supported:
+
+    >>> from sympy import diff
+    >>> diff(marcumq(m, a, b), a)
+    a*(-marcumq(m, a, b) + marcumq(m + 1, a, b))
+    >>> diff(marcumq(m, a, b), b)
+    -a**(1 - m)*b**m*exp(-a**2/2 - b**2/2)*besseli(m - 1, a*b)
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Marcum_Q-function
+    .. [2] https://mathworld.wolfram.com/MarcumQ-Function.html
+
+    """
+
+    @classmethod
+    def eval(cls, m, a, b):
+        if a is S.Zero:
+            if m is S.Zero and b is S.Zero:
+                return S.Zero
+            return uppergamma(m, b**2 * S.Half) / gamma(m)
+
+        if m is S.Zero and b is S.Zero:
+            return 1 - 1 / exp(a**2 * S.Half)
+
+        if a == b:
+            if m is S.One:
+                return (1 + exp(-a**2) * besseli(0, a**2))*S.Half
+            if m == 2:
+                return S.Half + S.Half * exp(-a**2) * besseli(0, a**2) + exp(-a**2) * besseli(1, a**2)
+
+        if a.is_zero:
+            if m.is_zero and b.is_zero:
+                return S.Zero
+            return uppergamma(m, b**2*S.Half) / gamma(m)
+
+        if m.is_zero and b.is_zero:
+            return 1 - 1 / exp(a**2*S.Half)
+
+    def fdiff(self, argindex=2):
+        m, a, b = self.args
+        if argindex == 2:
+            return a * (-marcumq(m, a, b) + marcumq(1+m, a, b))
+        elif argindex == 3:
+            return (-b**m / a**(m-1)) * exp(-(a**2 + b**2)/2) * besseli(m-1, a*b)
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    def _eval_rewrite_as_Integral(self, m, a, b, **kwargs):
+        from sympy.integrals.integrals import Integral
+        x = kwargs.get('x', Dummy('x'))
+        return a ** (1 - m) * \
+               Integral(x**m * exp(-(x**2 + a**2)/2) * besseli(m-1, a*x), [x, b, S.Infinity])
+
+    def _eval_rewrite_as_Sum(self, m, a, b, **kwargs):
+        from sympy.concrete.summations import Sum
+        k = kwargs.get('k', Dummy('k'))
+        return exp(-(a**2 + b**2) / 2) * Sum((a/b)**k * besseli(k, a*b), [k, 1-m, S.Infinity])
+
+    def _eval_rewrite_as_besseli(self, m, a, b, **kwargs):
+        if a == b:
+            if m == 1:
+                return (1 + exp(-a**2) * besseli(0, a**2)) / 2
+            if m.is_Integer and m >= 2:
+                s = sum([besseli(i, a**2) for i in range(1, m)])
+                return S.Half + exp(-a**2) * besseli(0, a**2) / 2 + exp(-a**2) * s
+
+    def _eval_is_zero(self):
+        if all(arg.is_zero for arg in self.args):
+            return True

llmeval-env/lib/python3.10/site-packages/sympy/functions/special/beta_functions.py

@@ -0,0 +1,389 @@
+from sympy.core import S
+from sympy.core.function import Function, ArgumentIndexError
+from sympy.core.symbol import Dummy
+from sympy.functions.special.gamma_functions import gamma, digamma
+from sympy.functions.combinatorial.numbers import catalan
+from sympy.functions.elementary.complexes import conjugate
+
+# See mpmath #569 and SymPy #20569
+def betainc_mpmath_fix(a, b, x1, x2, reg=0):
+    from mpmath import betainc, mpf
+    if x1 == x2:
+        return mpf(0)
+    else:
+        return betainc(a, b, x1, x2, reg)
+
+###############################################################################
+############################ COMPLETE BETA  FUNCTION ##########################
+###############################################################################
+
+class beta(Function):
+    r"""
+    The beta integral is called the Eulerian integral of the first kind by
+    Legendre:
+
+    .. math::
+        \mathrm{B}(x,y)  \int^{1}_{0} t^{x-1} (1-t)^{y-1} \mathrm{d}t.
+
+    Explanation
+    ===========
+
+    The Beta function or Euler's first integral is closely associated
+    with the gamma function. The Beta function is often used in probability
+    theory and mathematical statistics. It satisfies properties like:
+
+    .. math::
+        \mathrm{B}(a,1) = \frac{1}{a} \\
+        \mathrm{B}(a,b) = \mathrm{B}(b,a)  \\
+        \mathrm{B}(a,b) = \frac{\Gamma(a) \Gamma(b)}{\Gamma(a+b)}
+
+    Therefore for integral values of $a$ and $b$:
+
+    .. math::
+        \mathrm{B} = \frac{(a-1)! (b-1)!}{(a+b-1)!}
+
+    A special case of the Beta function when `x = y` is the
+    Central Beta function. It satisfies properties like:
+
+    .. math::
+        \mathrm{B}(x) = 2^{1 - 2x}\mathrm{B}(x, \frac{1}{2})
+        \mathrm{B}(x) = 2^{1 - 2x} cos(\pi x) \mathrm{B}(\frac{1}{2} - x, x)
+        \mathrm{B}(x) = \int_{0}^{1} \frac{t^x}{(1 + t)^{2x}} dt
+        \mathrm{B}(x) = \frac{2}{x} \prod_{n = 1}^{\infty} \frac{n(n + 2x)}{(n + x)^2}
+
+    Examples
+    ========
+
+    >>> from sympy import I, pi
+    >>> from sympy.abc import x, y
+
+    The Beta function obeys the mirror symmetry:
+
+    >>> from sympy import beta, conjugate
+    >>> conjugate(beta(x, y))
+    beta(conjugate(x), conjugate(y))
+
+    Differentiation with respect to both $x$ and $y$ is supported:
+
+    >>> from sympy import beta, diff
+    >>> diff(beta(x, y), x)
+    (polygamma(0, x) - polygamma(0, x + y))*beta(x, y)
+
+    >>> diff(beta(x, y), y)
+    (polygamma(0, y) - polygamma(0, x + y))*beta(x, y)
+
+    >>> diff(beta(x), x)
+    2*(polygamma(0, x) - polygamma(0, 2*x))*beta(x, x)
+
+    We can numerically evaluate the Beta function to
+    arbitrary precision for any complex numbers x and y:
+
+    >>> from sympy import beta
+    >>> beta(pi).evalf(40)
+    0.02671848900111377452242355235388489324562
+
+    >>> beta(1 + I).evalf(20)
+    -0.2112723729365330143 - 0.7655283165378005676*I
+
+    See Also
+    ========
+
+    gamma: Gamma function.
+    uppergamma: Upper incomplete gamma function.
+    lowergamma: Lower incomplete gamma function.
+    polygamma: Polygamma function.
+    loggamma: Log Gamma function.
+    digamma: Digamma function.
+    trigamma: Trigamma function.
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Beta_function
+    .. [2] https://mathworld.wolfram.com/BetaFunction.html
+    .. [3] https://dlmf.nist.gov/5.12
+
+    """
+    unbranched = True
+
+    def fdiff(self, argindex):
+        x, y = self.args
+        if argindex == 1:
+            # Diff wrt x
+            return beta(x, y)*(digamma(x) - digamma(x + y))
+        elif argindex == 2:
+            # Diff wrt y
+            return beta(x, y)*(digamma(y) - digamma(x + y))
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    @classmethod
+    def eval(cls, x, y=None):
+        if y is None:
+            return beta(x, x)
+        if x.is_Number and y.is_Number:
+            return beta(x, y, evaluate=False).doit()
+
+    def doit(self, **hints):
+        x = xold = self.args[0]
+        # Deal with unevaluated single argument beta
+        single_argument = len(self.args) == 1
+        y = yold = self.args[0] if single_argument else self.args[1]
+        if hints.get('deep', True):
+            x = x.doit(**hints)
+            y = y.doit(**hints)
+        if y.is_zero or x.is_zero:
+            return S.ComplexInfinity
+        if y is S.One:
+            return 1/x
+        if x is S.One:
+            return 1/y
+        if y == x + 1:
+            return 1/(x*y*catalan(x))
+        s = x + y
+        if (s.is_integer and s.is_negative and x.is_integer is False and
+            y.is_integer is False):
+            return S.Zero
+        if x == xold and y == yold and not single_argument:
+            return self
+        return beta(x, y)
+
+    def _eval_expand_func(self, **hints):
+        x, y = self.args
+        return gamma(x)*gamma(y) / gamma(x + y)
+
+    def _eval_is_real(self):
+        return self.args[0].is_real and self.args[1].is_real
+
+    def _eval_conjugate(self):
+        return self.func(self.args[0].conjugate(), self.args[1].conjugate())
+
+    def _eval_rewrite_as_gamma(self, x, y, piecewise=True, **kwargs):
+        return self._eval_expand_func(**kwargs)
+
+    def _eval_rewrite_as_Integral(self, x, y, **kwargs):
+        from sympy.integrals.integrals import Integral
+        t = Dummy('t')
+        return Integral(t**(x - 1)*(1 - t)**(y - 1), (t, 0, 1))
+
+###############################################################################
+########################## INCOMPLETE BETA FUNCTION ###########################
+###############################################################################
+
+class betainc(Function):
+    r"""
+    The Generalized Incomplete Beta function is defined as
+
+    .. math::
+        \mathrm{B}_{(x_1, x_2)}(a, b) = \int_{x_1}^{x_2} t^{a - 1} (1 - t)^{b - 1} dt
+
+    The Incomplete Beta function is a special case
+    of the Generalized Incomplete Beta function :
+
+    .. math:: \mathrm{B}_z (a, b) = \mathrm{B}_{(0, z)}(a, b)
+
+    The Incomplete Beta function satisfies :
+
+    .. math:: \mathrm{B}_z (a, b) = (-1)^a \mathrm{B}_{\frac{z}{z - 1}} (a, 1 - a - b)
+
+    The Beta function is a special case of the Incomplete Beta function :
+
+    .. math:: \mathrm{B}(a, b) = \mathrm{B}_{1}(a, b)
+
+    Examples
+    ========
+
+    >>> from sympy import betainc, symbols, conjugate
+    >>> a, b, x, x1, x2 = symbols('a b x x1 x2')
+
+    The Generalized Incomplete Beta function is given by:
+
+    >>> betainc(a, b, x1, x2)
+    betainc(a, b, x1, x2)
+
+    The Incomplete Beta function can be obtained as follows:
+
+    >>> betainc(a, b, 0, x)
+    betainc(a, b, 0, x)
+
+    The Incomplete Beta function obeys the mirror symmetry:
+
+    >>> conjugate(betainc(a, b, x1, x2))
+    betainc(conjugate(a), conjugate(b), conjugate(x1), conjugate(x2))
+
+    We can numerically evaluate the Incomplete Beta function to
+    arbitrary precision for any complex numbers a, b, x1 and x2:
+
+    >>> from sympy import betainc, I
+    >>> betainc(2, 3, 4, 5).evalf(10)
+    56.08333333
+    >>> betainc(0.75, 1 - 4*I, 0, 2 + 3*I).evalf(25)
+    0.2241657956955709603655887 + 0.3619619242700451992411724*I
+
+    The Generalized Incomplete Beta function can be expressed
+    in terms of the Generalized Hypergeometric function.
+
+    >>> from sympy import hyper
+    >>> betainc(a, b, x1, x2).rewrite(hyper)
+    (-x1**a*hyper((a, 1 - b), (a + 1,), x1) + x2**a*hyper((a, 1 - b), (a + 1,), x2))/a
+
+    See Also
+    ========
+
+    beta: Beta function
+    hyper: Generalized Hypergeometric function
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function
+    .. [2] https://dlmf.nist.gov/8.17
+    .. [3] https://functions.wolfram.com/GammaBetaErf/Beta4/
+    .. [4] https://functions.wolfram.com/GammaBetaErf/BetaRegularized4/02/
+
+    """
+    nargs = 4
+    unbranched = True
+
+    def fdiff(self, argindex):
+        a, b, x1, x2 = self.args
+        if argindex == 3:
+            # Diff wrt x1
+            return -(1 - x1)**(b - 1)*x1**(a - 1)
+        elif argindex == 4:
+            # Diff wrt x2
+            return (1 - x2)**(b - 1)*x2**(a - 1)
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    def _eval_mpmath(self):
+        return betainc_mpmath_fix, self.args
+
+    def _eval_is_real(self):
+        if all(arg.is_real for arg in self.args):
+            return True
+
+    def _eval_conjugate(self):
+        return self.func(*map(conjugate, self.args))
+
+    def _eval_rewrite_as_Integral(self, a, b, x1, x2, **kwargs):
+        from sympy.integrals.integrals import Integral
+        t = Dummy('t')
+        return Integral(t**(a - 1)*(1 - t)**(b - 1), (t, x1, x2))
+
+    def _eval_rewrite_as_hyper(self, a, b, x1, x2, **kwargs):
+        from sympy.functions.special.hyper import hyper
+        return (x2**a * hyper((a, 1 - b), (a + 1,), x2) - x1**a * hyper((a, 1 - b), (a + 1,), x1)) / a
+
+###############################################################################
+#################### REGULARIZED INCOMPLETE BETA FUNCTION #####################
+###############################################################################
+
+class betainc_regularized(Function):
+    r"""
+    The Generalized Regularized Incomplete Beta function is given by
+
+    .. math::
+        \mathrm{I}_{(x_1, x_2)}(a, b) = \frac{\mathrm{B}_{(x_1, x_2)}(a, b)}{\mathrm{B}(a, b)}
+
+    The Regularized Incomplete Beta function is a special case
+    of the Generalized Regularized Incomplete Beta function :
+
+    .. math:: \mathrm{I}_z (a, b) = \mathrm{I}_{(0, z)}(a, b)
+
+    The Regularized Incomplete Beta function is the cumulative distribution
+    function of the beta distribution.
+
+    Examples
+    ========
+
+    >>> from sympy import betainc_regularized, symbols, conjugate
+    >>> a, b, x, x1, x2 = symbols('a b x x1 x2')
+
+    The Generalized Regularized Incomplete Beta
+    function is given by:
+
+    >>> betainc_regularized(a, b, x1, x2)
+    betainc_regularized(a, b, x1, x2)
+
+    The Regularized Incomplete Beta function
+    can be obtained as follows:
+
+    >>> betainc_regularized(a, b, 0, x)
+    betainc_regularized(a, b, 0, x)
+
+    The Regularized Incomplete Beta function
+    obeys the mirror symmetry:
+
+    >>> conjugate(betainc_regularized(a, b, x1, x2))
+    betainc_regularized(conjugate(a), conjugate(b), conjugate(x1), conjugate(x2))
+
+    We can numerically evaluate the Regularized Incomplete Beta function
+    to arbitrary precision for any complex numbers a, b, x1 and x2:
+
+    >>> from sympy import betainc_regularized, pi, E
+    >>> betainc_regularized(1, 2, 0, 0.25).evalf(10)
+    0.4375000000
+    >>> betainc_regularized(pi, E, 0, 1).evalf(5)
+    1.00000
+
+    The Generalized Regularized Incomplete Beta function can be
+    expressed in terms of the Generalized Hypergeometric function.
+
+    >>> from sympy import hyper
+    >>> betainc_regularized(a, b, x1, x2).rewrite(hyper)
+    (-x1**a*hyper((a, 1 - b), (a + 1,), x1) + x2**a*hyper((a, 1 - b), (a + 1,), x2))/(a*beta(a, b))
+
+    See Also
+    ========
+
+    beta: Beta function
+    hyper: Generalized Hypergeometric function
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Beta_function#Incomplete_beta_function
+    .. [2] https://dlmf.nist.gov/8.17
+    .. [3] https://functions.wolfram.com/GammaBetaErf/Beta4/
+    .. [4] https://functions.wolfram.com/GammaBetaErf/BetaRegularized4/02/
+
+    """
+    nargs = 4
+    unbranched = True
+
+    def __new__(cls, a, b, x1, x2):
+        return Function.__new__(cls, a, b, x1, x2)
+
+    def _eval_mpmath(self):
+        return betainc_mpmath_fix, (*self.args, S(1))
+
+    def fdiff(self, argindex):
+        a, b, x1, x2 = self.args
+        if argindex == 3:
+            # Diff wrt x1
+            return -(1 - x1)**(b - 1)*x1**(a - 1) / beta(a, b)
+        elif argindex == 4:
+            # Diff wrt x2
+            return (1 - x2)**(b - 1)*x2**(a - 1) / beta(a, b)
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    def _eval_is_real(self):
+        if all(arg.is_real for arg in self.args):
+            return True
+
+    def _eval_conjugate(self):
+        return self.func(*map(conjugate, self.args))
+
+    def _eval_rewrite_as_Integral(self, a, b, x1, x2, **kwargs):
+        from sympy.integrals.integrals import Integral
+        t = Dummy('t')
+        integrand = t**(a - 1)*(1 - t)**(b - 1)
+        expr = Integral(integrand, (t, x1, x2))
+        return expr / Integral(integrand, (t, 0, 1))
+
+    def _eval_rewrite_as_hyper(self, a, b, x1, x2, **kwargs):
+        from sympy.functions.special.hyper import hyper
+        expr = (x2**a * hyper((a, 1 - b), (a + 1,), x2) - x1**a * hyper((a, 1 - b), (a + 1,), x1)) / a
+        return expr / beta(a, b)

llmeval-env/lib/python3.10/site-packages/sympy/functions/special/bsplines.py

@@ -0,0 +1,351 @@
+from sympy.core import S, sympify
+from sympy.core.symbol import (Dummy, symbols)
+from sympy.functions import Piecewise, piecewise_fold
+from sympy.logic.boolalg import And
+from sympy.sets.sets import Interval
+
+from functools import lru_cache
+
+
+def _ivl(cond, x):
+    """return the interval corresponding to the condition
+
+    Conditions in spline's Piecewise give the range over
+    which an expression is valid like (lo <= x) & (x <= hi).
+    This function returns (lo, hi).
+    """
+    if isinstance(cond, And) and len(cond.args) == 2:
+        a, b = cond.args
+        if a.lts == x:
+            a, b = b, a
+        return a.lts, b.gts
+    raise TypeError('unexpected cond type: %s' % cond)
+
+
+def _add_splines(c, b1, d, b2, x):
+    """Construct c*b1 + d*b2."""
+
+    if S.Zero in (b1, c):
+        rv = piecewise_fold(d * b2)
+    elif S.Zero in (b2, d):
+        rv = piecewise_fold(c * b1)
+    else:
+        new_args = []
+        # Just combining the Piecewise without any fancy optimization
+        p1 = piecewise_fold(c * b1)
+        p2 = piecewise_fold(d * b2)
+
+        # Search all Piecewise arguments except (0, True)
+        p2args = list(p2.args[:-1])
+
+        # This merging algorithm assumes the conditions in
+        # p1 and p2 are sorted
+        for arg in p1.args[:-1]:
+            expr = arg.expr
+            cond = arg.cond
+
+            lower = _ivl(cond, x)[0]
+
+            # Check p2 for matching conditions that can be merged
+            for i, arg2 in enumerate(p2args):
+                expr2 = arg2.expr
+                cond2 = arg2.cond
+
+                lower_2, upper_2 = _ivl(cond2, x)
+                if cond2 == cond:
+                    # Conditions match, join expressions
+                    expr += expr2
+                    # Remove matching element
+                    del p2args[i]
+                    # No need to check the rest
+                    break
+                elif lower_2 < lower and upper_2 <= lower:
+                    # Check if arg2 condition smaller than arg1,
+                    # add to new_args by itself (no match expected
+                    # in p1)
+                    new_args.append(arg2)
+                    del p2args[i]
+                    break
+
+            # Checked all, add expr and cond
+            new_args.append((expr, cond))
+
+        # Add remaining items from p2args
+        new_args.extend(p2args)
+
+        # Add final (0, True)
+        new_args.append((0, True))
+
+        rv = Piecewise(*new_args, evaluate=False)
+
+    return rv.expand()
+
+
+@lru_cache(maxsize=128)
+def bspline_basis(d, knots, n, x):
+    """
+    The $n$-th B-spline at $x$ of degree $d$ with knots.
+
+    Explanation
+    ===========
+
+    B-Splines are piecewise polynomials of degree $d$. They are defined on a
+    set of knots, which is a sequence of integers or floats.
+
+    Examples
+    ========
+
+    The 0th degree splines have a value of 1 on a single interval:
+
+        >>> from sympy import bspline_basis
+        >>> from sympy.abc import x
+        >>> d = 0
+        >>> knots = tuple(range(5))
+        >>> bspline_basis(d, knots, 0, x)
+        Piecewise((1, (x >= 0) & (x <= 1)), (0, True))
+
+    For a given ``(d, knots)`` there are ``len(knots)-d-1`` B-splines
+    defined, that are indexed by ``n`` (starting at 0).
+
+    Here is an example of a cubic B-spline:
+
+        >>> bspline_basis(3, tuple(range(5)), 0, x)
+        Piecewise((x**3/6, (x >= 0) & (x <= 1)),
+                  (-x**3/2 + 2*x**2 - 2*x + 2/3,
+                  (x >= 1) & (x <= 2)),
+                  (x**3/2 - 4*x**2 + 10*x - 22/3,
+                  (x >= 2) & (x <= 3)),
+                  (-x**3/6 + 2*x**2 - 8*x + 32/3,
+                  (x >= 3) & (x <= 4)),
+                  (0, True))
+
+    By repeating knot points, you can introduce discontinuities in the
+    B-splines and their derivatives:
+
+        >>> d = 1
+        >>> knots = (0, 0, 2, 3, 4)
+        >>> bspline_basis(d, knots, 0, x)
+        Piecewise((1 - x/2, (x >= 0) & (x <= 2)), (0, True))
+
+    It is quite time consuming to construct and evaluate B-splines. If
+    you need to evaluate a B-spline many times, it is best to lambdify them
+    first:
+
+        >>> from sympy import lambdify
+        >>> d = 3
+        >>> knots = tuple(range(10))
+        >>> b0 = bspline_basis(d, knots, 0, x)
+        >>> f = lambdify(x, b0)
+        >>> y = f(0.5)
+
+    Parameters
+    ==========
+
+    d : integer
+        degree of bspline
+
+    knots : list of integer values
+        list of knots points of bspline
+
+    n : integer
+        $n$-th B-spline
+
+    x : symbol
+
+    See Also
+    ========
+
+    bspline_basis_set
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/B-spline
+
+    """
+    # make sure x has no assumptions so conditions don't evaluate
+    xvar = x
+    x = Dummy()
+
+    knots = tuple(sympify(k) for k in knots)
+    d = int(d)
+    n = int(n)
+    n_knots = len(knots)
+    n_intervals = n_knots - 1
+    if n + d + 1 > n_intervals:
+        raise ValueError("n + d + 1 must not exceed len(knots) - 1")
+    if d == 0:
+        result = Piecewise(
+            (S.One, Interval(knots[n], knots[n + 1]).contains(x)), (0, True)
+        )
+    elif d > 0:
+        denom = knots[n + d + 1] - knots[n + 1]
+        if denom != S.Zero:
+            B = (knots[n + d + 1] - x) / denom
+            b2 = bspline_basis(d - 1, knots, n + 1, x)
+        else:
+            b2 = B = S.Zero
+
+        denom = knots[n + d] - knots[n]
+        if denom != S.Zero:
+            A = (x - knots[n]) / denom
+            b1 = bspline_basis(d - 1, knots, n, x)
+        else:
+            b1 = A = S.Zero
+
+        result = _add_splines(A, b1, B, b2, x)
+    else:
+        raise ValueError("degree must be non-negative: %r" % n)
+
+    # return result with user-given x
+    return result.xreplace({x: xvar})
+
+
+def bspline_basis_set(d, knots, x):
+    """
+    Return the ``len(knots)-d-1`` B-splines at *x* of degree *d*
+    with *knots*.
+
+    Explanation
+    ===========
+
+    This function returns a list of piecewise polynomials that are the
+    ``len(knots)-d-1`` B-splines of degree *d* for the given knots.
+    This function calls ``bspline_basis(d, knots, n, x)`` for different
+    values of *n*.
+
+    Examples
+    ========
+
+    >>> from sympy import bspline_basis_set
+    >>> from sympy.abc import x
+    >>> d = 2
+    >>> knots = range(5)
+    >>> splines = bspline_basis_set(d, knots, x)
+    >>> splines
+    [Piecewise((x**2/2, (x >= 0) & (x <= 1)),
+               (-x**2 + 3*x - 3/2, (x >= 1) & (x <= 2)),
+               (x**2/2 - 3*x + 9/2, (x >= 2) & (x <= 3)),
+               (0, True)),
+    Piecewise((x**2/2 - x + 1/2, (x >= 1) & (x <= 2)),
+              (-x**2 + 5*x - 11/2, (x >= 2) & (x <= 3)),
+              (x**2/2 - 4*x + 8, (x >= 3) & (x <= 4)),
+              (0, True))]
+
+    Parameters
+    ==========
+
+    d : integer
+        degree of bspline
+
+    knots : list of integers
+        list of knots points of bspline
+
+    x : symbol
+
+    See Also
+    ========
+
+    bspline_basis
+
+    """
+    n_splines = len(knots) - d - 1
+    return [bspline_basis(d, tuple(knots), i, x) for i in range(n_splines)]
+
+
+def interpolating_spline(d, x, X, Y):
+    """
+    Return spline of degree *d*, passing through the given *X*
+    and *Y* values.
+
+    Explanation
+    ===========
+
+    This function returns a piecewise function such that each part is
+    a polynomial of degree not greater than *d*. The value of *d*
+    must be 1 or greater and the values of *X* must be strictly
+    increasing.
+
+    Examples
+    ========
+
+    >>> from sympy import interpolating_spline
+    >>> from sympy.abc import x
+    >>> interpolating_spline(1, x, [1, 2, 4, 7], [3, 6, 5, 7])
+    Piecewise((3*x, (x >= 1) & (x <= 2)),
+            (7 - x/2, (x >= 2) & (x <= 4)),
+            (2*x/3 + 7/3, (x >= 4) & (x <= 7)))
+    >>> interpolating_spline(3, x, [-2, 0, 1, 3, 4], [4, 2, 1, 1, 3])
+    Piecewise((7*x**3/117 + 7*x**2/117 - 131*x/117 + 2, (x >= -2) & (x <= 1)),
+            (10*x**3/117 - 2*x**2/117 - 122*x/117 + 77/39, (x >= 1) & (x <= 4)))
+
+    Parameters
+    ==========
+
+    d : integer
+        Degree of Bspline strictly greater than equal to one
+
+    x : symbol
+
+    X : list of strictly increasing real values
+        list of X coordinates through which the spline passes
+
+    Y : list of real values
+        list of corresponding Y coordinates through which the spline passes
+
+    See Also
+    ========
+
+    bspline_basis_set, interpolating_poly
+
+    """
+    from sympy.solvers.solveset import linsolve
+    from sympy.matrices.dense import Matrix
+
+    # Input sanitization
+    d = sympify(d)
+    if not (d.is_Integer and d.is_positive):
+        raise ValueError("Spline degree must be a positive integer, not %s." % d)
+    if len(X) != len(Y):
+        raise ValueError("Number of X and Y coordinates must be the same.")
+    if len(X) < d + 1:
+        raise ValueError("Degree must be less than the number of control points.")
+    if not all(a < b for a, b in zip(X, X[1:])):
+        raise ValueError("The x-coordinates must be strictly increasing.")
+    X = [sympify(i) for i in X]
+
+    # Evaluating knots value
+    if d.is_odd:
+        j = (d + 1) // 2
+        interior_knots = X[j:-j]
+    else:
+        j = d // 2
+        interior_knots = [
+            (a + b)/2 for a, b in zip(X[j : -j - 1], X[j + 1 : -j])
+        ]
+
+    knots = [X[0]] * (d + 1) + list(interior_knots) + [X[-1]] * (d + 1)
+
+    basis = bspline_basis_set(d, knots, x)
+
+    A = [[b.subs(x, v) for b in basis] for v in X]
+
+    coeff = linsolve((Matrix(A), Matrix(Y)), symbols("c0:{}".format(len(X)), cls=Dummy))
+    coeff = list(coeff)[0]
+    intervals = {c for b in basis for (e, c) in b.args if c != True}
+
+    # Sorting the intervals
+    #  ival contains the end-points of each interval
+    ival = [_ivl(c, x) for c in intervals]
+    com = zip(ival, intervals)
+    com = sorted(com, key=lambda x: x[0])
+    intervals = [y for x, y in com]
+
+    basis_dicts = [{c: e for (e, c) in b.args} for b in basis]
+    spline = []
+    for i in intervals:
+        piece = sum(
+            [c * d.get(i, S.Zero) for (c, d) in zip(coeff, basis_dicts)], S.Zero
+        )
+        spline.append((piece, i))
+    return Piecewise(*spline)

llmeval-env/lib/python3.10/site-packages/sympy/functions/special/delta_functions.py

@@ -0,0 +1,664 @@
+from sympy.core import S, diff
+from sympy.core.function import Function, ArgumentIndexError
+from sympy.core.logic import fuzzy_not
+from sympy.core.relational import Eq, Ne
+from sympy.functions.elementary.complexes import im, sign
+from sympy.functions.elementary.piecewise import Piecewise
+from sympy.polys.polyerrors import PolynomialError
+from sympy.polys.polyroots import roots
+from sympy.utilities.misc import filldedent
+
+
+###############################################################################
+################################ DELTA FUNCTION ###############################
+###############################################################################
+
+
+class DiracDelta(Function):
+    r"""
+    The DiracDelta function and its derivatives.
+
+    Explanation
+    ===========
+
+    DiracDelta is not an ordinary function. It can be rigorously defined either
+    as a distribution or as a measure.
+
+    DiracDelta only makes sense in definite integrals, and in particular,
+    integrals of the form ``Integral(f(x)*DiracDelta(x - x0), (x, a, b))``,
+    where it equals ``f(x0)`` if ``a <= x0 <= b`` and ``0`` otherwise. Formally,
+    DiracDelta acts in some ways like a function that is ``0`` everywhere except
+    at ``0``, but in many ways it also does not. It can often be useful to treat
+    DiracDelta in formal ways, building up and manipulating expressions with
+    delta functions (which may eventually be integrated), but care must be taken
+    to not treat it as a real function. SymPy's ``oo`` is similar. It only
+    truly makes sense formally in certain contexts (such as integration limits),
+    but SymPy allows its use everywhere, and it tries to be consistent with
+    operations on it (like ``1/oo``), but it is easy to get into trouble and get
+    wrong results if ``oo`` is treated too much like a number. Similarly, if
+    DiracDelta is treated too much like a function, it is easy to get wrong or
+    nonsensical results.
+
+    DiracDelta function has the following properties:
+
+    1) $\frac{d}{d x} \theta(x) = \delta(x)$
+    2) $\int_{-\infty}^\infty \delta(x - a)f(x)\, dx = f(a)$ and $\int_{a-
+       \epsilon}^{a+\epsilon} \delta(x - a)f(x)\, dx = f(a)$
+    3) $\delta(x) = 0$ for all $x \neq 0$
+    4) $\delta(g(x)) = \sum_i \frac{\delta(x - x_i)}{\|g'(x_i)\|}$ where $x_i$
+       are the roots of $g$
+    5) $\delta(-x) = \delta(x)$
+
+    Derivatives of ``k``-th order of DiracDelta have the following properties:
+
+    6) $\delta(x, k) = 0$ for all $x \neq 0$
+    7) $\delta(-x, k) = -\delta(x, k)$ for odd $k$
+    8) $\delta(-x, k) = \delta(x, k)$ for even $k$
+
+    Examples
+    ========
+
+    >>> from sympy import DiracDelta, diff, pi
+    >>> from sympy.abc import x, y
+
+    >>> DiracDelta(x)
+    DiracDelta(x)
+    >>> DiracDelta(1)
+    0
+    >>> DiracDelta(-1)
+    0
+    >>> DiracDelta(pi)
+    0
+    >>> DiracDelta(x - 4).subs(x, 4)
+    DiracDelta(0)
+    >>> diff(DiracDelta(x))
+    DiracDelta(x, 1)
+    >>> diff(DiracDelta(x - 1), x, 2)
+    DiracDelta(x - 1, 2)
+    >>> diff(DiracDelta(x**2 - 1), x, 2)
+    2*(2*x**2*DiracDelta(x**2 - 1, 2) + DiracDelta(x**2 - 1, 1))
+    >>> DiracDelta(3*x).is_simple(x)
+    True
+    >>> DiracDelta(x**2).is_simple(x)
+    False
+    >>> DiracDelta((x**2 - 1)*y).expand(diracdelta=True, wrt=x)
+    DiracDelta(x - 1)/(2*Abs(y)) + DiracDelta(x + 1)/(2*Abs(y))
+
+    See Also
+    ========
+
+    Heaviside
+    sympy.simplify.simplify.simplify, is_simple
+    sympy.functions.special.tensor_functions.KroneckerDelta
+
+    References
+    ==========
+
+    .. [1] https://mathworld.wolfram.com/DeltaFunction.html
+
+    """
+
+    is_real = True
+
+    def fdiff(self, argindex=1):
+        """
+        Returns the first derivative of a DiracDelta Function.
+
+        Explanation
+        ===========
+
+        The difference between ``diff()`` and ``fdiff()`` is: ``diff()`` is the
+        user-level function and ``fdiff()`` is an object method. ``fdiff()`` is
+        a convenience method available in the ``Function`` class. It returns
+        the derivative of the function without considering the chain rule.
+        ``diff(function, x)`` calls ``Function._eval_derivative`` which in turn
+        calls ``fdiff()`` internally to compute the derivative of the function.
+
+        Examples
+        ========
+
+        >>> from sympy import DiracDelta, diff
+        >>> from sympy.abc import x
+
+        >>> DiracDelta(x).fdiff()
+        DiracDelta(x, 1)
+
+        >>> DiracDelta(x, 1).fdiff()
+        DiracDelta(x, 2)
+
+        >>> DiracDelta(x**2 - 1).fdiff()
+        DiracDelta(x**2 - 1, 1)
+
+        >>> diff(DiracDelta(x, 1)).fdiff()
+        DiracDelta(x, 3)
+
+        Parameters
+        ==========
+
+        argindex : integer
+            degree of derivative
+
+        """
+        if argindex == 1:
+            #I didn't know if there is a better way to handle default arguments
+            k = 0
+            if len(self.args) > 1:
+                k = self.args[1]
+            return self.func(self.args[0], k + 1)
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    @classmethod
+    def eval(cls, arg, k=S.Zero):
+        """
+        Returns a simplified form or a value of DiracDelta depending on the
+        argument passed by the DiracDelta object.
+
+        Explanation
+        ===========
+
+        The ``eval()`` method is automatically called when the ``DiracDelta``
+        class is about to be instantiated and it returns either some simplified
+        instance or the unevaluated instance depending on the argument passed.
+        In other words, ``eval()`` method is not needed to be called explicitly,
+        it is being called and evaluated once the object is called.
+
+        Examples
+        ========
+
+        >>> from sympy import DiracDelta, S
+        >>> from sympy.abc import x
+
+        >>> DiracDelta(x)
+        DiracDelta(x)
+
+        >>> DiracDelta(-x, 1)
+        -DiracDelta(x, 1)
+
+        >>> DiracDelta(1)
+        0
+
+        >>> DiracDelta(5, 1)
+        0
+
+        >>> DiracDelta(0)
+        DiracDelta(0)
+
+        >>> DiracDelta(-1)
+        0
+
+        >>> DiracDelta(S.NaN)
+        nan
+
+        >>> DiracDelta(x - 100).subs(x, 5)
+        0
+
+        >>> DiracDelta(x - 100).subs(x, 100)
+        DiracDelta(0)
+
+        Parameters
+        ==========
+
+        k : integer
+            order of derivative
+
+        arg : argument passed to DiracDelta
+
+        """
+        if not k.is_Integer or k.is_negative:
+            raise ValueError("Error: the second argument of DiracDelta must be \
+            a non-negative integer, %s given instead." % (k,))
+        if arg is S.NaN:
+            return S.NaN
+        if arg.is_nonzero:
+            return S.Zero
+        if fuzzy_not(im(arg).is_zero):
+            raise ValueError(filldedent('''
+                Function defined only for Real Values.
+                Complex part: %s  found in %s .''' % (
+                repr(im(arg)), repr(arg))))
+        c, nc = arg.args_cnc()
+        if c and c[0] is S.NegativeOne:
+            # keep this fast and simple instead of using
+            # could_extract_minus_sign
+            if k.is_odd:
+                return -cls(-arg, k)
+            elif k.is_even:
+                return cls(-arg, k) if k else cls(-arg)
+        elif k.is_zero:
+            return cls(arg, evaluate=False)
+
+    def _eval_expand_diracdelta(self, **hints):
+        """
+        Compute a simplified representation of the function using
+        property number 4. Pass ``wrt`` as a hint to expand the expression
+        with respect to a particular variable.
+
+        Explanation
+        ===========
+
+        ``wrt`` is:
+
+        - a variable with respect to which a DiracDelta expression will
+        get expanded.
+
+        Examples
+        ========
+
+        >>> from sympy import DiracDelta
+        >>> from sympy.abc import x, y
+
+        >>> DiracDelta(x*y).expand(diracdelta=True, wrt=x)
+        DiracDelta(x)/Abs(y)
+        >>> DiracDelta(x*y).expand(diracdelta=True, wrt=y)
+        DiracDelta(y)/Abs(x)
+
+        >>> DiracDelta(x**2 + x - 2).expand(diracdelta=True, wrt=x)
+        DiracDelta(x - 1)/3 + DiracDelta(x + 2)/3
+
+        See Also
+        ========
+
+        is_simple, Diracdelta
+
+        """
+        wrt = hints.get('wrt', None)
+        if wrt is None:
+            free = self.free_symbols
+            if len(free) == 1:
+                wrt = free.pop()
+            else:
+                raise TypeError(filldedent('''
+            When there is more than 1 free symbol or variable in the expression,
+            the 'wrt' keyword is required as a hint to expand when using the
+            DiracDelta hint.'''))
+
+        if not self.args[0].has(wrt) or (len(self.args) > 1 and self.args[1] != 0 ):
+            return self
+        try:
+            argroots = roots(self.args[0], wrt)
+            result = 0
+            valid = True
+            darg = abs(diff(self.args[0], wrt))
+            for r, m in argroots.items():
+                if r.is_real is not False and m == 1:
+                    result += self.func(wrt - r)/darg.subs(wrt, r)
+                else:
+                    # don't handle non-real and if m != 1 then
+                    # a polynomial will have a zero in the derivative (darg)
+                    # at r
+                    valid = False
+                    break
+            if valid:
+                return result
+        except PolynomialError:
+            pass
+        return self
+
+    def is_simple(self, x):
+        """
+        Tells whether the argument(args[0]) of DiracDelta is a linear
+        expression in *x*.
+
+        Examples
+        ========
+
+        >>> from sympy import DiracDelta, cos
+        >>> from sympy.abc import x, y
+
+        >>> DiracDelta(x*y).is_simple(x)
+        True
+        >>> DiracDelta(x*y).is_simple(y)
+        True
+
+        >>> DiracDelta(x**2 + x - 2).is_simple(x)
+        False
+
+        >>> DiracDelta(cos(x)).is_simple(x)
+        False
+
+        Parameters
+        ==========
+
+        x : can be a symbol
+
+        See Also
+        ========
+
+        sympy.simplify.simplify.simplify, DiracDelta
+
+        """
+        p = self.args[0].as_poly(x)
+        if p:
+            return p.degree() == 1
+        return False
+
+    def _eval_rewrite_as_Piecewise(self, *args, **kwargs):
+        """
+        Represents DiracDelta in a piecewise form.
+
+        Examples
+        ========
+
+        >>> from sympy import DiracDelta, Piecewise, Symbol
+        >>> x = Symbol('x')
+
+        >>> DiracDelta(x).rewrite(Piecewise)
+        Piecewise((DiracDelta(0), Eq(x, 0)), (0, True))
+
+        >>> DiracDelta(x - 5).rewrite(Piecewise)
+        Piecewise((DiracDelta(0), Eq(x, 5)), (0, True))
+
+        >>> DiracDelta(x**2 - 5).rewrite(Piecewise)
+           Piecewise((DiracDelta(0), Eq(x**2, 5)), (0, True))
+
+        >>> DiracDelta(x - 5, 4).rewrite(Piecewise)
+        DiracDelta(x - 5, 4)
+
+        """
+        if len(args) == 1:
+            return Piecewise((DiracDelta(0), Eq(args[0], 0)), (0, True))
+
+    def _eval_rewrite_as_SingularityFunction(self, *args, **kwargs):
+        """
+        Returns the DiracDelta expression written in the form of Singularity
+        Functions.
+
+        """
+        from sympy.solvers import solve
+        from sympy.functions.special.singularity_functions import SingularityFunction
+        if self == DiracDelta(0):
+            return SingularityFunction(0, 0, -1)
+        if self == DiracDelta(0, 1):
+            return SingularityFunction(0, 0, -2)
+        free = self.free_symbols
+        if len(free) == 1:
+            x = (free.pop())
+            if len(args) == 1:
+                return SingularityFunction(x, solve(args[0], x)[0], -1)
+            return SingularityFunction(x, solve(args[0], x)[0], -args[1] - 1)
+        else:
+            # I don't know how to handle the case for DiracDelta expressions
+            # having arguments with more than one variable.
+            raise TypeError(filldedent('''
+                rewrite(SingularityFunction) does not support
+                arguments with more that one variable.'''))
+
+
+###############################################################################
+############################## HEAVISIDE FUNCTION #############################
+###############################################################################
+
+
+class Heaviside(Function):
+    r"""
+    Heaviside step function.
+
+    Explanation
+    ===========
+
+    The Heaviside step function has the following properties:
+
+    1) $\frac{d}{d x} \theta(x) = \delta(x)$
+    2) $\theta(x) = \begin{cases} 0 & \text{for}\: x < 0 \\ \frac{1}{2} &
+       \text{for}\: x = 0 \\1 & \text{for}\: x > 0 \end{cases}$
+    3) $\frac{d}{d x} \max(x, 0) = \theta(x)$
+
+    Heaviside(x) is printed as $\theta(x)$ with the SymPy LaTeX printer.
+
+    The value at 0 is set differently in different fields. SymPy uses 1/2,
+    which is a convention from electronics and signal processing, and is
+    consistent with solving improper integrals by Fourier transform and
+    convolution.
+
+    To specify a different value of Heaviside at ``x=0``, a second argument
+    can be given. Using ``Heaviside(x, nan)`` gives an expression that will
+    evaluate to nan for x=0.
+
+    .. versionchanged:: 1.9 ``Heaviside(0)`` now returns 1/2 (before: undefined)
+
+    Examples
+    ========
+
+    >>> from sympy import Heaviside, nan
+    >>> from sympy.abc import x
+    >>> Heaviside(9)
+    1
+    >>> Heaviside(-9)
+    0
+    >>> Heaviside(0)
+    1/2
+    >>> Heaviside(0, nan)
+    nan
+    >>> (Heaviside(x) + 1).replace(Heaviside(x), Heaviside(x, 1))
+    Heaviside(x, 1) + 1
+
+    See Also
+    ========
+
+    DiracDelta
+
+    References
+    ==========
+
+    .. [1] https://mathworld.wolfram.com/HeavisideStepFunction.html
+    .. [2] https://dlmf.nist.gov/1.16#iv
+
+    """
+
+    is_real = True
+
+    def fdiff(self, argindex=1):
+        """
+        Returns the first derivative of a Heaviside Function.
+
+        Examples
+        ========
+
+        >>> from sympy import Heaviside, diff
+        >>> from sympy.abc import x
+
+        >>> Heaviside(x).fdiff()
+        DiracDelta(x)
+
+        >>> Heaviside(x**2 - 1).fdiff()
+        DiracDelta(x**2 - 1)
+
+        >>> diff(Heaviside(x)).fdiff()
+        DiracDelta(x, 1)
+
+        Parameters
+        ==========
+
+        argindex : integer
+            order of derivative
+
+        """
+        if argindex == 1:
+            return DiracDelta(self.args[0])
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    def __new__(cls, arg, H0=S.Half, **options):
+        if isinstance(H0, Heaviside) and len(H0.args) == 1:
+            H0 = S.Half
+        return super(cls, cls).__new__(cls, arg, H0, **options)
+
+    @property
+    def pargs(self):
+        """Args without default S.Half"""
+        args = self.args
+        if args[1] is S.Half:
+            args = args[:1]
+        return args
+
+    @classmethod
+    def eval(cls, arg, H0=S.Half):
+        """
+        Returns a simplified form or a value of Heaviside depending on the
+        argument passed by the Heaviside object.
+
+        Explanation
+        ===========
+
+        The ``eval()`` method is automatically called when the ``Heaviside``
+        class is about to be instantiated and it returns either some simplified
+        instance or the unevaluated instance depending on the argument passed.
+        In other words, ``eval()`` method is not needed to be called explicitly,
+        it is being called and evaluated once the object is called.
+
+        Examples
+        ========
+
+        >>> from sympy import Heaviside, S
+        >>> from sympy.abc import x
+
+        >>> Heaviside(x)
+        Heaviside(x)
+
+        >>> Heaviside(19)
+        1
+
+        >>> Heaviside(0)
+        1/2
+
+        >>> Heaviside(0, 1)
+        1
+
+        >>> Heaviside(-5)
+        0
+
+        >>> Heaviside(S.NaN)
+        nan
+
+        >>> Heaviside(x - 100).subs(x, 5)
+        0
+
+        >>> Heaviside(x - 100).subs(x, 105)
+        1
+
+        Parameters
+        ==========
+
+        arg : argument passed by Heaviside object
+
+        H0 : value of Heaviside(0)
+
+        """
+        if arg.is_extended_negative:
+            return S.Zero
+        elif arg.is_extended_positive:
+            return S.One
+        elif arg.is_zero:
+            return H0
+        elif arg is S.NaN:
+            return S.NaN
+        elif fuzzy_not(im(arg).is_zero):
+            raise ValueError("Function defined only for Real Values. Complex part: %s  found in %s ." % (repr(im(arg)), repr(arg)) )
+
+    def _eval_rewrite_as_Piecewise(self, arg, H0=None, **kwargs):
+        """
+        Represents Heaviside in a Piecewise form.
+
+        Examples
+        ========
+
+        >>> from sympy import Heaviside, Piecewise, Symbol, nan
+        >>> x = Symbol('x')
+
+        >>> Heaviside(x).rewrite(Piecewise)
+        Piecewise((0, x < 0), (1/2, Eq(x, 0)), (1, True))
+
+        >>> Heaviside(x,nan).rewrite(Piecewise)
+        Piecewise((0, x < 0), (nan, Eq(x, 0)), (1, True))
+
+        >>> Heaviside(x - 5).rewrite(Piecewise)
+        Piecewise((0, x < 5), (1/2, Eq(x, 5)), (1, True))
+
+        >>> Heaviside(x**2 - 1).rewrite(Piecewise)
+        Piecewise((0, x**2 < 1), (1/2, Eq(x**2, 1)), (1, True))
+
+        """
+        if H0 == 0:
+            return Piecewise((0, arg <= 0), (1, True))
+        if H0 == 1:
+            return Piecewise((0, arg < 0), (1, True))
+        return Piecewise((0, arg < 0), (H0, Eq(arg, 0)), (1, True))
+
+    def _eval_rewrite_as_sign(self, arg, H0=S.Half, **kwargs):
+        """
+        Represents the Heaviside function in the form of sign function.
+
+        Explanation
+        ===========
+
+        The value of Heaviside(0) must be 1/2 for rewriting as sign to be
+        strictly equivalent. For easier usage, we also allow this rewriting
+        when Heaviside(0) is undefined.
+
+        Examples
+        ========
+
+        >>> from sympy import Heaviside, Symbol, sign, nan
+        >>> x = Symbol('x', real=True)
+        >>> y = Symbol('y')
+
+        >>> Heaviside(x).rewrite(sign)
+        sign(x)/2 + 1/2
+
+        >>> Heaviside(x, 0).rewrite(sign)
+        Piecewise((sign(x)/2 + 1/2, Ne(x, 0)), (0, True))
+
+        >>> Heaviside(x, nan).rewrite(sign)
+        Piecewise((sign(x)/2 + 1/2, Ne(x, 0)), (nan, True))
+
+        >>> Heaviside(x - 2).rewrite(sign)
+        sign(x - 2)/2 + 1/2
+
+        >>> Heaviside(x**2 - 2*x + 1).rewrite(sign)
+        sign(x**2 - 2*x + 1)/2 + 1/2
+
+        >>> Heaviside(y).rewrite(sign)
+        Heaviside(y)
+
+        >>> Heaviside(y**2 - 2*y + 1).rewrite(sign)
+        Heaviside(y**2 - 2*y + 1)
+
+        See Also
+        ========
+
+        sign
+
+        """
+        if arg.is_extended_real:
+            pw1 = Piecewise(
+                ((sign(arg) + 1)/2, Ne(arg, 0)),
+                (Heaviside(0, H0=H0), True))
+            pw2 = Piecewise(
+                ((sign(arg) + 1)/2, Eq(Heaviside(0, H0=H0), S.Half)),
+                (pw1, True))
+            return pw2
+
+    def _eval_rewrite_as_SingularityFunction(self, args, H0=S.Half, **kwargs):
+        """
+        Returns the Heaviside expression written in the form of Singularity
+        Functions.
+
+        """
+        from sympy.solvers import solve
+        from sympy.functions.special.singularity_functions import SingularityFunction
+        if self == Heaviside(0):
+            return SingularityFunction(0, 0, 0)
+        free = self.free_symbols
+        if len(free) == 1:
+            x = (free.pop())
+            return SingularityFunction(x, solve(args, x)[0], 0)
+            # TODO
+            # ((x - 5)**3*Heaviside(x - 5)).rewrite(SingularityFunction) should output
+            # SingularityFunction(x, 5, 0) instead of (x - 5)**3*SingularityFunction(x, 5, 0)
+        else:
+            # I don't know how to handle the case for Heaviside expressions
+            # having arguments with more than one variable.
+            raise TypeError(filldedent('''
+                rewrite(SingularityFunction) does not
+                support arguments with more that one variable.'''))

llmeval-env/lib/python3.10/site-packages/sympy/functions/special/elliptic_integrals.py

@@ -0,0 +1,445 @@
+""" Elliptic Integrals. """
+
+from sympy.core import S, pi, I, Rational
+from sympy.core.function import Function, ArgumentIndexError
+from sympy.core.symbol import Dummy
+from sympy.functions.elementary.complexes import sign
+from sympy.functions.elementary.hyperbolic import atanh
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.trigonometric import sin, tan
+from sympy.functions.special.gamma_functions import gamma
+from sympy.functions.special.hyper import hyper, meijerg
+
+class elliptic_k(Function):
+    r"""
+    The complete elliptic integral of the first kind, defined by
+
+    .. math:: K(m) = F\left(\tfrac{\pi}{2}\middle| m\right)
+
+    where $F\left(z\middle| m\right)$ is the Legendre incomplete
+    elliptic integral of the first kind.
+
+    Explanation
+    ===========
+
+    The function $K(m)$ is a single-valued function on the complex
+    plane with branch cut along the interval $(1, \infty)$.
+
+    Note that our notation defines the incomplete elliptic integral
+    in terms of the parameter $m$ instead of the elliptic modulus
+    (eccentricity) $k$.
+    In this case, the parameter $m$ is defined as $m=k^2$.
+
+    Examples
+    ========
+
+    >>> from sympy import elliptic_k, I
+    >>> from sympy.abc import m
+    >>> elliptic_k(0)
+    pi/2
+    >>> elliptic_k(1.0 + I)
+    1.50923695405127 + 0.625146415202697*I
+    >>> elliptic_k(m).series(n=3)
+    pi/2 + pi*m/8 + 9*pi*m**2/128 + O(m**3)
+
+    See Also
+    ========
+
+    elliptic_f
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Elliptic_integrals
+    .. [2] https://functions.wolfram.com/EllipticIntegrals/EllipticK
+
+    """
+
+    @classmethod
+    def eval(cls, m):
+        if m.is_zero:
+            return pi*S.Half
+        elif m is S.Half:
+            return 8*pi**Rational(3, 2)/gamma(Rational(-1, 4))**2
+        elif m is S.One:
+            return S.ComplexInfinity
+        elif m is S.NegativeOne:
+            return gamma(Rational(1, 4))**2/(4*sqrt(2*pi))
+        elif m in (S.Infinity, S.NegativeInfinity, I*S.Infinity,
+                   I*S.NegativeInfinity, S.ComplexInfinity):
+            return S.Zero
+
+    def fdiff(self, argindex=1):
+        m = self.args[0]
+        return (elliptic_e(m) - (1 - m)*elliptic_k(m))/(2*m*(1 - m))
+
+    def _eval_conjugate(self):
+        m = self.args[0]
+        if (m.is_real and (m - 1).is_positive) is False:
+            return self.func(m.conjugate())
+
+    def _eval_nseries(self, x, n, logx, cdir=0):
+        from sympy.simplify import hyperexpand
+        return hyperexpand(self.rewrite(hyper)._eval_nseries(x, n=n, logx=logx))
+
+    def _eval_rewrite_as_hyper(self, m, **kwargs):
+        return pi*S.Half*hyper((S.Half, S.Half), (S.One,), m)
+
+    def _eval_rewrite_as_meijerg(self, m, **kwargs):
+        return meijerg(((S.Half, S.Half), []), ((S.Zero,), (S.Zero,)), -m)/2
+
+    def _eval_is_zero(self):
+        m = self.args[0]
+        if m.is_infinite:
+            return True
+
+    def _eval_rewrite_as_Integral(self, *args):
+        from sympy.integrals.integrals import Integral
+        t = Dummy('t')
+        m = self.args[0]
+        return Integral(1/sqrt(1 - m*sin(t)**2), (t, 0, pi/2))
+
+
+class elliptic_f(Function):
+    r"""
+    The Legendre incomplete elliptic integral of the first
+    kind, defined by
+
+    .. math:: F\left(z\middle| m\right) =
+              \int_0^z \frac{dt}{\sqrt{1 - m \sin^2 t}}
+
+    Explanation
+    ===========
+
+    This function reduces to a complete elliptic integral of
+    the first kind, $K(m)$, when $z = \pi/2$.
+
+    Note that our notation defines the incomplete elliptic integral
+    in terms of the parameter $m$ instead of the elliptic modulus
+    (eccentricity) $k$.
+    In this case, the parameter $m$ is defined as $m=k^2$.
+
+    Examples
+    ========
+
+    >>> from sympy import elliptic_f, I
+    >>> from sympy.abc import z, m
+    >>> elliptic_f(z, m).series(z)
+    z + z**5*(3*m**2/40 - m/30) + m*z**3/6 + O(z**6)
+    >>> elliptic_f(3.0 + I/2, 1.0 + I)
+    2.909449841483 + 1.74720545502474*I
+
+    See Also
+    ========
+
+    elliptic_k
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Elliptic_integrals
+    .. [2] https://functions.wolfram.com/EllipticIntegrals/EllipticF
+
+    """
+
+    @classmethod
+    def eval(cls, z, m):
+        if z.is_zero:
+            return S.Zero
+        if m.is_zero:
+            return z
+        k = 2*z/pi
+        if k.is_integer:
+            return k*elliptic_k(m)
+        elif m in (S.Infinity, S.NegativeInfinity):
+            return S.Zero
+        elif z.could_extract_minus_sign():
+            return -elliptic_f(-z, m)
+
+    def fdiff(self, argindex=1):
+        z, m = self.args
+        fm = sqrt(1 - m*sin(z)**2)
+        if argindex == 1:
+            return 1/fm
+        elif argindex == 2:
+            return (elliptic_e(z, m)/(2*m*(1 - m)) - elliptic_f(z, m)/(2*m) -
+                    sin(2*z)/(4*(1 - m)*fm))
+        raise ArgumentIndexError(self, argindex)
+
+    def _eval_conjugate(self):
+        z, m = self.args
+        if (m.is_real and (m - 1).is_positive) is False:
+            return self.func(z.conjugate(), m.conjugate())
+
+    def _eval_rewrite_as_Integral(self, *args):
+        from sympy.integrals.integrals import Integral
+        t = Dummy('t')
+        z, m = self.args[0], self.args[1]
+        return Integral(1/(sqrt(1 - m*sin(t)**2)), (t, 0, z))
+
+    def _eval_is_zero(self):
+        z, m = self.args
+        if z.is_zero:
+            return True
+        if m.is_extended_real and m.is_infinite:
+            return True
+
+
+class elliptic_e(Function):
+    r"""
+    Called with two arguments $z$ and $m$, evaluates the
+    incomplete elliptic integral of the second kind, defined by
+
+    .. math:: E\left(z\middle| m\right) = \int_0^z \sqrt{1 - m \sin^2 t} dt
+
+    Called with a single argument $m$, evaluates the Legendre complete
+    elliptic integral of the second kind
+
+    .. math:: E(m) = E\left(\tfrac{\pi}{2}\middle| m\right)
+
+    Explanation
+    ===========
+
+    The function $E(m)$ is a single-valued function on the complex
+    plane with branch cut along the interval $(1, \infty)$.
+
+    Note that our notation defines the incomplete elliptic integral
+    in terms of the parameter $m$ instead of the elliptic modulus
+    (eccentricity) $k$.
+    In this case, the parameter $m$ is defined as $m=k^2$.
+
+    Examples
+    ========
+
+    >>> from sympy import elliptic_e, I
+    >>> from sympy.abc import z, m
+    >>> elliptic_e(z, m).series(z)
+    z + z**5*(-m**2/40 + m/30) - m*z**3/6 + O(z**6)
+    >>> elliptic_e(m).series(n=4)
+    pi/2 - pi*m/8 - 3*pi*m**2/128 - 5*pi*m**3/512 + O(m**4)
+    >>> elliptic_e(1 + I, 2 - I/2).n()
+    1.55203744279187 + 0.290764986058437*I
+    >>> elliptic_e(0)
+    pi/2
+    >>> elliptic_e(2.0 - I)
+    0.991052601328069 + 0.81879421395609*I
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Elliptic_integrals
+    .. [2] https://functions.wolfram.com/EllipticIntegrals/EllipticE2
+    .. [3] https://functions.wolfram.com/EllipticIntegrals/EllipticE
+
+    """
+
+    @classmethod
+    def eval(cls, m, z=None):
+        if z is not None:
+            z, m = m, z
+            k = 2*z/pi
+            if m.is_zero:
+                return z
+            if z.is_zero:
+                return S.Zero
+            elif k.is_integer:
+                return k*elliptic_e(m)
+            elif m in (S.Infinity, S.NegativeInfinity):
+                return S.ComplexInfinity
+            elif z.could_extract_minus_sign():
+                return -elliptic_e(-z, m)
+        else:
+            if m.is_zero:
+                return pi/2
+            elif m is S.One:
+                return S.One
+            elif m is S.Infinity:
+                return I*S.Infinity
+            elif m is S.NegativeInfinity:
+                return S.Infinity
+            elif m is S.ComplexInfinity:
+                return S.ComplexInfinity
+
+    def fdiff(self, argindex=1):
+        if len(self.args) == 2:
+            z, m = self.args
+            if argindex == 1:
+                return sqrt(1 - m*sin(z)**2)
+            elif argindex == 2:
+                return (elliptic_e(z, m) - elliptic_f(z, m))/(2*m)
+        else:
+            m = self.args[0]
+            if argindex == 1:
+                return (elliptic_e(m) - elliptic_k(m))/(2*m)
+        raise ArgumentIndexError(self, argindex)
+
+    def _eval_conjugate(self):
+        if len(self.args) == 2:
+            z, m = self.args
+            if (m.is_real and (m - 1).is_positive) is False:
+                return self.func(z.conjugate(), m.conjugate())
+        else:
+            m = self.args[0]
+            if (m.is_real and (m - 1).is_positive) is False:
+                return self.func(m.conjugate())
+
+    def _eval_nseries(self, x, n, logx, cdir=0):
+        from sympy.simplify import hyperexpand
+        if len(self.args) == 1:
+            return hyperexpand(self.rewrite(hyper)._eval_nseries(x, n=n, logx=logx))
+        return super()._eval_nseries(x, n=n, logx=logx)
+
+    def _eval_rewrite_as_hyper(self, *args, **kwargs):
+        if len(args) == 1:
+            m = args[0]
+            return (pi/2)*hyper((Rational(-1, 2), S.Half), (S.One,), m)
+
+    def _eval_rewrite_as_meijerg(self, *args, **kwargs):
+        if len(args) == 1:
+            m = args[0]
+            return -meijerg(((S.Half, Rational(3, 2)), []), \
+                            ((S.Zero,), (S.Zero,)), -m)/4
+
+    def _eval_rewrite_as_Integral(self, *args):
+        from sympy.integrals.integrals import Integral
+        z, m = (pi/2, self.args[0]) if len(self.args) == 1 else self.args
+        t = Dummy('t')
+        return Integral(sqrt(1 - m*sin(t)**2), (t, 0, z))
+
+
+class elliptic_pi(Function):
+    r"""
+    Called with three arguments $n$, $z$ and $m$, evaluates the
+    Legendre incomplete elliptic integral of the third kind, defined by
+
+    .. math:: \Pi\left(n; z\middle| m\right) = \int_0^z \frac{dt}
+              {\left(1 - n \sin^2 t\right) \sqrt{1 - m \sin^2 t}}
+
+    Called with two arguments $n$ and $m$, evaluates the complete
+    elliptic integral of the third kind:
+
+    .. math:: \Pi\left(n\middle| m\right) =
+              \Pi\left(n; \tfrac{\pi}{2}\middle| m\right)
+
+    Explanation
+    ===========
+
+    Note that our notation defines the incomplete elliptic integral
+    in terms of the parameter $m$ instead of the elliptic modulus
+    (eccentricity) $k$.
+    In this case, the parameter $m$ is defined as $m=k^2$.
+
+    Examples
+    ========
+
+    >>> from sympy import elliptic_pi, I
+    >>> from sympy.abc import z, n, m
+    >>> elliptic_pi(n, z, m).series(z, n=4)
+    z + z**3*(m/6 + n/3) + O(z**4)
+    >>> elliptic_pi(0.5 + I, 1.0 - I, 1.2)
+    2.50232379629182 - 0.760939574180767*I
+    >>> elliptic_pi(0, 0)
+    pi/2
+    >>> elliptic_pi(1.0 - I/3, 2.0 + I)
+    3.29136443417283 + 0.32555634906645*I
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Elliptic_integrals
+    .. [2] https://functions.wolfram.com/EllipticIntegrals/EllipticPi3
+    .. [3] https://functions.wolfram.com/EllipticIntegrals/EllipticPi
+
+    """
+
+    @classmethod
+    def eval(cls, n, m, z=None):
+        if z is not None:
+            n, z, m = n, m, z
+            if n.is_zero:
+                return elliptic_f(z, m)
+            elif n is S.One:
+                return (elliptic_f(z, m) +
+                        (sqrt(1 - m*sin(z)**2)*tan(z) -
+                         elliptic_e(z, m))/(1 - m))
+            k = 2*z/pi
+            if k.is_integer:
+                return k*elliptic_pi(n, m)
+            elif m.is_zero:
+                return atanh(sqrt(n - 1)*tan(z))/sqrt(n - 1)
+            elif n == m:
+                return (elliptic_f(z, n) - elliptic_pi(1, z, n) +
+                        tan(z)/sqrt(1 - n*sin(z)**2))
+            elif n in (S.Infinity, S.NegativeInfinity):
+                return S.Zero
+            elif m in (S.Infinity, S.NegativeInfinity):
+                return S.Zero
+            elif z.could_extract_minus_sign():
+                return -elliptic_pi(n, -z, m)
+            if n.is_zero:
+                return elliptic_f(z, m)
+            if m.is_extended_real and m.is_infinite or \
+                    n.is_extended_real and n.is_infinite:
+                return S.Zero
+        else:
+            if n.is_zero:
+                return elliptic_k(m)
+            elif n is S.One:
+                return S.ComplexInfinity
+            elif m.is_zero:
+                return pi/(2*sqrt(1 - n))
+            elif m == S.One:
+                return S.NegativeInfinity/sign(n - 1)
+            elif n == m:
+                return elliptic_e(n)/(1 - n)
+            elif n in (S.Infinity, S.NegativeInfinity):
+                return S.Zero
+            elif m in (S.Infinity, S.NegativeInfinity):
+                return S.Zero
+            if n.is_zero:
+                return elliptic_k(m)
+            if m.is_extended_real and m.is_infinite or \
+                    n.is_extended_real and n.is_infinite:
+                return S.Zero
+
+    def _eval_conjugate(self):
+        if len(self.args) == 3:
+            n, z, m = self.args
+            if (n.is_real and (n - 1).is_positive) is False and \
+               (m.is_real and (m - 1).is_positive) is False:
+                return self.func(n.conjugate(), z.conjugate(), m.conjugate())
+        else:
+            n, m = self.args
+            return self.func(n.conjugate(), m.conjugate())
+
+    def fdiff(self, argindex=1):
+        if len(self.args) == 3:
+            n, z, m = self.args
+            fm, fn = sqrt(1 - m*sin(z)**2), 1 - n*sin(z)**2
+            if argindex == 1:
+                return (elliptic_e(z, m) + (m - n)*elliptic_f(z, m)/n +
+                        (n**2 - m)*elliptic_pi(n, z, m)/n -
+                        n*fm*sin(2*z)/(2*fn))/(2*(m - n)*(n - 1))
+            elif argindex == 2:
+                return 1/(fm*fn)
+            elif argindex == 3:
+                return (elliptic_e(z, m)/(m - 1) +
+                        elliptic_pi(n, z, m) -
+                        m*sin(2*z)/(2*(m - 1)*fm))/(2*(n - m))
+        else:
+            n, m = self.args
+            if argindex == 1:
+                return (elliptic_e(m) + (m - n)*elliptic_k(m)/n +
+                        (n**2 - m)*elliptic_pi(n, m)/n)/(2*(m - n)*(n - 1))
+            elif argindex == 2:
+                return (elliptic_e(m)/(m - 1) + elliptic_pi(n, m))/(2*(n - m))
+        raise ArgumentIndexError(self, argindex)
+
+    def _eval_rewrite_as_Integral(self, *args):
+        from sympy.integrals.integrals import Integral
+        if len(self.args) == 2:
+            n, m, z = self.args[0], self.args[1], pi/2
+        else:
+            n, z, m = self.args
+        t = Dummy('t')
+        return Integral(1/((1 - n*sin(t)**2)*sqrt(1 - m*sin(t)**2)), (t, 0, z))

llmeval-env/lib/python3.10/site-packages/sympy/functions/special/error_functions.py

@@ -0,0 +1,2741 @@
+""" This module contains various functions that are special cases
+    of incomplete gamma functions. It should probably be renamed. """
+
+from sympy.core import EulerGamma  # Must be imported from core, not core.numbers
+from sympy.core.add import Add
+from sympy.core.cache import cacheit
+from sympy.core.function import Function, ArgumentIndexError, expand_mul
+from sympy.core.numbers import I, pi, Rational
+from sympy.core.relational import is_eq
+from sympy.core.power import Pow
+from sympy.core.singleton import S
+from sympy.core.symbol import Symbol
+from sympy.core.sympify import sympify
+from sympy.functions.combinatorial.factorials import factorial, factorial2, RisingFactorial
+from sympy.functions.elementary.complexes import  polar_lift, re, unpolarify
+from sympy.functions.elementary.integers import ceiling, floor
+from sympy.functions.elementary.miscellaneous import sqrt, root
+from sympy.functions.elementary.exponential import exp, log, exp_polar
+from sympy.functions.elementary.hyperbolic import cosh, sinh
+from sympy.functions.elementary.trigonometric import cos, sin, sinc
+from sympy.functions.special.hyper import hyper, meijerg
+
+# TODO series expansions
+# TODO see the "Note:" in Ei
+
+# Helper function
+def real_to_real_as_real_imag(self, deep=True, **hints):
+    if self.args[0].is_extended_real:
+        if deep:
+            hints['complex'] = False
+            return (self.expand(deep, **hints), S.Zero)
+        else:
+            return (self, S.Zero)
+    if deep:
+        x, y = self.args[0].expand(deep, **hints).as_real_imag()
+    else:
+        x, y = self.args[0].as_real_imag()
+    re = (self.func(x + I*y) + self.func(x - I*y))/2
+    im = (self.func(x + I*y) - self.func(x - I*y))/(2*I)
+    return (re, im)
+
+
+###############################################################################
+################################ ERROR FUNCTION ###############################
+###############################################################################
+
+
+class erf(Function):
+    r"""
+    The Gauss error function.
+
+    Explanation
+    ===========
+
+    This function is defined as:
+
+    .. math ::
+        \mathrm{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} \mathrm{d}t.
+
+    Examples
+    ========
+
+    >>> from sympy import I, oo, erf
+    >>> from sympy.abc import z
+
+    Several special values are known:
+
+    >>> erf(0)
+    0
+    >>> erf(oo)
+    1
+    >>> erf(-oo)
+    -1
+    >>> erf(I*oo)
+    oo*I
+    >>> erf(-I*oo)
+    -oo*I
+
+    In general one can pull out factors of -1 and $I$ from the argument:
+
+    >>> erf(-z)
+    -erf(z)
+
+    The error function obeys the mirror symmetry:
+
+    >>> from sympy import conjugate
+    >>> conjugate(erf(z))
+    erf(conjugate(z))
+
+    Differentiation with respect to $z$ is supported:
+
+    >>> from sympy import diff
+    >>> diff(erf(z), z)
+    2*exp(-z**2)/sqrt(pi)
+
+    We can numerically evaluate the error function to arbitrary precision
+    on the whole complex plane:
+
+    >>> erf(4).evalf(30)
+    0.999999984582742099719981147840
+
+    >>> erf(-4*I).evalf(30)
+    -1296959.73071763923152794095062*I
+
+    See Also
+    ========
+
+    erfc: Complementary error function.
+    erfi: Imaginary error function.
+    erf2: Two-argument error function.
+    erfinv: Inverse error function.
+    erfcinv: Inverse Complementary error function.
+    erf2inv: Inverse two-argument error function.
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Error_function
+    .. [2] https://dlmf.nist.gov/7
+    .. [3] https://mathworld.wolfram.com/Erf.html
+    .. [4] https://functions.wolfram.com/GammaBetaErf/Erf
+
+    """
+
+    unbranched = True
+
+    def fdiff(self, argindex=1):
+        if argindex == 1:
+            return 2*exp(-self.args[0]**2)/sqrt(pi)
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+
+    def inverse(self, argindex=1):
+        """
+        Returns the inverse of this function.
+
+        """
+        return erfinv
+
+    @classmethod
+    def eval(cls, arg):
+        if arg.is_Number:
+            if arg is S.NaN:
+                return S.NaN
+            elif arg is S.Infinity:
+                return S.One
+            elif arg is S.NegativeInfinity:
+                return S.NegativeOne
+            elif arg.is_zero:
+                return S.Zero
+
+        if isinstance(arg, erfinv):
+             return arg.args[0]
+
+        if isinstance(arg, erfcinv):
+            return S.One - arg.args[0]
+
+        if arg.is_zero:
+            return S.Zero
+
+        # Only happens with unevaluated erf2inv
+        if isinstance(arg, erf2inv) and arg.args[0].is_zero:
+            return arg.args[1]
+
+        # Try to pull out factors of I
+        t = arg.extract_multiplicatively(I)
+        if t in (S.Infinity, S.NegativeInfinity):
+            return arg
+
+        # Try to pull out factors of -1
+        if arg.could_extract_minus_sign():
+            return -cls(-arg)
+
+    @staticmethod
+    @cacheit
+    def taylor_term(n, x, *previous_terms):
+        if n < 0 or n % 2 == 0:
+            return S.Zero
+        else:
+            x = sympify(x)
+            k = floor((n - 1)/S(2))
+            if len(previous_terms) > 2:
+                return -previous_terms[-2] * x**2 * (n - 2)/(n*k)
+            else:
+                return 2*S.NegativeOne**k * x**n/(n*factorial(k)*sqrt(pi))
+
+    def _eval_conjugate(self):
+        return self.func(self.args[0].conjugate())
+
+    def _eval_is_real(self):
+        return self.args[0].is_extended_real
+
+    def _eval_is_finite(self):
+        if self.args[0].is_finite:
+            return True
+        else:
+            return self.args[0].is_extended_real
+
+    def _eval_is_zero(self):
+        return self.args[0].is_zero
+
+    def _eval_rewrite_as_uppergamma(self, z, **kwargs):
+        from sympy.functions.special.gamma_functions import uppergamma
+        return sqrt(z**2)/z*(S.One - uppergamma(S.Half, z**2)/sqrt(pi))
+
+    def _eval_rewrite_as_fresnels(self, z, **kwargs):
+        arg = (S.One - I)*z/sqrt(pi)
+        return (S.One + I)*(fresnelc(arg) - I*fresnels(arg))
+
+    def _eval_rewrite_as_fresnelc(self, z, **kwargs):
+        arg = (S.One - I)*z/sqrt(pi)
+        return (S.One + I)*(fresnelc(arg) - I*fresnels(arg))
+
+    def _eval_rewrite_as_meijerg(self, z, **kwargs):
+        return z/sqrt(pi)*meijerg([S.Half], [], [0], [Rational(-1, 2)], z**2)
+
+    def _eval_rewrite_as_hyper(self, z, **kwargs):
+        return 2*z/sqrt(pi)*hyper([S.Half], [3*S.Half], -z**2)
+
+    def _eval_rewrite_as_expint(self, z, **kwargs):
+        return sqrt(z**2)/z - z*expint(S.Half, z**2)/sqrt(pi)
+
+    def _eval_rewrite_as_tractable(self, z, limitvar=None, **kwargs):
+        from sympy.series.limits import limit
+        if limitvar:
+            lim = limit(z, limitvar, S.Infinity)
+            if lim is S.NegativeInfinity:
+                return S.NegativeOne + _erfs(-z)*exp(-z**2)
+        return S.One - _erfs(z)*exp(-z**2)
+
+    def _eval_rewrite_as_erfc(self, z, **kwargs):
+        return S.One - erfc(z)
+
+    def _eval_rewrite_as_erfi(self, z, **kwargs):
+        return -I*erfi(I*z)
+
+    def _eval_as_leading_term(self, x, logx=None, cdir=0):
+        arg = self.args[0].as_leading_term(x, logx=logx, cdir=cdir)
+        arg0 = arg.subs(x, 0)
+
+        if arg0 is S.ComplexInfinity:
+            arg0 = arg.limit(x, 0, dir='-' if cdir == -1 else '+')
+        if x in arg.free_symbols and arg0.is_zero:
+            return 2*arg/sqrt(pi)
+        else:
+            return self.func(arg0)
+
+    def _eval_aseries(self, n, args0, x, logx):
+        from sympy.series.order import Order
+        point = args0[0]
+
+        if point in [S.Infinity, S.NegativeInfinity]:
+            z = self.args[0]
+
+            try:
+                _, ex = z.leadterm(x)
+            except (ValueError, NotImplementedError):
+                return self
+
+            ex = -ex # as x->1/x for aseries
+            if ex.is_positive:
+                newn = ceiling(n/ex)
+                s = [S.NegativeOne**k * factorial2(2*k - 1) / (z**(2*k + 1) * 2**k)
+                     for k in range(newn)] + [Order(1/z**newn, x)]
+                return S.One - (exp(-z**2)/sqrt(pi)) * Add(*s)
+
+        return super(erf, self)._eval_aseries(n, args0, x, logx)
+
+    as_real_imag = real_to_real_as_real_imag
+
+
+class erfc(Function):
+    r"""
+    Complementary Error Function.
+
+    Explanation
+    ===========
+
+    The function is defined as:
+
+    .. math ::
+        \mathrm{erfc}(x) = \frac{2}{\sqrt{\pi}} \int_x^\infty e^{-t^2} \mathrm{d}t
+
+    Examples
+    ========
+
+    >>> from sympy import I, oo, erfc
+    >>> from sympy.abc import z
+
+    Several special values are known:
+
+    >>> erfc(0)
+    1
+    >>> erfc(oo)
+    0
+    >>> erfc(-oo)
+    2
+    >>> erfc(I*oo)
+    -oo*I
+    >>> erfc(-I*oo)
+    oo*I
+
+    The error function obeys the mirror symmetry:
+
+    >>> from sympy import conjugate
+    >>> conjugate(erfc(z))
+    erfc(conjugate(z))
+
+    Differentiation with respect to $z$ is supported:
+
+    >>> from sympy import diff
+    >>> diff(erfc(z), z)
+    -2*exp(-z**2)/sqrt(pi)
+
+    It also follows
+
+    >>> erfc(-z)
+    2 - erfc(z)
+
+    We can numerically evaluate the complementary error function to arbitrary
+    precision on the whole complex plane:
+
+    >>> erfc(4).evalf(30)
+    0.0000000154172579002800188521596734869
+
+    >>> erfc(4*I).evalf(30)
+    1.0 - 1296959.73071763923152794095062*I
+
+    See Also
+    ========
+
+    erf: Gaussian error function.
+    erfi: Imaginary error function.
+    erf2: Two-argument error function.
+    erfinv: Inverse error function.
+    erfcinv: Inverse Complementary error function.
+    erf2inv: Inverse two-argument error function.
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Error_function
+    .. [2] https://dlmf.nist.gov/7
+    .. [3] https://mathworld.wolfram.com/Erfc.html
+    .. [4] https://functions.wolfram.com/GammaBetaErf/Erfc
+
+    """
+
+    unbranched = True
+
+    def fdiff(self, argindex=1):
+        if argindex == 1:
+            return -2*exp(-self.args[0]**2)/sqrt(pi)
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    def inverse(self, argindex=1):
+        """
+        Returns the inverse of this function.
+
+        """
+        return erfcinv
+
+    @classmethod
+    def eval(cls, arg):
+        if arg.is_Number:
+            if arg is S.NaN:
+                return S.NaN
+            elif arg is S.Infinity:
+                return S.Zero
+            elif arg.is_zero:
+                return S.One
+
+        if isinstance(arg, erfinv):
+            return S.One - arg.args[0]
+
+        if isinstance(arg, erfcinv):
+            return arg.args[0]
+
+        if arg.is_zero:
+            return S.One
+
+        # Try to pull out factors of I
+        t = arg.extract_multiplicatively(I)
+        if t in (S.Infinity, S.NegativeInfinity):
+            return -arg
+
+        # Try to pull out factors of -1
+        if arg.could_extract_minus_sign():
+            return 2 - cls(-arg)
+
+    @staticmethod
+    @cacheit
+    def taylor_term(n, x, *previous_terms):
+        if n == 0:
+            return S.One
+        elif n < 0 or n % 2 == 0:
+            return S.Zero
+        else:
+            x = sympify(x)
+            k = floor((n - 1)/S(2))
+            if len(previous_terms) > 2:
+                return -previous_terms[-2] * x**2 * (n - 2)/(n*k)
+            else:
+                return -2*S.NegativeOne**k * x**n/(n*factorial(k)*sqrt(pi))
+
+    def _eval_conjugate(self):
+        return self.func(self.args[0].conjugate())
+
+    def _eval_is_real(self):
+        return self.args[0].is_extended_real
+
+    def _eval_rewrite_as_tractable(self, z, limitvar=None, **kwargs):
+        return self.rewrite(erf).rewrite("tractable", deep=True, limitvar=limitvar)
+
+    def _eval_rewrite_as_erf(self, z, **kwargs):
+        return S.One - erf(z)
+
+    def _eval_rewrite_as_erfi(self, z, **kwargs):
+        return S.One + I*erfi(I*z)
+
+    def _eval_rewrite_as_fresnels(self, z, **kwargs):
+        arg = (S.One - I)*z/sqrt(pi)
+        return S.One - (S.One + I)*(fresnelc(arg) - I*fresnels(arg))
+
+    def _eval_rewrite_as_fresnelc(self, z, **kwargs):
+        arg = (S.One-I)*z/sqrt(pi)
+        return S.One - (S.One + I)*(fresnelc(arg) - I*fresnels(arg))
+
+    def _eval_rewrite_as_meijerg(self, z, **kwargs):
+        return S.One - z/sqrt(pi)*meijerg([S.Half], [], [0], [Rational(-1, 2)], z**2)
+
+    def _eval_rewrite_as_hyper(self, z, **kwargs):
+        return S.One - 2*z/sqrt(pi)*hyper([S.Half], [3*S.Half], -z**2)
+
+    def _eval_rewrite_as_uppergamma(self, z, **kwargs):
+        from sympy.functions.special.gamma_functions import uppergamma
+        return S.One - sqrt(z**2)/z*(S.One - uppergamma(S.Half, z**2)/sqrt(pi))
+
+    def _eval_rewrite_as_expint(self, z, **kwargs):
+        return S.One - sqrt(z**2)/z + z*expint(S.Half, z**2)/sqrt(pi)
+
+    def _eval_expand_func(self, **hints):
+        return self.rewrite(erf)
+
+    def _eval_as_leading_term(self, x, logx=None, cdir=0):
+        arg = self.args[0].as_leading_term(x, logx=logx, cdir=cdir)
+        arg0 = arg.subs(x, 0)
+
+        if arg0 is S.ComplexInfinity:
+            arg0 = arg.limit(x, 0, dir='-' if cdir == -1 else '+')
+        if arg0.is_zero:
+            return S.One
+        else:
+            return self.func(arg0)
+
+    as_real_imag = real_to_real_as_real_imag
+
+    def _eval_aseries(self, n, args0, x, logx):
+        return S.One - erf(*self.args)._eval_aseries(n, args0, x, logx)
+
+
+class erfi(Function):
+    r"""
+    Imaginary error function.
+
+    Explanation
+    ===========
+
+    The function erfi is defined as:
+
+    .. math ::
+        \mathrm{erfi}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{t^2} \mathrm{d}t
+
+    Examples
+    ========
+
+    >>> from sympy import I, oo, erfi
+    >>> from sympy.abc import z
+
+    Several special values are known:
+
+    >>> erfi(0)
+    0
+    >>> erfi(oo)
+    oo
+    >>> erfi(-oo)
+    -oo
+    >>> erfi(I*oo)
+    I
+    >>> erfi(-I*oo)
+    -I
+
+    In general one can pull out factors of -1 and $I$ from the argument:
+
+    >>> erfi(-z)
+    -erfi(z)
+
+    >>> from sympy import conjugate
+    >>> conjugate(erfi(z))
+    erfi(conjugate(z))
+
+    Differentiation with respect to $z$ is supported:
+
+    >>> from sympy import diff
+    >>> diff(erfi(z), z)
+    2*exp(z**2)/sqrt(pi)
+
+    We can numerically evaluate the imaginary error function to arbitrary
+    precision on the whole complex plane:
+
+    >>> erfi(2).evalf(30)
+    18.5648024145755525987042919132
+
+    >>> erfi(-2*I).evalf(30)
+    -0.995322265018952734162069256367*I
+
+    See Also
+    ========
+
+    erf: Gaussian error function.
+    erfc: Complementary error function.
+    erf2: Two-argument error function.
+    erfinv: Inverse error function.
+    erfcinv: Inverse Complementary error function.
+    erf2inv: Inverse two-argument error function.
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Error_function
+    .. [2] https://mathworld.wolfram.com/Erfi.html
+    .. [3] https://functions.wolfram.com/GammaBetaErf/Erfi
+
+    """
+
+    unbranched = True
+
+    def fdiff(self, argindex=1):
+        if argindex == 1:
+            return 2*exp(self.args[0]**2)/sqrt(pi)
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    @classmethod
+    def eval(cls, z):
+        if z.is_Number:
+            if z is S.NaN:
+                return S.NaN
+            elif z.is_zero:
+                return S.Zero
+            elif z is S.Infinity:
+                return S.Infinity
+
+        if z.is_zero:
+            return S.Zero
+
+        # Try to pull out factors of -1
+        if z.could_extract_minus_sign():
+            return -cls(-z)
+
+        # Try to pull out factors of I
+        nz = z.extract_multiplicatively(I)
+        if nz is not None:
+            if nz is S.Infinity:
+                return I
+            if isinstance(nz, erfinv):
+                return I*nz.args[0]
+            if isinstance(nz, erfcinv):
+                return I*(S.One - nz.args[0])
+            # Only happens with unevaluated erf2inv
+            if isinstance(nz, erf2inv) and nz.args[0].is_zero:
+                return I*nz.args[1]
+
+    @staticmethod
+    @cacheit
+    def taylor_term(n, x, *previous_terms):
+        if n < 0 or n % 2 == 0:
+            return S.Zero
+        else:
+            x = sympify(x)
+            k = floor((n - 1)/S(2))
+            if len(previous_terms) > 2:
+                return previous_terms[-2] * x**2 * (n - 2)/(n*k)
+            else:
+                return 2 * x**n/(n*factorial(k)*sqrt(pi))
+
+    def _eval_conjugate(self):
+        return self.func(self.args[0].conjugate())
+
+    def _eval_is_extended_real(self):
+        return self.args[0].is_extended_real
+
+    def _eval_is_zero(self):
+        return self.args[0].is_zero
+
+    def _eval_rewrite_as_tractable(self, z, limitvar=None, **kwargs):
+        return self.rewrite(erf).rewrite("tractable", deep=True, limitvar=limitvar)
+
+    def _eval_rewrite_as_erf(self, z, **kwargs):
+        return -I*erf(I*z)
+
+    def _eval_rewrite_as_erfc(self, z, **kwargs):
+        return I*erfc(I*z) - I
+
+    def _eval_rewrite_as_fresnels(self, z, **kwargs):
+        arg = (S.One + I)*z/sqrt(pi)
+        return (S.One - I)*(fresnelc(arg) - I*fresnels(arg))
+
+    def _eval_rewrite_as_fresnelc(self, z, **kwargs):
+        arg = (S.One + I)*z/sqrt(pi)
+        return (S.One - I)*(fresnelc(arg) - I*fresnels(arg))
+
+    def _eval_rewrite_as_meijerg(self, z, **kwargs):
+        return z/sqrt(pi)*meijerg([S.Half], [], [0], [Rational(-1, 2)], -z**2)
+
+    def _eval_rewrite_as_hyper(self, z, **kwargs):
+        return 2*z/sqrt(pi)*hyper([S.Half], [3*S.Half], z**2)
+
+    def _eval_rewrite_as_uppergamma(self, z, **kwargs):
+        from sympy.functions.special.gamma_functions import uppergamma
+        return sqrt(-z**2)/z*(uppergamma(S.Half, -z**2)/sqrt(pi) - S.One)
+
+    def _eval_rewrite_as_expint(self, z, **kwargs):
+        return sqrt(-z**2)/z - z*expint(S.Half, -z**2)/sqrt(pi)
+
+    def _eval_expand_func(self, **hints):
+        return self.rewrite(erf)
+
+    as_real_imag = real_to_real_as_real_imag
+
+    def _eval_as_leading_term(self, x, logx=None, cdir=0):
+        arg = self.args[0].as_leading_term(x, logx=logx, cdir=cdir)
+        arg0 = arg.subs(x, 0)
+
+        if x in arg.free_symbols and arg0.is_zero:
+            return 2*arg/sqrt(pi)
+        elif arg0.is_finite:
+            return self.func(arg0)
+        return self.func(arg)
+
+    def _eval_aseries(self, n, args0, x, logx):
+        from sympy.series.order import Order
+        point = args0[0]
+
+        if point is S.Infinity:
+            z = self.args[0]
+            s = [factorial2(2*k - 1) / (2**k * z**(2*k + 1))
+                    for k in range(n)] + [Order(1/z**n, x)]
+            return -I + (exp(z**2)/sqrt(pi)) * Add(*s)
+
+        return super(erfi, self)._eval_aseries(n, args0, x, logx)
+
+
+class erf2(Function):
+    r"""
+    Two-argument error function.
+
+    Explanation
+    ===========
+
+    This function is defined as:
+
+    .. math ::
+        \mathrm{erf2}(x, y) = \frac{2}{\sqrt{\pi}} \int_x^y e^{-t^2} \mathrm{d}t
+
+    Examples
+    ========
+
+    >>> from sympy import oo, erf2
+    >>> from sympy.abc import x, y
+
+    Several special values are known:
+
+    >>> erf2(0, 0)
+    0
+    >>> erf2(x, x)
+    0
+    >>> erf2(x, oo)
+    1 - erf(x)
+    >>> erf2(x, -oo)
+    -erf(x) - 1
+    >>> erf2(oo, y)
+    erf(y) - 1
+    >>> erf2(-oo, y)
+    erf(y) + 1
+
+    In general one can pull out factors of -1:
+
+    >>> erf2(-x, -y)
+    -erf2(x, y)
+
+    The error function obeys the mirror symmetry:
+
+    >>> from sympy import conjugate
+    >>> conjugate(erf2(x, y))
+    erf2(conjugate(x), conjugate(y))
+
+    Differentiation with respect to $x$, $y$ is supported:
+
+    >>> from sympy import diff
+    >>> diff(erf2(x, y), x)
+    -2*exp(-x**2)/sqrt(pi)
+    >>> diff(erf2(x, y), y)
+    2*exp(-y**2)/sqrt(pi)
+
+    See Also
+    ========
+
+    erf: Gaussian error function.
+    erfc: Complementary error function.
+    erfi: Imaginary error function.
+    erfinv: Inverse error function.
+    erfcinv: Inverse Complementary error function.
+    erf2inv: Inverse two-argument error function.
+
+    References
+    ==========
+
+    .. [1] https://functions.wolfram.com/GammaBetaErf/Erf2/
+
+    """
+
+
+    def fdiff(self, argindex):
+        x, y = self.args
+        if argindex == 1:
+            return -2*exp(-x**2)/sqrt(pi)
+        elif argindex == 2:
+            return 2*exp(-y**2)/sqrt(pi)
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    @classmethod
+    def eval(cls, x, y):
+        chk = (S.Infinity, S.NegativeInfinity, S.Zero)
+        if x is S.NaN or y is S.NaN:
+            return S.NaN
+        elif x == y:
+            return S.Zero
+        elif x in chk or y in chk:
+            return erf(y) - erf(x)
+
+        if isinstance(y, erf2inv) and y.args[0] == x:
+            return y.args[1]
+
+        if x.is_zero or y.is_zero or x.is_extended_real and x.is_infinite or \
+                y.is_extended_real and y.is_infinite:
+            return erf(y) - erf(x)
+
+        #Try to pull out -1 factor
+        sign_x = x.could_extract_minus_sign()
+        sign_y = y.could_extract_minus_sign()
+        if (sign_x and sign_y):
+            return -cls(-x, -y)
+        elif (sign_x or sign_y):
+            return erf(y)-erf(x)
+
+    def _eval_conjugate(self):
+        return self.func(self.args[0].conjugate(), self.args[1].conjugate())
+
+    def _eval_is_extended_real(self):
+        return self.args[0].is_extended_real and self.args[1].is_extended_real
+
+    def _eval_rewrite_as_erf(self, x, y, **kwargs):
+        return erf(y) - erf(x)
+
+    def _eval_rewrite_as_erfc(self, x, y, **kwargs):
+        return erfc(x) - erfc(y)
+
+    def _eval_rewrite_as_erfi(self, x, y, **kwargs):
+        return I*(erfi(I*x)-erfi(I*y))
+
+    def _eval_rewrite_as_fresnels(self, x, y, **kwargs):
+        return erf(y).rewrite(fresnels) - erf(x).rewrite(fresnels)
+
+    def _eval_rewrite_as_fresnelc(self, x, y, **kwargs):
+        return erf(y).rewrite(fresnelc) - erf(x).rewrite(fresnelc)
+
+    def _eval_rewrite_as_meijerg(self, x, y, **kwargs):
+        return erf(y).rewrite(meijerg) - erf(x).rewrite(meijerg)
+
+    def _eval_rewrite_as_hyper(self, x, y, **kwargs):
+        return erf(y).rewrite(hyper) - erf(x).rewrite(hyper)
+
+    def _eval_rewrite_as_uppergamma(self, x, y, **kwargs):
+        from sympy.functions.special.gamma_functions import uppergamma
+        return (sqrt(y**2)/y*(S.One - uppergamma(S.Half, y**2)/sqrt(pi)) -
+            sqrt(x**2)/x*(S.One - uppergamma(S.Half, x**2)/sqrt(pi)))
+
+    def _eval_rewrite_as_expint(self, x, y, **kwargs):
+        return erf(y).rewrite(expint) - erf(x).rewrite(expint)
+
+    def _eval_expand_func(self, **hints):
+        return self.rewrite(erf)
+
+    def _eval_is_zero(self):
+        return is_eq(*self.args)
+
+class erfinv(Function):
+    r"""
+    Inverse Error Function. The erfinv function is defined as:
+
+    .. math ::
+        \mathrm{erf}(x) = y \quad \Rightarrow \quad \mathrm{erfinv}(y) = x
+
+    Examples
+    ========
+
+    >>> from sympy import erfinv
+    >>> from sympy.abc import x
+
+    Several special values are known:
+
+    >>> erfinv(0)
+    0
+    >>> erfinv(1)
+    oo
+
+    Differentiation with respect to $x$ is supported:
+
+    >>> from sympy import diff
+    >>> diff(erfinv(x), x)
+    sqrt(pi)*exp(erfinv(x)**2)/2
+
+    We can numerically evaluate the inverse error function to arbitrary
+    precision on [-1, 1]:
+
+    >>> erfinv(0.2).evalf(30)
+    0.179143454621291692285822705344
+
+    See Also
+    ========
+
+    erf: Gaussian error function.
+    erfc: Complementary error function.
+    erfi: Imaginary error function.
+    erf2: Two-argument error function.
+    erfcinv: Inverse Complementary error function.
+    erf2inv: Inverse two-argument error function.
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Error_function#Inverse_functions
+    .. [2] https://functions.wolfram.com/GammaBetaErf/InverseErf/
+
+    """
+
+
+    def fdiff(self, argindex =1):
+        if argindex == 1:
+            return sqrt(pi)*exp(self.func(self.args[0])**2)*S.Half
+        else :
+            raise ArgumentIndexError(self, argindex)
+
+    def inverse(self, argindex=1):
+        """
+        Returns the inverse of this function.
+
+        """
+        return erf
+
+    @classmethod
+    def eval(cls, z):
+        if z is S.NaN:
+            return S.NaN
+        elif z is S.NegativeOne:
+            return S.NegativeInfinity
+        elif z.is_zero:
+            return S.Zero
+        elif z is S.One:
+            return S.Infinity
+
+        if isinstance(z, erf) and z.args[0].is_extended_real:
+            return z.args[0]
+
+        if z.is_zero:
+            return S.Zero
+
+        # Try to pull out factors of -1
+        nz = z.extract_multiplicatively(-1)
+        if nz is not None and (isinstance(nz, erf) and (nz.args[0]).is_extended_real):
+            return -nz.args[0]
+
+    def _eval_rewrite_as_erfcinv(self, z, **kwargs):
+       return erfcinv(1-z)
+
+    def _eval_is_zero(self):
+        return self.args[0].is_zero
+
+
+class erfcinv (Function):
+    r"""
+    Inverse Complementary Error Function. The erfcinv function is defined as:
+
+    .. math ::
+        \mathrm{erfc}(x) = y \quad \Rightarrow \quad \mathrm{erfcinv}(y) = x
+
+    Examples
+    ========
+
+    >>> from sympy import erfcinv
+    >>> from sympy.abc import x
+
+    Several special values are known:
+
+    >>> erfcinv(1)
+    0
+    >>> erfcinv(0)
+    oo
+
+    Differentiation with respect to $x$ is supported:
+
+    >>> from sympy import diff
+    >>> diff(erfcinv(x), x)
+    -sqrt(pi)*exp(erfcinv(x)**2)/2
+
+    See Also
+    ========
+
+    erf: Gaussian error function.
+    erfc: Complementary error function.
+    erfi: Imaginary error function.
+    erf2: Two-argument error function.
+    erfinv: Inverse error function.
+    erf2inv: Inverse two-argument error function.
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Error_function#Inverse_functions
+    .. [2] https://functions.wolfram.com/GammaBetaErf/InverseErfc/
+
+    """
+
+
+    def fdiff(self, argindex =1):
+        if argindex == 1:
+            return -sqrt(pi)*exp(self.func(self.args[0])**2)*S.Half
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    def inverse(self, argindex=1):
+        """
+        Returns the inverse of this function.
+
+        """
+        return erfc
+
+    @classmethod
+    def eval(cls, z):
+        if z is S.NaN:
+            return S.NaN
+        elif z.is_zero:
+            return S.Infinity
+        elif z is S.One:
+            return S.Zero
+        elif z == 2:
+            return S.NegativeInfinity
+
+        if z.is_zero:
+            return S.Infinity
+
+    def _eval_rewrite_as_erfinv(self, z, **kwargs):
+        return erfinv(1-z)
+
+    def _eval_is_zero(self):
+        return (self.args[0] - 1).is_zero
+
+    def _eval_is_infinite(self):
+        return self.args[0].is_zero
+
+
+class erf2inv(Function):
+    r"""
+    Two-argument Inverse error function. The erf2inv function is defined as:
+
+    .. math ::
+        \mathrm{erf2}(x, w) = y \quad \Rightarrow \quad \mathrm{erf2inv}(x, y) = w
+
+    Examples
+    ========
+
+    >>> from sympy import erf2inv, oo
+    >>> from sympy.abc import x, y
+
+    Several special values are known:
+
+    >>> erf2inv(0, 0)
+    0
+    >>> erf2inv(1, 0)
+    1
+    >>> erf2inv(0, 1)
+    oo
+    >>> erf2inv(0, y)
+    erfinv(y)
+    >>> erf2inv(oo, y)
+    erfcinv(-y)
+
+    Differentiation with respect to $x$ and $y$ is supported:
+
+    >>> from sympy import diff
+    >>> diff(erf2inv(x, y), x)
+    exp(-x**2 + erf2inv(x, y)**2)
+    >>> diff(erf2inv(x, y), y)
+    sqrt(pi)*exp(erf2inv(x, y)**2)/2
+
+    See Also
+    ========
+
+    erf: Gaussian error function.
+    erfc: Complementary error function.
+    erfi: Imaginary error function.
+    erf2: Two-argument error function.
+    erfinv: Inverse error function.
+    erfcinv: Inverse complementary error function.
+
+    References
+    ==========
+
+    .. [1] https://functions.wolfram.com/GammaBetaErf/InverseErf2/
+
+    """
+
+
+    def fdiff(self, argindex):
+        x, y = self.args
+        if argindex == 1:
+            return exp(self.func(x,y)**2-x**2)
+        elif argindex == 2:
+            return sqrt(pi)*S.Half*exp(self.func(x,y)**2)
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    @classmethod
+    def eval(cls, x, y):
+        if x is S.NaN or y is S.NaN:
+            return S.NaN
+        elif x.is_zero and y.is_zero:
+            return S.Zero
+        elif x.is_zero and y is S.One:
+            return S.Infinity
+        elif x is S.One and y.is_zero:
+            return S.One
+        elif x.is_zero:
+            return erfinv(y)
+        elif x is S.Infinity:
+            return erfcinv(-y)
+        elif y.is_zero:
+            return x
+        elif y is S.Infinity:
+            return erfinv(x)
+
+        if x.is_zero:
+            if y.is_zero:
+                return S.Zero
+            else:
+                return erfinv(y)
+        if y.is_zero:
+            return x
+
+    def _eval_is_zero(self):
+        x, y = self.args
+        if x.is_zero and y.is_zero:
+            return True
+
+###############################################################################
+#################### EXPONENTIAL INTEGRALS ####################################
+###############################################################################
+
+class Ei(Function):
+    r"""
+    The classical exponential integral.
+
+    Explanation
+    ===========
+
+    For use in SymPy, this function is defined as
+
+    .. math:: \operatorname{Ei}(x) = \sum_{n=1}^\infty \frac{x^n}{n\, n!}
+                                     + \log(x) + \gamma,
+
+    where $\gamma$ is the Euler-Mascheroni constant.
+
+    If $x$ is a polar number, this defines an analytic function on the
+    Riemann surface of the logarithm. Otherwise this defines an analytic
+    function in the cut plane $\mathbb{C} \setminus (-\infty, 0]$.
+
+    **Background**
+
+    The name exponential integral comes from the following statement:
+
+    .. math:: \operatorname{Ei}(x) = \int_{-\infty}^x \frac{e^t}{t} \mathrm{d}t
+
+    If the integral is interpreted as a Cauchy principal value, this statement
+    holds for $x > 0$ and $\operatorname{Ei}(x)$ as defined above.
+
+    Examples
+    ========
+
+    >>> from sympy import Ei, polar_lift, exp_polar, I, pi
+    >>> from sympy.abc import x
+
+    >>> Ei(-1)
+    Ei(-1)
+
+    This yields a real value:
+
+    >>> Ei(-1).n(chop=True)
+    -0.219383934395520
+
+    On the other hand the analytic continuation is not real:
+
+    >>> Ei(polar_lift(-1)).n(chop=True)
+    -0.21938393439552 + 3.14159265358979*I
+
+    The exponential integral has a logarithmic branch point at the origin:
+
+    >>> Ei(x*exp_polar(2*I*pi))
+    Ei(x) + 2*I*pi
+
+    Differentiation is supported:
+
+    >>> Ei(x).diff(x)
+    exp(x)/x
+
+    The exponential integral is related to many other special functions.
+    For example:
+
+    >>> from sympy import expint, Shi
+    >>> Ei(x).rewrite(expint)
+    -expint(1, x*exp_polar(I*pi)) - I*pi
+    >>> Ei(x).rewrite(Shi)
+    Chi(x) + Shi(x)
+
+    See Also
+    ========
+
+    expint: Generalised exponential integral.
+    E1: Special case of the generalised exponential integral.
+    li: Logarithmic integral.
+    Li: Offset logarithmic integral.
+    Si: Sine integral.
+    Ci: Cosine integral.
+    Shi: Hyperbolic sine integral.
+    Chi: Hyperbolic cosine integral.
+    uppergamma: Upper incomplete gamma function.
+
+    References
+    ==========
+
+    .. [1] https://dlmf.nist.gov/6.6
+    .. [2] https://en.wikipedia.org/wiki/Exponential_integral
+    .. [3] Abramowitz & Stegun, section 5: https://web.archive.org/web/20201128173312/http://people.math.sfu.ca/~cbm/aands/page_228.htm
+
+    """
+
+
+    @classmethod
+    def eval(cls, z):
+        if z.is_zero:
+            return S.NegativeInfinity
+        elif z is S.Infinity:
+            return S.Infinity
+        elif z is S.NegativeInfinity:
+            return S.Zero
+
+        if z.is_zero:
+            return S.NegativeInfinity
+
+        nz, n = z.extract_branch_factor()
+        if n:
+            return Ei(nz) + 2*I*pi*n
+
+    def fdiff(self, argindex=1):
+        arg = unpolarify(self.args[0])
+        if argindex == 1:
+            return exp(arg)/arg
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    def _eval_evalf(self, prec):
+        if (self.args[0]/polar_lift(-1)).is_positive:
+            return Function._eval_evalf(self, prec) + (I*pi)._eval_evalf(prec)
+        return Function._eval_evalf(self, prec)
+
+    def _eval_rewrite_as_uppergamma(self, z, **kwargs):
+        from sympy.functions.special.gamma_functions import uppergamma
+        # XXX this does not currently work usefully because uppergamma
+        #     immediately turns into expint
+        return -uppergamma(0, polar_lift(-1)*z) - I*pi
+
+    def _eval_rewrite_as_expint(self, z, **kwargs):
+        return -expint(1, polar_lift(-1)*z) - I*pi
+
+    def _eval_rewrite_as_li(self, z, **kwargs):
+        if isinstance(z, log):
+            return li(z.args[0])
+        # TODO:
+        # Actually it only holds that:
+        #  Ei(z) = li(exp(z))
+        # for -pi < imag(z) <= pi
+        return li(exp(z))
+
+    def _eval_rewrite_as_Si(self, z, **kwargs):
+        if z.is_negative:
+            return Shi(z) + Chi(z) - I*pi
+        else:
+            return Shi(z) + Chi(z)
+    _eval_rewrite_as_Ci = _eval_rewrite_as_Si
+    _eval_rewrite_as_Chi = _eval_rewrite_as_Si
+    _eval_rewrite_as_Shi = _eval_rewrite_as_Si
+
+    def _eval_rewrite_as_tractable(self, z, limitvar=None, **kwargs):
+        return exp(z) * _eis(z)
+
+    def _eval_as_leading_term(self, x, logx=None, cdir=0):
+        from sympy import re
+        x0 = self.args[0].limit(x, 0)
+        arg = self.args[0].as_leading_term(x, cdir=cdir)
+        cdir = arg.dir(x, cdir)
+        if x0.is_zero:
+            c, e = arg.as_coeff_exponent(x)
+            logx = log(x) if logx is None else logx
+            return log(c) + e*logx + EulerGamma - (
+                I*pi if re(cdir).is_negative else S.Zero)
+        return super()._eval_as_leading_term(x, logx=logx, cdir=cdir)
+
+    def _eval_nseries(self, x, n, logx, cdir=0):
+        x0 = self.args[0].limit(x, 0)
+        if x0.is_zero:
+            f = self._eval_rewrite_as_Si(*self.args)
+            return f._eval_nseries(x, n, logx)
+        return super()._eval_nseries(x, n, logx)
+
+    def _eval_aseries(self, n, args0, x, logx):
+        from sympy.series.order import Order
+        point = args0[0]
+
+        if point is S.Infinity:
+            z = self.args[0]
+            s = [factorial(k) / (z)**k for k in range(n)] + \
+                    [Order(1/z**n, x)]
+            return (exp(z)/z) * Add(*s)
+
+        return super(Ei, self)._eval_aseries(n, args0, x, logx)
+
+
+class expint(Function):
+    r"""
+    Generalized exponential integral.
+
+    Explanation
+    ===========
+
+    This function is defined as
+
+    .. math:: \operatorname{E}_\nu(z) = z^{\nu - 1} \Gamma(1 - \nu, z),
+
+    where $\Gamma(1 - \nu, z)$ is the upper incomplete gamma function
+    (``uppergamma``).
+
+    Hence for $z$ with positive real part we have
+
+    .. math:: \operatorname{E}_\nu(z)
+              =   \int_1^\infty \frac{e^{-zt}}{t^\nu} \mathrm{d}t,
+
+    which explains the name.
+
+    The representation as an incomplete gamma function provides an analytic
+    continuation for $\operatorname{E}_\nu(z)$. If $\nu$ is a
+    non-positive integer, the exponential integral is thus an unbranched
+    function of $z$, otherwise there is a branch point at the origin.
+    Refer to the incomplete gamma function documentation for details of the
+    branching behavior.
+
+    Examples
+    ========
+
+    >>> from sympy import expint, S
+    >>> from sympy.abc import nu, z
+
+    Differentiation is supported. Differentiation with respect to $z$ further
+    explains the name: for integral orders, the exponential integral is an
+    iterated integral of the exponential function.
+
+    >>> expint(nu, z).diff(z)
+    -expint(nu - 1, z)
+
+    Differentiation with respect to $\nu$ has no classical expression:
+
+    >>> expint(nu, z).diff(nu)
+    -z**(nu - 1)*meijerg(((), (1, 1)), ((0, 0, 1 - nu), ()), z)
+
+    At non-postive integer orders, the exponential integral reduces to the
+    exponential function:
+
+    >>> expint(0, z)
+    exp(-z)/z
+    >>> expint(-1, z)
+    exp(-z)/z + exp(-z)/z**2
+
+    At half-integers it reduces to error functions:
+
+    >>> expint(S(1)/2, z)
+    sqrt(pi)*erfc(sqrt(z))/sqrt(z)
+
+    At positive integer orders it can be rewritten in terms of exponentials
+    and ``expint(1, z)``. Use ``expand_func()`` to do this:
+
+    >>> from sympy import expand_func
+    >>> expand_func(expint(5, z))
+    z**4*expint(1, z)/24 + (-z**3 + z**2 - 2*z + 6)*exp(-z)/24
+
+    The generalised exponential integral is essentially equivalent to the
+    incomplete gamma function:
+
+    >>> from sympy import uppergamma
+    >>> expint(nu, z).rewrite(uppergamma)
+    z**(nu - 1)*uppergamma(1 - nu, z)
+
+    As such it is branched at the origin:
+
+    >>> from sympy import exp_polar, pi, I
+    >>> expint(4, z*exp_polar(2*pi*I))
+    I*pi*z**3/3 + expint(4, z)
+    >>> expint(nu, z*exp_polar(2*pi*I))
+    z**(nu - 1)*(exp(2*I*pi*nu) - 1)*gamma(1 - nu) + expint(nu, z)
+
+    See Also
+    ========
+
+    Ei: Another related function called exponential integral.
+    E1: The classical case, returns expint(1, z).
+    li: Logarithmic integral.
+    Li: Offset logarithmic integral.
+    Si: Sine integral.
+    Ci: Cosine integral.
+    Shi: Hyperbolic sine integral.
+    Chi: Hyperbolic cosine integral.
+    uppergamma
+
+    References
+    ==========
+
+    .. [1] https://dlmf.nist.gov/8.19
+    .. [2] https://functions.wolfram.com/GammaBetaErf/ExpIntegralE/
+    .. [3] https://en.wikipedia.org/wiki/Exponential_integral
+
+    """
+
+
+    @classmethod
+    def eval(cls, nu, z):
+        from sympy.functions.special.gamma_functions import (gamma, uppergamma)
+        nu2 = unpolarify(nu)
+        if nu != nu2:
+            return expint(nu2, z)
+        if nu.is_Integer and nu <= 0 or (not nu.is_Integer and (2*nu).is_Integer):
+            return unpolarify(expand_mul(z**(nu - 1)*uppergamma(1 - nu, z)))
+
+        # Extract branching information. This can be deduced from what is
+        # explained in lowergamma.eval().
+        z, n = z.extract_branch_factor()
+        if n is S.Zero:
+            return
+        if nu.is_integer:
+            if not nu > 0:
+                return
+            return expint(nu, z) \
+                - 2*pi*I*n*S.NegativeOne**(nu - 1)/factorial(nu - 1)*unpolarify(z)**(nu - 1)
+        else:
+            return (exp(2*I*pi*nu*n) - 1)*z**(nu - 1)*gamma(1 - nu) + expint(nu, z)
+
+    def fdiff(self, argindex):
+        nu, z = self.args
+        if argindex == 1:
+            return -z**(nu - 1)*meijerg([], [1, 1], [0, 0, 1 - nu], [], z)
+        elif argindex == 2:
+            return -expint(nu - 1, z)
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    def _eval_rewrite_as_uppergamma(self, nu, z, **kwargs):
+        from sympy.functions.special.gamma_functions import uppergamma
+        return z**(nu - 1)*uppergamma(1 - nu, z)
+
+    def _eval_rewrite_as_Ei(self, nu, z, **kwargs):
+        if nu == 1:
+            return -Ei(z*exp_polar(-I*pi)) - I*pi
+        elif nu.is_Integer and nu > 1:
+            # DLMF, 8.19.7
+            x = -unpolarify(z)
+            return x**(nu - 1)/factorial(nu - 1)*E1(z).rewrite(Ei) + \
+                exp(x)/factorial(nu - 1) * \
+                Add(*[factorial(nu - k - 2)*x**k for k in range(nu - 1)])
+        else:
+            return self
+
+    def _eval_expand_func(self, **hints):
+        return self.rewrite(Ei).rewrite(expint, **hints)
+
+    def _eval_rewrite_as_Si(self, nu, z, **kwargs):
+        if nu != 1:
+            return self
+        return Shi(z) - Chi(z)
+    _eval_rewrite_as_Ci = _eval_rewrite_as_Si
+    _eval_rewrite_as_Chi = _eval_rewrite_as_Si
+    _eval_rewrite_as_Shi = _eval_rewrite_as_Si
+
+    def _eval_nseries(self, x, n, logx, cdir=0):
+        if not self.args[0].has(x):
+            nu = self.args[0]
+            if nu == 1:
+                f = self._eval_rewrite_as_Si(*self.args)
+                return f._eval_nseries(x, n, logx)
+            elif nu.is_Integer and nu > 1:
+                f = self._eval_rewrite_as_Ei(*self.args)
+                return f._eval_nseries(x, n, logx)
+        return super()._eval_nseries(x, n, logx)
+
+    def _eval_aseries(self, n, args0, x, logx):
+        from sympy.series.order import Order
+        point = args0[1]
+        nu = self.args[0]
+
+        if point is S.Infinity:
+            z = self.args[1]
+            s = [S.NegativeOne**k * RisingFactorial(nu, k) / z**k for k in range(n)] + [Order(1/z**n, x)]
+            return (exp(-z)/z) * Add(*s)
+
+        return super(expint, self)._eval_aseries(n, args0, x, logx)
+
+
+def E1(z):
+    """
+    Classical case of the generalized exponential integral.
+
+    Explanation
+    ===========
+
+    This is equivalent to ``expint(1, z)``.
+
+    Examples
+    ========
+
+    >>> from sympy import E1
+    >>> E1(0)
+    expint(1, 0)
+
+    >>> E1(5)
+    expint(1, 5)
+
+    See Also
+    ========
+
+    Ei: Exponential integral.
+    expint: Generalised exponential integral.
+    li: Logarithmic integral.
+    Li: Offset logarithmic integral.
+    Si: Sine integral.
+    Ci: Cosine integral.
+    Shi: Hyperbolic sine integral.
+    Chi: Hyperbolic cosine integral.
+
+    """
+    return expint(1, z)
+
+
+class li(Function):
+    r"""
+    The classical logarithmic integral.
+
+    Explanation
+    ===========
+
+    For use in SymPy, this function is defined as
+
+    .. math:: \operatorname{li}(x) = \int_0^x \frac{1}{\log(t)} \mathrm{d}t \,.
+
+    Examples
+    ========
+
+    >>> from sympy import I, oo, li
+    >>> from sympy.abc import z
+
+    Several special values are known:
+
+    >>> li(0)
+    0
+    >>> li(1)
+    -oo
+    >>> li(oo)
+    oo
+
+    Differentiation with respect to $z$ is supported:
+
+    >>> from sympy import diff
+    >>> diff(li(z), z)
+    1/log(z)
+
+    Defining the ``li`` function via an integral:
+    >>> from sympy import integrate
+    >>> integrate(li(z))
+    z*li(z) - Ei(2*log(z))
+
+    >>> integrate(li(z),z)
+    z*li(z) - Ei(2*log(z))
+
+
+    The logarithmic integral can also be defined in terms of ``Ei``:
+
+    >>> from sympy import Ei
+    >>> li(z).rewrite(Ei)
+    Ei(log(z))
+    >>> diff(li(z).rewrite(Ei), z)
+    1/log(z)
+
+    We can numerically evaluate the logarithmic integral to arbitrary precision
+    on the whole complex plane (except the singular points):
+
+    >>> li(2).evalf(30)
+    1.04516378011749278484458888919
+
+    >>> li(2*I).evalf(30)
+    1.0652795784357498247001125598 + 3.08346052231061726610939702133*I
+
+    We can even compute Soldner's constant by the help of mpmath:
+
+    >>> from mpmath import findroot
+    >>> findroot(li, 2)
+    1.45136923488338
+
+    Further transformations include rewriting ``li`` in terms of
+    the trigonometric integrals ``Si``, ``Ci``, ``Shi`` and ``Chi``:
+
+    >>> from sympy import Si, Ci, Shi, Chi
+    >>> li(z).rewrite(Si)
+    -log(I*log(z)) - log(1/log(z))/2 + log(log(z))/2 + Ci(I*log(z)) + Shi(log(z))
+    >>> li(z).rewrite(Ci)
+    -log(I*log(z)) - log(1/log(z))/2 + log(log(z))/2 + Ci(I*log(z)) + Shi(log(z))
+    >>> li(z).rewrite(Shi)
+    -log(1/log(z))/2 + log(log(z))/2 + Chi(log(z)) - Shi(log(z))
+    >>> li(z).rewrite(Chi)
+    -log(1/log(z))/2 + log(log(z))/2 + Chi(log(z)) - Shi(log(z))
+
+    See Also
+    ========
+
+    Li: Offset logarithmic integral.
+    Ei: Exponential integral.
+    expint: Generalised exponential integral.
+    E1: Special case of the generalised exponential integral.
+    Si: Sine integral.
+    Ci: Cosine integral.
+    Shi: Hyperbolic sine integral.
+    Chi: Hyperbolic cosine integral.
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Logarithmic_integral
+    .. [2] https://mathworld.wolfram.com/LogarithmicIntegral.html
+    .. [3] https://dlmf.nist.gov/6
+    .. [4] https://mathworld.wolfram.com/SoldnersConstant.html
+
+    """
+
+
+    @classmethod
+    def eval(cls, z):
+        if z.is_zero:
+            return S.Zero
+        elif z is S.One:
+            return S.NegativeInfinity
+        elif z is S.Infinity:
+            return S.Infinity
+        if z.is_zero:
+            return S.Zero
+
+    def fdiff(self, argindex=1):
+        arg = self.args[0]
+        if argindex == 1:
+            return S.One / log(arg)
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    def _eval_conjugate(self):
+        z = self.args[0]
+        # Exclude values on the branch cut (-oo, 0)
+        if not z.is_extended_negative:
+            return self.func(z.conjugate())
+
+    def _eval_rewrite_as_Li(self, z, **kwargs):
+        return Li(z) + li(2)
+
+    def _eval_rewrite_as_Ei(self, z, **kwargs):
+        return Ei(log(z))
+
+    def _eval_rewrite_as_uppergamma(self, z, **kwargs):
+        from sympy.functions.special.gamma_functions import uppergamma
+        return (-uppergamma(0, -log(z)) +
+                S.Half*(log(log(z)) - log(S.One/log(z))) - log(-log(z)))
+
+    def _eval_rewrite_as_Si(self, z, **kwargs):
+        return (Ci(I*log(z)) - I*Si(I*log(z)) -
+                S.Half*(log(S.One/log(z)) - log(log(z))) - log(I*log(z)))
+
+    _eval_rewrite_as_Ci = _eval_rewrite_as_Si
+
+    def _eval_rewrite_as_Shi(self, z, **kwargs):
+        return (Chi(log(z)) - Shi(log(z)) - S.Half*(log(S.One/log(z)) - log(log(z))))
+
+    _eval_rewrite_as_Chi = _eval_rewrite_as_Shi
+
+    def _eval_rewrite_as_hyper(self, z, **kwargs):
+        return (log(z)*hyper((1, 1), (2, 2), log(z)) +
+                S.Half*(log(log(z)) - log(S.One/log(z))) + EulerGamma)
+
+    def _eval_rewrite_as_meijerg(self, z, **kwargs):
+        return (-log(-log(z)) - S.Half*(log(S.One/log(z)) - log(log(z)))
+                - meijerg(((), (1,)), ((0, 0), ()), -log(z)))
+
+    def _eval_rewrite_as_tractable(self, z, limitvar=None, **kwargs):
+        return z * _eis(log(z))
+
+    def _eval_nseries(self, x, n, logx, cdir=0):
+        z = self.args[0]
+        s = [(log(z))**k / (factorial(k) * k) for k in range(1, n)]
+        return EulerGamma + log(log(z)) + Add(*s)
+
+    def _eval_is_zero(self):
+        z = self.args[0]
+        if z.is_zero:
+            return True
+
+class Li(Function):
+    r"""
+    The offset logarithmic integral.
+
+    Explanation
+    ===========
+
+    For use in SymPy, this function is defined as
+
+    .. math:: \operatorname{Li}(x) = \operatorname{li}(x) - \operatorname{li}(2)
+
+    Examples
+    ========
+
+    >>> from sympy import Li
+    >>> from sympy.abc import z
+
+    The following special value is known:
+
+    >>> Li(2)
+    0
+
+    Differentiation with respect to $z$ is supported:
+
+    >>> from sympy import diff
+    >>> diff(Li(z), z)
+    1/log(z)
+
+    The shifted logarithmic integral can be written in terms of $li(z)$:
+
+    >>> from sympy import li
+    >>> Li(z).rewrite(li)
+    li(z) - li(2)
+
+    We can numerically evaluate the logarithmic integral to arbitrary precision
+    on the whole complex plane (except the singular points):
+
+    >>> Li(2).evalf(30)
+    0
+
+    >>> Li(4).evalf(30)
+    1.92242131492155809316615998938
+
+    See Also
+    ========
+
+    li: Logarithmic integral.
+    Ei: Exponential integral.
+    expint: Generalised exponential integral.
+    E1: Special case of the generalised exponential integral.
+    Si: Sine integral.
+    Ci: Cosine integral.
+    Shi: Hyperbolic sine integral.
+    Chi: Hyperbolic cosine integral.
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Logarithmic_integral
+    .. [2] https://mathworld.wolfram.com/LogarithmicIntegral.html
+    .. [3] https://dlmf.nist.gov/6
+
+    """
+
+
+    @classmethod
+    def eval(cls, z):
+        if z is S.Infinity:
+            return S.Infinity
+        elif z == S(2):
+            return S.Zero
+
+    def fdiff(self, argindex=1):
+        arg = self.args[0]
+        if argindex == 1:
+            return S.One / log(arg)
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    def _eval_evalf(self, prec):
+        return self.rewrite(li).evalf(prec)
+
+    def _eval_rewrite_as_li(self, z, **kwargs):
+        return li(z) - li(2)
+
+    def _eval_rewrite_as_tractable(self, z, limitvar=None, **kwargs):
+        return self.rewrite(li).rewrite("tractable", deep=True)
+
+    def _eval_nseries(self, x, n, logx, cdir=0):
+        f = self._eval_rewrite_as_li(*self.args)
+        return f._eval_nseries(x, n, logx)
+
+###############################################################################
+#################### TRIGONOMETRIC INTEGRALS ##################################
+###############################################################################
+
+class TrigonometricIntegral(Function):
+    """ Base class for trigonometric integrals. """
+
+
+    @classmethod
+    def eval(cls, z):
+        if z is S.Zero:
+            return cls._atzero
+        elif z is S.Infinity:
+            return cls._atinf()
+        elif z is S.NegativeInfinity:
+            return cls._atneginf()
+
+        if z.is_zero:
+            return cls._atzero
+
+        nz = z.extract_multiplicatively(polar_lift(I))
+        if nz is None and cls._trigfunc(0) == 0:
+            nz = z.extract_multiplicatively(I)
+        if nz is not None:
+            return cls._Ifactor(nz, 1)
+        nz = z.extract_multiplicatively(polar_lift(-I))
+        if nz is not None:
+            return cls._Ifactor(nz, -1)
+
+        nz = z.extract_multiplicatively(polar_lift(-1))
+        if nz is None and cls._trigfunc(0) == 0:
+            nz = z.extract_multiplicatively(-1)
+        if nz is not None:
+            return cls._minusfactor(nz)
+
+        nz, n = z.extract_branch_factor()
+        if n == 0 and nz == z:
+            return
+        return 2*pi*I*n*cls._trigfunc(0) + cls(nz)
+
+    def fdiff(self, argindex=1):
+        arg = unpolarify(self.args[0])
+        if argindex == 1:
+            return self._trigfunc(arg)/arg
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    def _eval_rewrite_as_Ei(self, z, **kwargs):
+        return self._eval_rewrite_as_expint(z).rewrite(Ei)
+
+    def _eval_rewrite_as_uppergamma(self, z, **kwargs):
+        from sympy.functions.special.gamma_functions import uppergamma
+        return self._eval_rewrite_as_expint(z).rewrite(uppergamma)
+
+    def _eval_nseries(self, x, n, logx, cdir=0):
+        # NOTE this is fairly inefficient
+        n += 1
+        if self.args[0].subs(x, 0) != 0:
+            return super()._eval_nseries(x, n, logx)
+        baseseries = self._trigfunc(x)._eval_nseries(x, n, logx)
+        if self._trigfunc(0) != 0:
+            baseseries -= 1
+        baseseries = baseseries.replace(Pow, lambda t, n: t**n/n, simultaneous=False)
+        if self._trigfunc(0) != 0:
+            baseseries += EulerGamma + log(x)
+        return baseseries.subs(x, self.args[0])._eval_nseries(x, n, logx)
+
+
+class Si(TrigonometricIntegral):
+    r"""
+    Sine integral.
+
+    Explanation
+    ===========
+
+    This function is defined by
+
+    .. math:: \operatorname{Si}(z) = \int_0^z \frac{\sin{t}}{t} \mathrm{d}t.
+
+    It is an entire function.
+
+    Examples
+    ========
+
+    >>> from sympy import Si
+    >>> from sympy.abc import z
+
+    The sine integral is an antiderivative of $sin(z)/z$:
+
+    >>> Si(z).diff(z)
+    sin(z)/z
+
+    It is unbranched:
+
+    >>> from sympy import exp_polar, I, pi
+    >>> Si(z*exp_polar(2*I*pi))
+    Si(z)
+
+    Sine integral behaves much like ordinary sine under multiplication by ``I``:
+
+    >>> Si(I*z)
+    I*Shi(z)
+    >>> Si(-z)
+    -Si(z)
+
+    It can also be expressed in terms of exponential integrals, but beware
+    that the latter is branched:
+
+    >>> from sympy import expint
+    >>> Si(z).rewrite(expint)
+    -I*(-expint(1, z*exp_polar(-I*pi/2))/2 +
+         expint(1, z*exp_polar(I*pi/2))/2) + pi/2
+
+    It can be rewritten in the form of sinc function (by definition):
+
+    >>> from sympy import sinc
+    >>> Si(z).rewrite(sinc)
+    Integral(sinc(t), (t, 0, z))
+
+    See Also
+    ========
+
+    Ci: Cosine integral.
+    Shi: Hyperbolic sine integral.
+    Chi: Hyperbolic cosine integral.
+    Ei: Exponential integral.
+    expint: Generalised exponential integral.
+    sinc: unnormalized sinc function
+    E1: Special case of the generalised exponential integral.
+    li: Logarithmic integral.
+    Li: Offset logarithmic integral.
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Trigonometric_integral
+
+    """
+
+    _trigfunc = sin
+    _atzero = S.Zero
+
+    @classmethod
+    def _atinf(cls):
+        return pi*S.Half
+
+    @classmethod
+    def _atneginf(cls):
+        return -pi*S.Half
+
+    @classmethod
+    def _minusfactor(cls, z):
+        return -Si(z)
+
+    @classmethod
+    def _Ifactor(cls, z, sign):
+        return I*Shi(z)*sign
+
+    def _eval_rewrite_as_expint(self, z, **kwargs):
+        # XXX should we polarify z?
+        return pi/2 + (E1(polar_lift(I)*z) - E1(polar_lift(-I)*z))/2/I
+
+    def _eval_rewrite_as_sinc(self, z, **kwargs):
+        from sympy.integrals.integrals import Integral
+        t = Symbol('t', Dummy=True)
+        return Integral(sinc(t), (t, 0, z))
+
+    def _eval_aseries(self, n, args0, x, logx):
+        from sympy.series.order import Order
+        point = args0[0]
+
+        # Expansion at oo
+        if point is S.Infinity:
+            z = self.args[0]
+            p = [S.NegativeOne**k * factorial(2*k) / z**(2*k)
+                    for k in range(int((n - 1)/2))] + [Order(1/z**n, x)]
+            q = [S.NegativeOne**k * factorial(2*k + 1) / z**(2*k + 1)
+                    for k in range(int(n/2) - 1)] + [Order(1/z**n, x)]
+            return pi/2 - (cos(z)/z)*Add(*p) - (sin(z)/z)*Add(*q)
+
+        # All other points are not handled
+        return super(Si, self)._eval_aseries(n, args0, x, logx)
+
+    def _eval_is_zero(self):
+        z = self.args[0]
+        if z.is_zero:
+            return True
+
+
+class Ci(TrigonometricIntegral):
+    r"""
+    Cosine integral.
+
+    Explanation
+    ===========
+
+    This function is defined for positive $x$ by
+
+    .. math:: \operatorname{Ci}(x) = \gamma + \log{x}
+                         + \int_0^x \frac{\cos{t} - 1}{t} \mathrm{d}t
+           = -\int_x^\infty \frac{\cos{t}}{t} \mathrm{d}t,
+
+    where $\gamma$ is the Euler-Mascheroni constant.
+
+    We have
+
+    .. math:: \operatorname{Ci}(z) =
+        -\frac{\operatorname{E}_1\left(e^{i\pi/2} z\right)
+               + \operatorname{E}_1\left(e^{-i \pi/2} z\right)}{2}
+
+    which holds for all polar $z$ and thus provides an analytic
+    continuation to the Riemann surface of the logarithm.
+
+    The formula also holds as stated
+    for $z \in \mathbb{C}$ with $\Re(z) > 0$.
+    By lifting to the principal branch, we obtain an analytic function on the
+    cut complex plane.
+
+    Examples
+    ========
+
+    >>> from sympy import Ci
+    >>> from sympy.abc import z
+
+    The cosine integral is a primitive of $\cos(z)/z$:
+
+    >>> Ci(z).diff(z)
+    cos(z)/z
+
+    It has a logarithmic branch point at the origin:
+
+    >>> from sympy import exp_polar, I, pi
+    >>> Ci(z*exp_polar(2*I*pi))
+    Ci(z) + 2*I*pi
+
+    The cosine integral behaves somewhat like ordinary $\cos$ under
+    multiplication by $i$:
+
+    >>> from sympy import polar_lift
+    >>> Ci(polar_lift(I)*z)
+    Chi(z) + I*pi/2
+    >>> Ci(polar_lift(-1)*z)
+    Ci(z) + I*pi
+
+    It can also be expressed in terms of exponential integrals:
+
+    >>> from sympy import expint
+    >>> Ci(z).rewrite(expint)
+    -expint(1, z*exp_polar(-I*pi/2))/2 - expint(1, z*exp_polar(I*pi/2))/2
+
+    See Also
+    ========
+
+    Si: Sine integral.
+    Shi: Hyperbolic sine integral.
+    Chi: Hyperbolic cosine integral.
+    Ei: Exponential integral.
+    expint: Generalised exponential integral.
+    E1: Special case of the generalised exponential integral.
+    li: Logarithmic integral.
+    Li: Offset logarithmic integral.
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Trigonometric_integral
+
+    """
+
+    _trigfunc = cos
+    _atzero = S.ComplexInfinity
+
+    @classmethod
+    def _atinf(cls):
+        return S.Zero
+
+    @classmethod
+    def _atneginf(cls):
+        return I*pi
+
+    @classmethod
+    def _minusfactor(cls, z):
+        return Ci(z) + I*pi
+
+    @classmethod
+    def _Ifactor(cls, z, sign):
+        return Chi(z) + I*pi/2*sign
+
+    def _eval_rewrite_as_expint(self, z, **kwargs):
+        return -(E1(polar_lift(I)*z) + E1(polar_lift(-I)*z))/2
+
+    def _eval_as_leading_term(self, x, logx=None, cdir=0):
+        arg = self.args[0].as_leading_term(x, logx=logx, cdir=cdir)
+        arg0 = arg.subs(x, 0)
+
+        if arg0 is S.NaN:
+            arg0 = arg.limit(x, 0, dir='-' if re(cdir).is_negative else '+')
+        if arg0.is_zero:
+            c, e = arg.as_coeff_exponent(x)
+            logx = log(x) if logx is None else logx
+            return log(c) + e*logx + EulerGamma
+        elif arg0.is_finite:
+            return self.func(arg0)
+        else:
+            return self
+
+    def _eval_aseries(self, n, args0, x, logx):
+        from sympy.series.order import Order
+        point = args0[0]
+
+        # Expansion at oo
+        if point is S.Infinity:
+            z = self.args[0]
+            p = [S.NegativeOne**k * factorial(2*k) / z**(2*k)
+                    for k in range(int((n - 1)/2))] + [Order(1/z**n, x)]
+            q = [S.NegativeOne**k * factorial(2*k + 1) / z**(2*k + 1)
+                    for k in range(int(n/2) - 1)] + [Order(1/z**n, x)]
+            return (sin(z)/z)*Add(*p) - (cos(z)/z)*Add(*q)
+
+        # All other points are not handled
+        return super(Ci, self)._eval_aseries(n, args0, x, logx)
+
+
+class Shi(TrigonometricIntegral):
+    r"""
+    Sinh integral.
+
+    Explanation
+    ===========
+
+    This function is defined by
+
+    .. math:: \operatorname{Shi}(z) = \int_0^z \frac{\sinh{t}}{t} \mathrm{d}t.
+
+    It is an entire function.
+
+    Examples
+    ========
+
+    >>> from sympy import Shi
+    >>> from sympy.abc import z
+
+    The Sinh integral is a primitive of $\sinh(z)/z$:
+
+    >>> Shi(z).diff(z)
+    sinh(z)/z
+
+    It is unbranched:
+
+    >>> from sympy import exp_polar, I, pi
+    >>> Shi(z*exp_polar(2*I*pi))
+    Shi(z)
+
+    The $\sinh$ integral behaves much like ordinary $\sinh$ under
+    multiplication by $i$:
+
+    >>> Shi(I*z)
+    I*Si(z)
+    >>> Shi(-z)
+    -Shi(z)
+
+    It can also be expressed in terms of exponential integrals, but beware
+    that the latter is branched:
+
+    >>> from sympy import expint
+    >>> Shi(z).rewrite(expint)
+    expint(1, z)/2 - expint(1, z*exp_polar(I*pi))/2 - I*pi/2
+
+    See Also
+    ========
+
+    Si: Sine integral.
+    Ci: Cosine integral.
+    Chi: Hyperbolic cosine integral.
+    Ei: Exponential integral.
+    expint: Generalised exponential integral.
+    E1: Special case of the generalised exponential integral.
+    li: Logarithmic integral.
+    Li: Offset logarithmic integral.
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Trigonometric_integral
+
+    """
+
+    _trigfunc = sinh
+    _atzero = S.Zero
+
+    @classmethod
+    def _atinf(cls):
+        return S.Infinity
+
+    @classmethod
+    def _atneginf(cls):
+        return S.NegativeInfinity
+
+    @classmethod
+    def _minusfactor(cls, z):
+        return -Shi(z)
+
+    @classmethod
+    def _Ifactor(cls, z, sign):
+        return I*Si(z)*sign
+
+    def _eval_rewrite_as_expint(self, z, **kwargs):
+        # XXX should we polarify z?
+        return (E1(z) - E1(exp_polar(I*pi)*z))/2 - I*pi/2
+
+    def _eval_is_zero(self):
+        z = self.args[0]
+        if z.is_zero:
+            return True
+
+    def _eval_as_leading_term(self, x, logx=None, cdir=0):
+        arg = self.args[0].as_leading_term(x)
+        arg0 = arg.subs(x, 0)
+
+        if arg0 is S.NaN:
+            arg0 = arg.limit(x, 0, dir='-' if re(cdir).is_negative else '+')
+        if arg0.is_zero:
+            return arg
+        elif not arg0.is_infinite:
+            return self.func(arg0)
+        else:
+            return self
+
+
+class Chi(TrigonometricIntegral):
+    r"""
+    Cosh integral.
+
+    Explanation
+    ===========
+
+    This function is defined for positive $x$ by
+
+    .. math:: \operatorname{Chi}(x) = \gamma + \log{x}
+                         + \int_0^x \frac{\cosh{t} - 1}{t} \mathrm{d}t,
+
+    where $\gamma$ is the Euler-Mascheroni constant.
+
+    We have
+
+    .. math:: \operatorname{Chi}(z) = \operatorname{Ci}\left(e^{i \pi/2}z\right)
+                         - i\frac{\pi}{2},
+
+    which holds for all polar $z$ and thus provides an analytic
+    continuation to the Riemann surface of the logarithm.
+    By lifting to the principal branch we obtain an analytic function on the
+    cut complex plane.
+
+    Examples
+    ========
+
+    >>> from sympy import Chi
+    >>> from sympy.abc import z
+
+    The $\cosh$ integral is a primitive of $\cosh(z)/z$:
+
+    >>> Chi(z).diff(z)
+    cosh(z)/z
+
+    It has a logarithmic branch point at the origin:
+
+    >>> from sympy import exp_polar, I, pi
+    >>> Chi(z*exp_polar(2*I*pi))
+    Chi(z) + 2*I*pi
+
+    The $\cosh$ integral behaves somewhat like ordinary $\cosh$ under
+    multiplication by $i$:
+
+    >>> from sympy import polar_lift
+    >>> Chi(polar_lift(I)*z)
+    Ci(z) + I*pi/2
+    >>> Chi(polar_lift(-1)*z)
+    Chi(z) + I*pi
+
+    It can also be expressed in terms of exponential integrals:
+
+    >>> from sympy import expint
+    >>> Chi(z).rewrite(expint)
+    -expint(1, z)/2 - expint(1, z*exp_polar(I*pi))/2 - I*pi/2
+
+    See Also
+    ========
+
+    Si: Sine integral.
+    Ci: Cosine integral.
+    Shi: Hyperbolic sine integral.
+    Ei: Exponential integral.
+    expint: Generalised exponential integral.
+    E1: Special case of the generalised exponential integral.
+    li: Logarithmic integral.
+    Li: Offset logarithmic integral.
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Trigonometric_integral
+
+    """
+
+    _trigfunc = cosh
+    _atzero = S.ComplexInfinity
+
+    @classmethod
+    def _atinf(cls):
+        return S.Infinity
+
+    @classmethod
+    def _atneginf(cls):
+        return S.Infinity
+
+    @classmethod
+    def _minusfactor(cls, z):
+        return Chi(z) + I*pi
+
+    @classmethod
+    def _Ifactor(cls, z, sign):
+        return Ci(z) + I*pi/2*sign
+
+    def _eval_rewrite_as_expint(self, z, **kwargs):
+        return -I*pi/2 - (E1(z) + E1(exp_polar(I*pi)*z))/2
+
+    def _eval_as_leading_term(self, x, logx=None, cdir=0):
+        arg = self.args[0].as_leading_term(x, logx=logx, cdir=cdir)
+        arg0 = arg.subs(x, 0)
+
+        if arg0 is S.NaN:
+            arg0 = arg.limit(x, 0, dir='-' if re(cdir).is_negative else '+')
+        if arg0.is_zero:
+            c, e = arg.as_coeff_exponent(x)
+            logx = log(x) if logx is None else logx
+            return log(c) + e*logx + EulerGamma
+        elif arg0.is_finite:
+            return self.func(arg0)
+        else:
+            return self
+
+
+###############################################################################
+#################### FRESNEL INTEGRALS ########################################
+###############################################################################
+
+class FresnelIntegral(Function):
+    """ Base class for the Fresnel integrals."""
+
+    unbranched = True
+
+    @classmethod
+    def eval(cls, z):
+        # Values at positive infinities signs
+        # if any were extracted automatically
+        if z is S.Infinity:
+            return S.Half
+
+        # Value at zero
+        if z.is_zero:
+            return S.Zero
+
+        # Try to pull out factors of -1 and I
+        prefact = S.One
+        newarg = z
+        changed = False
+
+        nz = newarg.extract_multiplicatively(-1)
+        if nz is not None:
+            prefact = -prefact
+            newarg = nz
+            changed = True
+
+        nz = newarg.extract_multiplicatively(I)
+        if nz is not None:
+            prefact = cls._sign*I*prefact
+            newarg = nz
+            changed = True
+
+        if changed:
+            return prefact*cls(newarg)
+
+    def fdiff(self, argindex=1):
+        if argindex == 1:
+            return self._trigfunc(S.Half*pi*self.args[0]**2)
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    def _eval_is_extended_real(self):
+        return self.args[0].is_extended_real
+
+    _eval_is_finite = _eval_is_extended_real
+
+    def _eval_is_zero(self):
+        return self.args[0].is_zero
+
+    def _eval_conjugate(self):
+        return self.func(self.args[0].conjugate())
+
+    as_real_imag = real_to_real_as_real_imag
+
+
+class fresnels(FresnelIntegral):
+    r"""
+    Fresnel integral S.
+
+    Explanation
+    ===========
+
+    This function is defined by
+
+    .. math:: \operatorname{S}(z) = \int_0^z \sin{\frac{\pi}{2} t^2} \mathrm{d}t.
+
+    It is an entire function.
+
+    Examples
+    ========
+
+    >>> from sympy import I, oo, fresnels
+    >>> from sympy.abc import z
+
+    Several special values are known:
+
+    >>> fresnels(0)
+    0
+    >>> fresnels(oo)
+    1/2
+    >>> fresnels(-oo)
+    -1/2
+    >>> fresnels(I*oo)
+    -I/2
+    >>> fresnels(-I*oo)
+    I/2
+
+    In general one can pull out factors of -1 and $i$ from the argument:
+
+    >>> fresnels(-z)
+    -fresnels(z)
+    >>> fresnels(I*z)
+    -I*fresnels(z)
+
+    The Fresnel S integral obeys the mirror symmetry
+    $\overline{S(z)} = S(\bar{z})$:
+
+    >>> from sympy import conjugate
+    >>> conjugate(fresnels(z))
+    fresnels(conjugate(z))
+
+    Differentiation with respect to $z$ is supported:
+
+    >>> from sympy import diff
+    >>> diff(fresnels(z), z)
+    sin(pi*z**2/2)
+
+    Defining the Fresnel functions via an integral:
+
+    >>> from sympy import integrate, pi, sin, expand_func
+    >>> integrate(sin(pi*z**2/2), z)
+    3*fresnels(z)*gamma(3/4)/(4*gamma(7/4))
+    >>> expand_func(integrate(sin(pi*z**2/2), z))
+    fresnels(z)
+
+    We can numerically evaluate the Fresnel integral to arbitrary precision
+    on the whole complex plane:
+
+    >>> fresnels(2).evalf(30)
+    0.343415678363698242195300815958
+
+    >>> fresnels(-2*I).evalf(30)
+    0.343415678363698242195300815958*I
+
+    See Also
+    ========
+
+    fresnelc: Fresnel cosine integral.
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Fresnel_integral
+    .. [2] https://dlmf.nist.gov/7
+    .. [3] https://mathworld.wolfram.com/FresnelIntegrals.html
+    .. [4] https://functions.wolfram.com/GammaBetaErf/FresnelS
+    .. [5] The converging factors for the fresnel integrals
+            by John W. Wrench Jr. and Vicki Alley
+
+    """
+    _trigfunc = sin
+    _sign = -S.One
+
+    @staticmethod
+    @cacheit
+    def taylor_term(n, x, *previous_terms):
+        if n < 0:
+            return S.Zero
+        else:
+            x = sympify(x)
+            if len(previous_terms) > 1:
+                p = previous_terms[-1]
+                return (-pi**2*x**4*(4*n - 1)/(8*n*(2*n + 1)*(4*n + 3))) * p
+            else:
+                return x**3 * (-x**4)**n * (S(2)**(-2*n - 1)*pi**(2*n + 1)) / ((4*n + 3)*factorial(2*n + 1))
+
+    def _eval_rewrite_as_erf(self, z, **kwargs):
+        return (S.One + I)/4 * (erf((S.One + I)/2*sqrt(pi)*z) - I*erf((S.One - I)/2*sqrt(pi)*z))
+
+    def _eval_rewrite_as_hyper(self, z, **kwargs):
+        return pi*z**3/6 * hyper([Rational(3, 4)], [Rational(3, 2), Rational(7, 4)], -pi**2*z**4/16)
+
+    def _eval_rewrite_as_meijerg(self, z, **kwargs):
+        return (pi*z**Rational(9, 4) / (sqrt(2)*(z**2)**Rational(3, 4)*(-z)**Rational(3, 4))
+                * meijerg([], [1], [Rational(3, 4)], [Rational(1, 4), 0], -pi**2*z**4/16))
+
+    def _eval_as_leading_term(self, x, logx=None, cdir=0):
+        from sympy.series.order import Order
+        arg = self.args[0].as_leading_term(x, logx=logx, cdir=cdir)
+        arg0 = arg.subs(x, 0)
+
+        if arg0 is S.ComplexInfinity:
+            arg0 = arg.limit(x, 0, dir='-' if re(cdir).is_negative else '+')
+        if arg0.is_zero:
+            return pi*arg**3/6
+        elif arg0 in [S.Infinity, S.NegativeInfinity]:
+            s = 1 if arg0 is S.Infinity else -1
+            return s*S.Half + Order(x, x)
+        else:
+            return self.func(arg0)
+
+    def _eval_aseries(self, n, args0, x, logx):
+        from sympy.series.order import Order
+        point = args0[0]
+
+        # Expansion at oo and -oo
+        if point in [S.Infinity, -S.Infinity]:
+            z = self.args[0]
+
+            # expansion of S(x) = S1(x*sqrt(pi/2)), see reference[5] page 1-8
+            # as only real infinities are dealt with, sin and cos are O(1)
+            p = [S.NegativeOne**k * factorial(4*k + 1) /
+                 (2**(2*k + 2) * z**(4*k + 3) * 2**(2*k)*factorial(2*k))
+                 for k in range(0, n) if 4*k + 3 < n]
+            q = [1/(2*z)] + [S.NegativeOne**k * factorial(4*k - 1) /
+                 (2**(2*k + 1) * z**(4*k + 1) * 2**(2*k - 1)*factorial(2*k - 1))
+                 for k in range(1, n) if 4*k + 1 < n]
+
+            p = [-sqrt(2/pi)*t for t in p]
+            q = [-sqrt(2/pi)*t for t in q]
+            s = 1 if point is S.Infinity else -1
+            # The expansion at oo is 1/2 + some odd powers of z
+            # To get the expansion at -oo, replace z by -z and flip the sign
+            # The result -1/2 + the same odd powers of z as before.
+            return s*S.Half + (sin(z**2)*Add(*p) + cos(z**2)*Add(*q)
+                ).subs(x, sqrt(2/pi)*x) + Order(1/z**n, x)
+
+        # All other points are not handled
+        return super()._eval_aseries(n, args0, x, logx)
+
+
+class fresnelc(FresnelIntegral):
+    r"""
+    Fresnel integral C.
+
+    Explanation
+    ===========
+
+    This function is defined by
+
+    .. math:: \operatorname{C}(z) = \int_0^z \cos{\frac{\pi}{2} t^2} \mathrm{d}t.
+
+    It is an entire function.
+
+    Examples
+    ========
+
+    >>> from sympy import I, oo, fresnelc
+    >>> from sympy.abc import z
+
+    Several special values are known:
+
+    >>> fresnelc(0)
+    0
+    >>> fresnelc(oo)
+    1/2
+    >>> fresnelc(-oo)
+    -1/2
+    >>> fresnelc(I*oo)
+    I/2
+    >>> fresnelc(-I*oo)
+    -I/2
+
+    In general one can pull out factors of -1 and $i$ from the argument:
+
+    >>> fresnelc(-z)
+    -fresnelc(z)
+    >>> fresnelc(I*z)
+    I*fresnelc(z)
+
+    The Fresnel C integral obeys the mirror symmetry
+    $\overline{C(z)} = C(\bar{z})$:
+
+    >>> from sympy import conjugate
+    >>> conjugate(fresnelc(z))
+    fresnelc(conjugate(z))
+
+    Differentiation with respect to $z$ is supported:
+
+    >>> from sympy import diff
+    >>> diff(fresnelc(z), z)
+    cos(pi*z**2/2)
+
+    Defining the Fresnel functions via an integral:
+
+    >>> from sympy import integrate, pi, cos, expand_func
+    >>> integrate(cos(pi*z**2/2), z)
+    fresnelc(z)*gamma(1/4)/(4*gamma(5/4))
+    >>> expand_func(integrate(cos(pi*z**2/2), z))
+    fresnelc(z)
+
+    We can numerically evaluate the Fresnel integral to arbitrary precision
+    on the whole complex plane:
+
+    >>> fresnelc(2).evalf(30)
+    0.488253406075340754500223503357
+
+    >>> fresnelc(-2*I).evalf(30)
+    -0.488253406075340754500223503357*I
+
+    See Also
+    ========
+
+    fresnels: Fresnel sine integral.
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Fresnel_integral
+    .. [2] https://dlmf.nist.gov/7
+    .. [3] https://mathworld.wolfram.com/FresnelIntegrals.html
+    .. [4] https://functions.wolfram.com/GammaBetaErf/FresnelC
+    .. [5] The converging factors for the fresnel integrals
+            by John W. Wrench Jr. and Vicki Alley
+
+    """
+    _trigfunc = cos
+    _sign = S.One
+
+    @staticmethod
+    @cacheit
+    def taylor_term(n, x, *previous_terms):
+        if n < 0:
+            return S.Zero
+        else:
+            x = sympify(x)
+            if len(previous_terms) > 1:
+                p = previous_terms[-1]
+                return (-pi**2*x**4*(4*n - 3)/(8*n*(2*n - 1)*(4*n + 1))) * p
+            else:
+                return x * (-x**4)**n * (S(2)**(-2*n)*pi**(2*n)) / ((4*n + 1)*factorial(2*n))
+
+    def _eval_rewrite_as_erf(self, z, **kwargs):
+        return (S.One - I)/4 * (erf((S.One + I)/2*sqrt(pi)*z) + I*erf((S.One - I)/2*sqrt(pi)*z))
+
+    def _eval_rewrite_as_hyper(self, z, **kwargs):
+        return z * hyper([Rational(1, 4)], [S.Half, Rational(5, 4)], -pi**2*z**4/16)
+
+    def _eval_rewrite_as_meijerg(self, z, **kwargs):
+        return (pi*z**Rational(3, 4) / (sqrt(2)*root(z**2, 4)*root(-z, 4))
+                * meijerg([], [1], [Rational(1, 4)], [Rational(3, 4), 0], -pi**2*z**4/16))
+
+    def _eval_as_leading_term(self, x, logx=None, cdir=0):
+        from sympy.series.order import Order
+        arg = self.args[0].as_leading_term(x, logx=logx, cdir=cdir)
+        arg0 = arg.subs(x, 0)
+
+        if arg0 is S.ComplexInfinity:
+            arg0 = arg.limit(x, 0, dir='-' if re(cdir).is_negative else '+')
+        if arg0.is_zero:
+            return arg
+        elif arg0 in [S.Infinity, S.NegativeInfinity]:
+            s = 1 if arg0 is S.Infinity else -1
+            return s*S.Half + Order(x, x)
+        else:
+            return self.func(arg0)
+
+    def _eval_aseries(self, n, args0, x, logx):
+        from sympy.series.order import Order
+        point = args0[0]
+
+        # Expansion at oo
+        if point in [S.Infinity, -S.Infinity]:
+            z = self.args[0]
+
+            # expansion of C(x) = C1(x*sqrt(pi/2)), see reference[5] page 1-8
+            # as only real infinities are dealt with, sin and cos are O(1)
+            p = [S.NegativeOne**k * factorial(4*k + 1) /
+                 (2**(2*k + 2) * z**(4*k + 3) * 2**(2*k)*factorial(2*k))
+                 for k in range(n) if 4*k + 3 < n]
+            q = [1/(2*z)] + [S.NegativeOne**k * factorial(4*k - 1) /
+                 (2**(2*k + 1) * z**(4*k + 1) * 2**(2*k - 1)*factorial(2*k - 1))
+                 for k in range(1, n) if 4*k + 1 < n]
+
+            p = [-sqrt(2/pi)*t for t in p]
+            q = [ sqrt(2/pi)*t for t in q]
+            s = 1 if point is S.Infinity else -1
+            # The expansion at oo is 1/2 + some odd powers of z
+            # To get the expansion at -oo, replace z by -z and flip the sign
+            # The result -1/2 + the same odd powers of z as before.
+            return s*S.Half + (cos(z**2)*Add(*p) + sin(z**2)*Add(*q)
+                ).subs(x, sqrt(2/pi)*x) + Order(1/z**n, x)
+
+        # All other points are not handled
+        return super()._eval_aseries(n, args0, x, logx)
+
+
+###############################################################################
+#################### HELPER FUNCTIONS #########################################
+###############################################################################
+
+
+class _erfs(Function):
+    """
+    Helper function to make the $\\mathrm{erf}(z)$ function
+    tractable for the Gruntz algorithm.
+
+    """
+    @classmethod
+    def eval(cls, arg):
+        if arg.is_zero:
+            return S.One
+
+    def _eval_aseries(self, n, args0, x, logx):
+        from sympy.series.order import Order
+        point = args0[0]
+
+        # Expansion at oo
+        if point is S.Infinity:
+            z = self.args[0]
+            l = [1/sqrt(pi) * factorial(2*k)*(-S(
+                 4))**(-k)/factorial(k) * (1/z)**(2*k + 1) for k in range(n)]
+            o = Order(1/z**(2*n + 1), x)
+            # It is very inefficient to first add the order and then do the nseries
+            return (Add(*l))._eval_nseries(x, n, logx) + o
+
+        # Expansion at I*oo
+        t = point.extract_multiplicatively(I)
+        if t is S.Infinity:
+            z = self.args[0]
+            # TODO: is the series really correct?
+            l = [1/sqrt(pi) * factorial(2*k)*(-S(
+                 4))**(-k)/factorial(k) * (1/z)**(2*k + 1) for k in range(n)]
+            o = Order(1/z**(2*n + 1), x)
+            # It is very inefficient to first add the order and then do the nseries
+            return (Add(*l))._eval_nseries(x, n, logx) + o
+
+        # All other points are not handled
+        return super()._eval_aseries(n, args0, x, logx)
+
+    def fdiff(self, argindex=1):
+        if argindex == 1:
+            z = self.args[0]
+            return -2/sqrt(pi) + 2*z*_erfs(z)
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    def _eval_rewrite_as_intractable(self, z, **kwargs):
+        return (S.One - erf(z))*exp(z**2)
+
+
+class _eis(Function):
+    """
+    Helper function to make the $\\mathrm{Ei}(z)$ and $\\mathrm{li}(z)$
+    functions tractable for the Gruntz algorithm.
+
+    """
+
+
+    def _eval_aseries(self, n, args0, x, logx):
+        from sympy.series.order import Order
+        if args0[0] != S.Infinity:
+            return super(_erfs, self)._eval_aseries(n, args0, x, logx)
+
+        z = self.args[0]
+        l = [factorial(k) * (1/z)**(k + 1) for k in range(n)]
+        o = Order(1/z**(n + 1), x)
+        # It is very inefficient to first add the order and then do the nseries
+        return (Add(*l))._eval_nseries(x, n, logx) + o
+
+
+    def fdiff(self, argindex=1):
+        if argindex == 1:
+            z = self.args[0]
+            return S.One / z - _eis(z)
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    def _eval_rewrite_as_intractable(self, z, **kwargs):
+        return exp(-z)*Ei(z)
+
+    def _eval_as_leading_term(self, x, logx=None, cdir=0):
+        x0 = self.args[0].limit(x, 0)
+        if x0.is_zero:
+            f = self._eval_rewrite_as_intractable(*self.args)
+            return f._eval_as_leading_term(x, logx=logx, cdir=cdir)
+        return super()._eval_as_leading_term(x, logx=logx, cdir=cdir)
+
+    def _eval_nseries(self, x, n, logx, cdir=0):
+        x0 = self.args[0].limit(x, 0)
+        if x0.is_zero:
+            f = self._eval_rewrite_as_intractable(*self.args)
+            return f._eval_nseries(x, n, logx)
+        return super()._eval_nseries(x, n, logx)

llmeval-env/lib/python3.10/site-packages/sympy/functions/special/gamma_functions.py

@@ -0,0 +1,1344 @@
+from math import prod
+
+from sympy.core import Add, S, Dummy, expand_func
+from sympy.core.expr import Expr
+from sympy.core.function import Function, ArgumentIndexError, PoleError
+from sympy.core.logic import fuzzy_and, fuzzy_not
+from sympy.core.numbers import Rational, pi, oo, I
+from sympy.core.power import Pow
+from sympy.functions.special.zeta_functions import zeta
+from sympy.functions.special.error_functions import erf, erfc, Ei
+from sympy.functions.elementary.complexes import re, unpolarify
+from sympy.functions.elementary.exponential import exp, log
+from sympy.functions.elementary.integers import ceiling, floor
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.trigonometric import sin, cos, cot
+from sympy.functions.combinatorial.numbers import bernoulli, harmonic
+from sympy.functions.combinatorial.factorials import factorial, rf, RisingFactorial
+from sympy.utilities.misc import as_int
+
+from mpmath import mp, workprec
+from mpmath.libmp.libmpf import prec_to_dps
+
+def intlike(n):
+    try:
+        as_int(n, strict=False)
+        return True
+    except ValueError:
+        return False
+
+###############################################################################
+############################ COMPLETE GAMMA FUNCTION ##########################
+###############################################################################
+
+class gamma(Function):
+    r"""
+    The gamma function
+
+    .. math::
+        \Gamma(x) := \int^{\infty}_{0} t^{x-1} e^{-t} \mathrm{d}t.
+
+    Explanation
+    ===========
+
+    The ``gamma`` function implements the function which passes through the
+    values of the factorial function (i.e., $\Gamma(n) = (n - 1)!$ when n is
+    an integer). More generally, $\Gamma(z)$ is defined in the whole complex
+    plane except at the negative integers where there are simple poles.
+
+    Examples
+    ========
+
+    >>> from sympy import S, I, pi, gamma
+    >>> from sympy.abc import x
+
+    Several special values are known:
+
+    >>> gamma(1)
+    1
+    >>> gamma(4)
+    6
+    >>> gamma(S(3)/2)
+    sqrt(pi)/2
+
+    The ``gamma`` function obeys the mirror symmetry:
+
+    >>> from sympy import conjugate
+    >>> conjugate(gamma(x))
+    gamma(conjugate(x))
+
+    Differentiation with respect to $x$ is supported:
+
+    >>> from sympy import diff
+    >>> diff(gamma(x), x)
+    gamma(x)*polygamma(0, x)
+
+    Series expansion is also supported:
+
+    >>> from sympy import series
+    >>> series(gamma(x), x, 0, 3)
+    1/x - EulerGamma + x*(EulerGamma**2/2 + pi**2/12) + x**2*(-EulerGamma*pi**2/12 - zeta(3)/3 - EulerGamma**3/6) + O(x**3)
+
+    We can numerically evaluate the ``gamma`` function to arbitrary precision
+    on the whole complex plane:
+
+    >>> gamma(pi).evalf(40)
+    2.288037795340032417959588909060233922890
+    >>> gamma(1+I).evalf(20)
+    0.49801566811835604271 - 0.15494982830181068512*I
+
+    See Also
+    ========
+
+    lowergamma: Lower incomplete gamma function.
+    uppergamma: Upper incomplete gamma function.
+    polygamma: Polygamma function.
+    loggamma: Log Gamma function.
+    digamma: Digamma function.
+    trigamma: Trigamma function.
+    beta: Euler Beta function.
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Gamma_function
+    .. [2] https://dlmf.nist.gov/5
+    .. [3] https://mathworld.wolfram.com/GammaFunction.html
+    .. [4] https://functions.wolfram.com/GammaBetaErf/Gamma/
+
+    """
+
+    unbranched = True
+    _singularities = (S.ComplexInfinity,)
+
+    def fdiff(self, argindex=1):
+        if argindex == 1:
+            return self.func(self.args[0])*polygamma(0, self.args[0])
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    @classmethod
+    def eval(cls, arg):
+        if arg.is_Number:
+            if arg is S.NaN:
+                return S.NaN
+            elif arg is oo:
+                return oo
+            elif intlike(arg):
+                if arg.is_positive:
+                    return factorial(arg - 1)
+                else:
+                    return S.ComplexInfinity
+            elif arg.is_Rational:
+                if arg.q == 2:
+                    n = abs(arg.p) // arg.q
+
+                    if arg.is_positive:
+                        k, coeff = n, S.One
+                    else:
+                        n = k = n + 1
+
+                        if n & 1 == 0:
+                            coeff = S.One
+                        else:
+                            coeff = S.NegativeOne
+
+                    coeff *= prod(range(3, 2*k, 2))
+
+                    if arg.is_positive:
+                        return coeff*sqrt(pi) / 2**n
+                    else:
+                        return 2**n*sqrt(pi) / coeff
+
+    def _eval_expand_func(self, **hints):
+        arg = self.args[0]
+        if arg.is_Rational:
+            if abs(arg.p) > arg.q:
+                x = Dummy('x')
+                n = arg.p // arg.q
+                p = arg.p - n*arg.q
+                return self.func(x + n)._eval_expand_func().subs(x, Rational(p, arg.q))
+
+        if arg.is_Add:
+            coeff, tail = arg.as_coeff_add()
+            if coeff and coeff.q != 1:
+                intpart = floor(coeff)
+                tail = (coeff - intpart,) + tail
+                coeff = intpart
+            tail = arg._new_rawargs(*tail, reeval=False)
+            return self.func(tail)*RisingFactorial(tail, coeff)
+
+        return self.func(*self.args)
+
+    def _eval_conjugate(self):
+        return self.func(self.args[0].conjugate())
+
+    def _eval_is_real(self):
+        x = self.args[0]
+        if x.is_nonpositive and x.is_integer:
+            return False
+        if intlike(x) and x <= 0:
+            return False
+        if x.is_positive or x.is_noninteger:
+            return True
+
+    def _eval_is_positive(self):
+        x = self.args[0]
+        if x.is_positive:
+            return True
+        elif x.is_noninteger:
+            return floor(x).is_even
+
+    def _eval_rewrite_as_tractable(self, z, limitvar=None, **kwargs):
+        return exp(loggamma(z))
+
+    def _eval_rewrite_as_factorial(self, z, **kwargs):
+        return factorial(z - 1)
+
+    def _eval_nseries(self, x, n, logx, cdir=0):
+        x0 = self.args[0].limit(x, 0)
+        if not (x0.is_Integer and x0 <= 0):
+            return super()._eval_nseries(x, n, logx)
+        t = self.args[0] - x0
+        return (self.func(t + 1)/rf(self.args[0], -x0 + 1))._eval_nseries(x, n, logx)
+
+    def _eval_as_leading_term(self, x, logx=None, cdir=0):
+        arg = self.args[0]
+        x0 = arg.subs(x, 0)
+
+        if x0.is_integer and x0.is_nonpositive:
+            n = -x0
+            res = S.NegativeOne**n/self.func(n + 1)
+            return res/(arg + n).as_leading_term(x)
+        elif not x0.is_infinite:
+            return self.func(x0)
+        raise PoleError()
+
+
+###############################################################################
+################## LOWER and UPPER INCOMPLETE GAMMA FUNCTIONS #################
+###############################################################################
+
+class lowergamma(Function):
+    r"""
+    The lower incomplete gamma function.
+
+    Explanation
+    ===========
+
+    It can be defined as the meromorphic continuation of
+
+    .. math::
+        \gamma(s, x) := \int_0^x t^{s-1} e^{-t} \mathrm{d}t = \Gamma(s) - \Gamma(s, x).
+
+    This can be shown to be the same as
+
+    .. math::
+        \gamma(s, x) = \frac{x^s}{s} {}_1F_1\left({s \atop s+1} \middle| -x\right),
+
+    where ${}_1F_1$ is the (confluent) hypergeometric function.
+
+    Examples
+    ========
+
+    >>> from sympy import lowergamma, S
+    >>> from sympy.abc import s, x
+    >>> lowergamma(s, x)
+    lowergamma(s, x)
+    >>> lowergamma(3, x)
+    -2*(x**2/2 + x + 1)*exp(-x) + 2
+    >>> lowergamma(-S(1)/2, x)
+    -2*sqrt(pi)*erf(sqrt(x)) - 2*exp(-x)/sqrt(x)
+
+    See Also
+    ========
+
+    gamma: Gamma function.
+    uppergamma: Upper incomplete gamma function.
+    polygamma: Polygamma function.
+    loggamma: Log Gamma function.
+    digamma: Digamma function.
+    trigamma: Trigamma function.
+    beta: Euler Beta function.
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Incomplete_gamma_function#Lower_incomplete_gamma_function
+    .. [2] Abramowitz, Milton; Stegun, Irene A., eds. (1965), Chapter 6,
+           Section 5, Handbook of Mathematical Functions with Formulas, Graphs,
+           and Mathematical Tables
+    .. [3] https://dlmf.nist.gov/8
+    .. [4] https://functions.wolfram.com/GammaBetaErf/Gamma2/
+    .. [5] https://functions.wolfram.com/GammaBetaErf/Gamma3/
+
+    """
+
+
+    def fdiff(self, argindex=2):
+        from sympy.functions.special.hyper import meijerg
+        if argindex == 2:
+            a, z = self.args
+            return exp(-unpolarify(z))*z**(a - 1)
+        elif argindex == 1:
+            a, z = self.args
+            return gamma(a)*digamma(a) - log(z)*uppergamma(a, z) \
+                - meijerg([], [1, 1], [0, 0, a], [], z)
+
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    @classmethod
+    def eval(cls, a, x):
+        # For lack of a better place, we use this one to extract branching
+        # information. The following can be
+        # found in the literature (c/f references given above), albeit scattered:
+        # 1) For fixed x != 0, lowergamma(s, x) is an entire function of s
+        # 2) For fixed positive integers s, lowergamma(s, x) is an entire
+        #    function of x.
+        # 3) For fixed non-positive integers s,
+        #    lowergamma(s, exp(I*2*pi*n)*x) =
+        #              2*pi*I*n*(-1)**(-s)/factorial(-s) + lowergamma(s, x)
+        #    (this follows from lowergamma(s, x).diff(x) = x**(s-1)*exp(-x)).
+        # 4) For fixed non-integral s,
+        #    lowergamma(s, x) = x**s*gamma(s)*lowergamma_unbranched(s, x),
+        #    where lowergamma_unbranched(s, x) is an entire function (in fact
+        #    of both s and x), i.e.
+        #    lowergamma(s, exp(2*I*pi*n)*x) = exp(2*pi*I*n*a)*lowergamma(a, x)
+        if x is S.Zero:
+            return S.Zero
+        nx, n = x.extract_branch_factor()
+        if a.is_integer and a.is_positive:
+            nx = unpolarify(x)
+            if nx != x:
+                return lowergamma(a, nx)
+        elif a.is_integer and a.is_nonpositive:
+            if n != 0:
+                return 2*pi*I*n*S.NegativeOne**(-a)/factorial(-a) + lowergamma(a, nx)
+        elif n != 0:
+            return exp(2*pi*I*n*a)*lowergamma(a, nx)
+
+        # Special values.
+        if a.is_Number:
+            if a is S.One:
+                return S.One - exp(-x)
+            elif a is S.Half:
+                return sqrt(pi)*erf(sqrt(x))
+            elif a.is_Integer or (2*a).is_Integer:
+                b = a - 1
+                if b.is_positive:
+                    if a.is_integer:
+                        return factorial(b) - exp(-x) * factorial(b) * Add(*[x ** k / factorial(k) for k in range(a)])
+                    else:
+                        return gamma(a)*(lowergamma(S.Half, x)/sqrt(pi) - exp(-x)*Add(*[x**(k - S.Half)/gamma(S.Half + k) for k in range(1, a + S.Half)]))
+
+                if not a.is_Integer:
+                    return S.NegativeOne**(S.Half - a)*pi*erf(sqrt(x))/gamma(1 - a) + exp(-x)*Add(*[x**(k + a - 1)*gamma(a)/gamma(a + k) for k in range(1, Rational(3, 2) - a)])
+
+        if x.is_zero:
+            return S.Zero
+
+    def _eval_evalf(self, prec):
+        if all(x.is_number for x in self.args):
+            a = self.args[0]._to_mpmath(prec)
+            z = self.args[1]._to_mpmath(prec)
+            with workprec(prec):
+                res = mp.gammainc(a, 0, z)
+            return Expr._from_mpmath(res, prec)
+        else:
+            return self
+
+    def _eval_conjugate(self):
+        x = self.args[1]
+        if x not in (S.Zero, S.NegativeInfinity):
+            return self.func(self.args[0].conjugate(), x.conjugate())
+
+    def _eval_is_meromorphic(self, x, a):
+        # By https://en.wikipedia.org/wiki/Incomplete_gamma_function#Holomorphic_extension,
+        #    lowergamma(s, z) = z**s*gamma(s)*gammastar(s, z),
+        # where gammastar(s, z) is holomorphic for all s and z.
+        # Hence the singularities of lowergamma are z = 0  (branch
+        # point) and nonpositive integer values of s (poles of gamma(s)).
+        s, z = self.args
+        args_merom = fuzzy_and([z._eval_is_meromorphic(x, a),
+            s._eval_is_meromorphic(x, a)])
+        if not args_merom:
+            return args_merom
+        z0 = z.subs(x, a)
+        if s.is_integer:
+            return fuzzy_and([s.is_positive, z0.is_finite])
+        s0 = s.subs(x, a)
+        return fuzzy_and([s0.is_finite, z0.is_finite, fuzzy_not(z0.is_zero)])
+
+    def _eval_aseries(self, n, args0, x, logx):
+        from sympy.series.order import O
+        s, z = self.args
+        if args0[0] is oo and not z.has(x):
+            coeff = z**s*exp(-z)
+            sum_expr = sum(z**k/rf(s, k + 1) for k in range(n - 1))
+            o = O(z**s*s**(-n))
+            return coeff*sum_expr + o
+        return super()._eval_aseries(n, args0, x, logx)
+
+    def _eval_rewrite_as_uppergamma(self, s, x, **kwargs):
+        return gamma(s) - uppergamma(s, x)
+
+    def _eval_rewrite_as_expint(self, s, x, **kwargs):
+        from sympy.functions.special.error_functions import expint
+        if s.is_integer and s.is_nonpositive:
+            return self
+        return self.rewrite(uppergamma).rewrite(expint)
+
+    def _eval_is_zero(self):
+        x = self.args[1]
+        if x.is_zero:
+            return True
+
+
+class uppergamma(Function):
+    r"""
+    The upper incomplete gamma function.
+
+    Explanation
+    ===========
+
+    It can be defined as the meromorphic continuation of
+
+    .. math::
+        \Gamma(s, x) := \int_x^\infty t^{s-1} e^{-t} \mathrm{d}t = \Gamma(s) - \gamma(s, x).
+
+    where $\gamma(s, x)$ is the lower incomplete gamma function,
+    :class:`lowergamma`. This can be shown to be the same as
+
+    .. math::
+        \Gamma(s, x) = \Gamma(s) - \frac{x^s}{s} {}_1F_1\left({s \atop s+1} \middle| -x\right),
+
+    where ${}_1F_1$ is the (confluent) hypergeometric function.
+
+    The upper incomplete gamma function is also essentially equivalent to the
+    generalized exponential integral:
+
+    .. math::
+        \operatorname{E}_{n}(x) = \int_{1}^{\infty}{\frac{e^{-xt}}{t^n} \, dt} = x^{n-1}\Gamma(1-n,x).
+
+    Examples
+    ========
+
+    >>> from sympy import uppergamma, S
+    >>> from sympy.abc import s, x
+    >>> uppergamma(s, x)
+    uppergamma(s, x)
+    >>> uppergamma(3, x)
+    2*(x**2/2 + x + 1)*exp(-x)
+    >>> uppergamma(-S(1)/2, x)
+    -2*sqrt(pi)*erfc(sqrt(x)) + 2*exp(-x)/sqrt(x)
+    >>> uppergamma(-2, x)
+    expint(3, x)/x**2
+
+    See Also
+    ========
+
+    gamma: Gamma function.
+    lowergamma: Lower incomplete gamma function.
+    polygamma: Polygamma function.
+    loggamma: Log Gamma function.
+    digamma: Digamma function.
+    trigamma: Trigamma function.
+    beta: Euler Beta function.
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Incomplete_gamma_function#Upper_incomplete_gamma_function
+    .. [2] Abramowitz, Milton; Stegun, Irene A., eds. (1965), Chapter 6,
+           Section 5, Handbook of Mathematical Functions with Formulas, Graphs,
+           and Mathematical Tables
+    .. [3] https://dlmf.nist.gov/8
+    .. [4] https://functions.wolfram.com/GammaBetaErf/Gamma2/
+    .. [5] https://functions.wolfram.com/GammaBetaErf/Gamma3/
+    .. [6] https://en.wikipedia.org/wiki/Exponential_integral#Relation_with_other_functions
+
+    """
+
+
+    def fdiff(self, argindex=2):
+        from sympy.functions.special.hyper import meijerg
+        if argindex == 2:
+            a, z = self.args
+            return -exp(-unpolarify(z))*z**(a - 1)
+        elif argindex == 1:
+            a, z = self.args
+            return uppergamma(a, z)*log(z) + meijerg([], [1, 1], [0, 0, a], [], z)
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    def _eval_evalf(self, prec):
+        if all(x.is_number for x in self.args):
+            a = self.args[0]._to_mpmath(prec)
+            z = self.args[1]._to_mpmath(prec)
+            with workprec(prec):
+                res = mp.gammainc(a, z, mp.inf)
+            return Expr._from_mpmath(res, prec)
+        return self
+
+    @classmethod
+    def eval(cls, a, z):
+        from sympy.functions.special.error_functions import expint
+        if z.is_Number:
+            if z is S.NaN:
+                return S.NaN
+            elif z is oo:
+                return S.Zero
+            elif z.is_zero:
+                if re(a).is_positive:
+                    return gamma(a)
+
+        # We extract branching information here. C/f lowergamma.
+        nx, n = z.extract_branch_factor()
+        if a.is_integer and a.is_positive:
+            nx = unpolarify(z)
+            if z != nx:
+                return uppergamma(a, nx)
+        elif a.is_integer and a.is_nonpositive:
+            if n != 0:
+                return -2*pi*I*n*S.NegativeOne**(-a)/factorial(-a) + uppergamma(a, nx)
+        elif n != 0:
+            return gamma(a)*(1 - exp(2*pi*I*n*a)) + exp(2*pi*I*n*a)*uppergamma(a, nx)
+
+        # Special values.
+        if a.is_Number:
+            if a is S.Zero and z.is_positive:
+                return -Ei(-z)
+            elif a is S.One:
+                return exp(-z)
+            elif a is S.Half:
+                return sqrt(pi)*erfc(sqrt(z))
+            elif a.is_Integer or (2*a).is_Integer:
+                b = a - 1
+                if b.is_positive:
+                    if a.is_integer:
+                        return exp(-z) * factorial(b) * Add(*[z**k / factorial(k)
+                                                              for k in range(a)])
+                    else:
+                        return (gamma(a) * erfc(sqrt(z)) +
+                                S.NegativeOne**(a - S(3)/2) * exp(-z) * sqrt(z)
+                                * Add(*[gamma(-S.Half - k) * (-z)**k / gamma(1-a)
+                                        for k in range(a - S.Half)]))
+                elif b.is_Integer:
+                    return expint(-b, z)*unpolarify(z)**(b + 1)
+
+                if not a.is_Integer:
+                    return (S.NegativeOne**(S.Half - a) * pi*erfc(sqrt(z))/gamma(1-a)
+                            - z**a * exp(-z) * Add(*[z**k * gamma(a) / gamma(a+k+1)
+                                                     for k in range(S.Half - a)]))
+
+        if a.is_zero and z.is_positive:
+            return -Ei(-z)
+
+        if z.is_zero and re(a).is_positive:
+            return gamma(a)
+
+    def _eval_conjugate(self):
+        z = self.args[1]
+        if z not in (S.Zero, S.NegativeInfinity):
+            return self.func(self.args[0].conjugate(), z.conjugate())
+
+    def _eval_is_meromorphic(self, x, a):
+        return lowergamma._eval_is_meromorphic(self, x, a)
+
+    def _eval_rewrite_as_lowergamma(self, s, x, **kwargs):
+        return gamma(s) - lowergamma(s, x)
+
+    def _eval_rewrite_as_tractable(self, s, x, **kwargs):
+        return exp(loggamma(s)) - lowergamma(s, x)
+
+    def _eval_rewrite_as_expint(self, s, x, **kwargs):
+        from sympy.functions.special.error_functions import expint
+        return expint(1 - s, x)*x**s
+
+
+###############################################################################
+###################### POLYGAMMA and LOGGAMMA FUNCTIONS #######################
+###############################################################################
+
+class polygamma(Function):
+    r"""
+    The function ``polygamma(n, z)`` returns ``log(gamma(z)).diff(n + 1)``.
+
+    Explanation
+    ===========
+
+    It is a meromorphic function on $\mathbb{C}$ and defined as the $(n+1)$-th
+    derivative of the logarithm of the gamma function:
+
+    .. math::
+        \psi^{(n)} (z) := \frac{\mathrm{d}^{n+1}}{\mathrm{d} z^{n+1}} \log\Gamma(z).
+
+    For `n` not a nonnegative integer the generalization by Espinosa and Moll [5]_
+    is used:
+
+    .. math:: \psi(s,z) = \frac{\zeta'(s+1, z) + (\gamma + \psi(-s)) \zeta(s+1, z)}
+        {\Gamma(-s)}
+
+    Examples
+    ========
+
+    Several special values are known:
+
+    >>> from sympy import S, polygamma
+    >>> polygamma(0, 1)
+    -EulerGamma
+    >>> polygamma(0, 1/S(2))
+    -2*log(2) - EulerGamma
+    >>> polygamma(0, 1/S(3))
+    -log(3) - sqrt(3)*pi/6 - EulerGamma - log(sqrt(3))
+    >>> polygamma(0, 1/S(4))
+    -pi/2 - log(4) - log(2) - EulerGamma
+    >>> polygamma(0, 2)
+    1 - EulerGamma
+    >>> polygamma(0, 23)
+    19093197/5173168 - EulerGamma
+
+    >>> from sympy import oo, I
+    >>> polygamma(0, oo)
+    oo
+    >>> polygamma(0, -oo)
+    oo
+    >>> polygamma(0, I*oo)
+    oo
+    >>> polygamma(0, -I*oo)
+    oo
+
+    Differentiation with respect to $x$ is supported:
+
+    >>> from sympy import Symbol, diff
+    >>> x = Symbol("x")
+    >>> diff(polygamma(0, x), x)
+    polygamma(1, x)
+    >>> diff(polygamma(0, x), x, 2)
+    polygamma(2, x)
+    >>> diff(polygamma(0, x), x, 3)
+    polygamma(3, x)
+    >>> diff(polygamma(1, x), x)
+    polygamma(2, x)
+    >>> diff(polygamma(1, x), x, 2)
+    polygamma(3, x)
+    >>> diff(polygamma(2, x), x)
+    polygamma(3, x)
+    >>> diff(polygamma(2, x), x, 2)
+    polygamma(4, x)
+
+    >>> n = Symbol("n")
+    >>> diff(polygamma(n, x), x)
+    polygamma(n + 1, x)
+    >>> diff(polygamma(n, x), x, 2)
+    polygamma(n + 2, x)
+
+    We can rewrite ``polygamma`` functions in terms of harmonic numbers:
+
+    >>> from sympy import harmonic
+    >>> polygamma(0, x).rewrite(harmonic)
+    harmonic(x - 1) - EulerGamma
+    >>> polygamma(2, x).rewrite(harmonic)
+    2*harmonic(x - 1, 3) - 2*zeta(3)
+    >>> ni = Symbol("n", integer=True)
+    >>> polygamma(ni, x).rewrite(harmonic)
+    (-1)**(n + 1)*(-harmonic(x - 1, n + 1) + zeta(n + 1))*factorial(n)
+
+    See Also
+    ========
+
+    gamma: Gamma function.
+    lowergamma: Lower incomplete gamma function.
+    uppergamma: Upper incomplete gamma function.
+    loggamma: Log Gamma function.
+    digamma: Digamma function.
+    trigamma: Trigamma function.
+    beta: Euler Beta function.
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Polygamma_function
+    .. [2] https://mathworld.wolfram.com/PolygammaFunction.html
+    .. [3] https://functions.wolfram.com/GammaBetaErf/PolyGamma/
+    .. [4] https://functions.wolfram.com/GammaBetaErf/PolyGamma2/
+    .. [5] O. Espinosa and V. Moll, "A generalized polygamma function",
+           *Integral Transforms and Special Functions* (2004), 101-115.
+
+    """
+
+    @classmethod
+    def eval(cls, n, z):
+        if n is S.NaN or z is S.NaN:
+            return S.NaN
+        elif z is oo:
+            return oo if n.is_zero else S.Zero
+        elif z.is_Integer and z.is_nonpositive:
+            return S.ComplexInfinity
+        elif n is S.NegativeOne:
+            return loggamma(z) - log(2*pi) / 2
+        elif n.is_zero:
+            if z is -oo or z.extract_multiplicatively(I) in (oo, -oo):
+                return oo
+            elif z.is_Integer:
+                return harmonic(z-1) - S.EulerGamma
+            elif z.is_Rational:
+                # TODO n == 1 also can do some rational z
+                p, q = z.as_numer_denom()
+                # only expand for small denominators to avoid creating long expressions
+                if q <= 6:
+                    return expand_func(polygamma(S.Zero, z, evaluate=False))
+        elif n.is_integer and n.is_nonnegative:
+            nz = unpolarify(z)
+            if z != nz:
+                return polygamma(n, nz)
+            if z.is_Integer:
+                return S.NegativeOne**(n+1) * factorial(n) * zeta(n+1, z)
+            elif z is S.Half:
+                return S.NegativeOne**(n+1) * factorial(n) * (2**(n+1)-1) * zeta(n+1)
+
+    def _eval_is_real(self):
+        if self.args[0].is_positive and self.args[1].is_positive:
+            return True
+
+    def _eval_is_complex(self):
+        z = self.args[1]
+        is_negative_integer = fuzzy_and([z.is_negative, z.is_integer])
+        return fuzzy_and([z.is_complex, fuzzy_not(is_negative_integer)])
+
+    def _eval_is_positive(self):
+        n, z = self.args
+        if n.is_positive:
+            if n.is_odd and z.is_real:
+                return True
+            if n.is_even and z.is_positive:
+                return False
+
+    def _eval_is_negative(self):
+        n, z = self.args
+        if n.is_positive:
+            if n.is_even and z.is_positive:
+                return True
+            if n.is_odd and z.is_real:
+                return False
+
+    def _eval_expand_func(self, **hints):
+        n, z = self.args
+
+        if n.is_Integer and n.is_nonnegative:
+            if z.is_Add:
+                coeff = z.args[0]
+                if coeff.is_Integer:
+                    e = -(n + 1)
+                    if coeff > 0:
+                        tail = Add(*[Pow(
+                            z - i, e) for i in range(1, int(coeff) + 1)])
+                    else:
+                        tail = -Add(*[Pow(
+                            z + i, e) for i in range(int(-coeff))])
+                    return polygamma(n, z - coeff) + S.NegativeOne**n*factorial(n)*tail
+
+            elif z.is_Mul:
+                coeff, z = z.as_two_terms()
+                if coeff.is_Integer and coeff.is_positive:
+                    tail = [polygamma(n, z + Rational(
+                        i, coeff)) for i in range(int(coeff))]
+                    if n == 0:
+                        return Add(*tail)/coeff + log(coeff)
+                    else:
+                        return Add(*tail)/coeff**(n + 1)
+                z *= coeff
+
+        if n == 0 and z.is_Rational:
+            p, q = z.as_numer_denom()
+
+            # Reference:
+            #   Values of the polygamma functions at rational arguments, J. Choi, 2007
+            part_1 = -S.EulerGamma - pi * cot(p * pi / q) / 2 - log(q) + Add(
+                *[cos(2 * k * pi * p / q) * log(2 * sin(k * pi / q)) for k in range(1, q)])
+
+            if z > 0:
+                n = floor(z)
+                z0 = z - n
+                return part_1 + Add(*[1 / (z0 + k) for k in range(n)])
+            elif z < 0:
+                n = floor(1 - z)
+                z0 = z + n
+                return part_1 - Add(*[1 / (z0 - 1 - k) for k in range(n)])
+
+        if n == -1:
+            return loggamma(z) - log(2*pi) / 2
+        if n.is_integer is False or n.is_nonnegative is False:
+            s = Dummy("s")
+            dzt = zeta(s, z).diff(s).subs(s, n+1)
+            return (dzt + (S.EulerGamma + digamma(-n)) * zeta(n+1, z)) / gamma(-n)
+
+        return polygamma(n, z)
+
+    def _eval_rewrite_as_zeta(self, n, z, **kwargs):
+        if n.is_integer and n.is_positive:
+            return S.NegativeOne**(n + 1)*factorial(n)*zeta(n + 1, z)
+
+    def _eval_rewrite_as_harmonic(self, n, z, **kwargs):
+        if n.is_integer:
+            if n.is_zero:
+                return harmonic(z - 1) - S.EulerGamma
+            else:
+                return S.NegativeOne**(n+1) * factorial(n) * (zeta(n+1) - harmonic(z-1, n+1))
+
+    def _eval_as_leading_term(self, x, logx=None, cdir=0):
+        from sympy.series.order import Order
+        n, z = [a.as_leading_term(x) for a in self.args]
+        o = Order(z, x)
+        if n == 0 and o.contains(1/x):
+            logx = log(x) if logx is None else logx
+            return o.getn() * logx
+        else:
+            return self.func(n, z)
+
+    def fdiff(self, argindex=2):
+        if argindex == 2:
+            n, z = self.args[:2]
+            return polygamma(n + 1, z)
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    def _eval_aseries(self, n, args0, x, logx):
+        from sympy.series.order import Order
+        if args0[1] != oo or not \
+                (self.args[0].is_Integer and self.args[0].is_nonnegative):
+            return super()._eval_aseries(n, args0, x, logx)
+        z = self.args[1]
+        N = self.args[0]
+
+        if N == 0:
+            # digamma function series
+            # Abramowitz & Stegun, p. 259, 6.3.18
+            r = log(z) - 1/(2*z)
+            o = None
+            if n < 2:
+                o = Order(1/z, x)
+            else:
+                m = ceiling((n + 1)//2)
+                l = [bernoulli(2*k) / (2*k*z**(2*k)) for k in range(1, m)]
+                r -= Add(*l)
+                o = Order(1/z**n, x)
+            return r._eval_nseries(x, n, logx) + o
+        else:
+            # proper polygamma function
+            # Abramowitz & Stegun, p. 260, 6.4.10
+            # We return terms to order higher than O(x**n) on purpose
+            # -- otherwise we would not be able to return any terms for
+            #    quite a long time!
+            fac = gamma(N)
+            e0 = fac + N*fac/(2*z)
+            m = ceiling((n + 1)//2)
+            for k in range(1, m):
+                fac = fac*(2*k + N - 1)*(2*k + N - 2) / ((2*k)*(2*k - 1))
+                e0 += bernoulli(2*k)*fac/z**(2*k)
+            o = Order(1/z**(2*m), x)
+            if n == 0:
+                o = Order(1/z, x)
+            elif n == 1:
+                o = Order(1/z**2, x)
+            r = e0._eval_nseries(z, n, logx) + o
+            return (-1 * (-1/z)**N * r)._eval_nseries(x, n, logx)
+
+    def _eval_evalf(self, prec):
+        if not all(i.is_number for i in self.args):
+            return
+        s = self.args[0]._to_mpmath(prec+12)
+        z = self.args[1]._to_mpmath(prec+12)
+        if mp.isint(z) and z <= 0:
+            return S.ComplexInfinity
+        with workprec(prec+12):
+            if mp.isint(s) and s >= 0:
+                res = mp.polygamma(s, z)
+            else:
+                zt = mp.zeta(s+1, z)
+                dzt = mp.zeta(s+1, z, 1)
+                res = (dzt + (mp.euler + mp.digamma(-s)) * zt) * mp.rgamma(-s)
+        return Expr._from_mpmath(res, prec)
+
+
+class loggamma(Function):
+    r"""
+    The ``loggamma`` function implements the logarithm of the
+    gamma function (i.e., $\log\Gamma(x)$).
+
+    Examples
+    ========
+
+    Several special values are known. For numerical integral
+    arguments we have:
+
+    >>> from sympy import loggamma
+    >>> loggamma(-2)
+    oo
+    >>> loggamma(0)
+    oo
+    >>> loggamma(1)
+    0
+    >>> loggamma(2)
+    0
+    >>> loggamma(3)
+    log(2)
+
+    And for symbolic values:
+
+    >>> from sympy import Symbol
+    >>> n = Symbol("n", integer=True, positive=True)
+    >>> loggamma(n)
+    log(gamma(n))
+    >>> loggamma(-n)
+    oo
+
+    For half-integral values:
+
+    >>> from sympy import S
+    >>> loggamma(S(5)/2)
+    log(3*sqrt(pi)/4)
+    >>> loggamma(n/2)
+    log(2**(1 - n)*sqrt(pi)*gamma(n)/gamma(n/2 + 1/2))
+
+    And general rational arguments:
+
+    >>> from sympy import expand_func
+    >>> L = loggamma(S(16)/3)
+    >>> expand_func(L).doit()
+    -5*log(3) + loggamma(1/3) + log(4) + log(7) + log(10) + log(13)
+    >>> L = loggamma(S(19)/4)
+    >>> expand_func(L).doit()
+    -4*log(4) + loggamma(3/4) + log(3) + log(7) + log(11) + log(15)
+    >>> L = loggamma(S(23)/7)
+    >>> expand_func(L).doit()
+    -3*log(7) + log(2) + loggamma(2/7) + log(9) + log(16)
+
+    The ``loggamma`` function has the following limits towards infinity:
+
+    >>> from sympy import oo
+    >>> loggamma(oo)
+    oo
+    >>> loggamma(-oo)
+    zoo
+
+    The ``loggamma`` function obeys the mirror symmetry
+    if $x \in \mathbb{C} \setminus \{-\infty, 0\}$:
+
+    >>> from sympy.abc import x
+    >>> from sympy import conjugate
+    >>> conjugate(loggamma(x))
+    loggamma(conjugate(x))
+
+    Differentiation with respect to $x$ is supported:
+
+    >>> from sympy import diff
+    >>> diff(loggamma(x), x)
+    polygamma(0, x)
+
+    Series expansion is also supported:
+
+    >>> from sympy import series
+    >>> series(loggamma(x), x, 0, 4).cancel()
+    -log(x) - EulerGamma*x + pi**2*x**2/12 - x**3*zeta(3)/3 + O(x**4)
+
+    We can numerically evaluate the ``loggamma`` function
+    to arbitrary precision on the whole complex plane:
+
+    >>> from sympy import I
+    >>> loggamma(5).evalf(30)
+    3.17805383034794561964694160130
+    >>> loggamma(I).evalf(20)
+    -0.65092319930185633889 - 1.8724366472624298171*I
+
+    See Also
+    ========
+
+    gamma: Gamma function.
+    lowergamma: Lower incomplete gamma function.
+    uppergamma: Upper incomplete gamma function.
+    polygamma: Polygamma function.
+    digamma: Digamma function.
+    trigamma: Trigamma function.
+    beta: Euler Beta function.
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Gamma_function
+    .. [2] https://dlmf.nist.gov/5
+    .. [3] https://mathworld.wolfram.com/LogGammaFunction.html
+    .. [4] https://functions.wolfram.com/GammaBetaErf/LogGamma/
+
+    """
+    @classmethod
+    def eval(cls, z):
+        if z.is_integer:
+            if z.is_nonpositive:
+                return oo
+            elif z.is_positive:
+                return log(gamma(z))
+        elif z.is_rational:
+            p, q = z.as_numer_denom()
+            # Half-integral values:
+            if p.is_positive and q == 2:
+                return log(sqrt(pi) * 2**(1 - p) * gamma(p) / gamma((p + 1)*S.Half))
+
+        if z is oo:
+            return oo
+        elif abs(z) is oo:
+            return S.ComplexInfinity
+        if z is S.NaN:
+            return S.NaN
+
+    def _eval_expand_func(self, **hints):
+        from sympy.concrete.summations import Sum
+        z = self.args[0]
+
+        if z.is_Rational:
+            p, q = z.as_numer_denom()
+            # General rational arguments (u + p/q)
+            # Split z as n + p/q with p < q
+            n = p // q
+            p = p - n*q
+            if p.is_positive and q.is_positive and p < q:
+                k = Dummy("k")
+                if n.is_positive:
+                    return loggamma(p / q) - n*log(q) + Sum(log((k - 1)*q + p), (k, 1, n))
+                elif n.is_negative:
+                    return loggamma(p / q) - n*log(q) + pi*I*n - Sum(log(k*q - p), (k, 1, -n))
+                elif n.is_zero:
+                    return loggamma(p / q)
+
+        return self
+
+    def _eval_nseries(self, x, n, logx=None, cdir=0):
+        x0 = self.args[0].limit(x, 0)
+        if x0.is_zero:
+            f = self._eval_rewrite_as_intractable(*self.args)
+            return f._eval_nseries(x, n, logx)
+        return super()._eval_nseries(x, n, logx)
+
+    def _eval_aseries(self, n, args0, x, logx):
+        from sympy.series.order import Order
+        if args0[0] != oo:
+            return super()._eval_aseries(n, args0, x, logx)
+        z = self.args[0]
+        r = log(z)*(z - S.Half) - z + log(2*pi)/2
+        l = [bernoulli(2*k) / (2*k*(2*k - 1)*z**(2*k - 1)) for k in range(1, n)]
+        o = None
+        if n == 0:
+            o = Order(1, x)
+        else:
+            o = Order(1/z**n, x)
+        # It is very inefficient to first add the order and then do the nseries
+        return (r + Add(*l))._eval_nseries(x, n, logx) + o
+
+    def _eval_rewrite_as_intractable(self, z, **kwargs):
+        return log(gamma(z))
+
+    def _eval_is_real(self):
+        z = self.args[0]
+        if z.is_positive:
+            return True
+        elif z.is_nonpositive:
+            return False
+
+    def _eval_conjugate(self):
+        z = self.args[0]
+        if z not in (S.Zero, S.NegativeInfinity):
+            return self.func(z.conjugate())
+
+    def fdiff(self, argindex=1):
+        if argindex == 1:
+            return polygamma(0, self.args[0])
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+
+class digamma(Function):
+    r"""
+    The ``digamma`` function is the first derivative of the ``loggamma``
+    function
+
+    .. math::
+        \psi(x) := \frac{\mathrm{d}}{\mathrm{d} z} \log\Gamma(z)
+                = \frac{\Gamma'(z)}{\Gamma(z) }.
+
+    In this case, ``digamma(z) = polygamma(0, z)``.
+
+    Examples
+    ========
+
+    >>> from sympy import digamma
+    >>> digamma(0)
+    zoo
+    >>> from sympy import Symbol
+    >>> z = Symbol('z')
+    >>> digamma(z)
+    polygamma(0, z)
+
+    To retain ``digamma`` as it is:
+
+    >>> digamma(0, evaluate=False)
+    digamma(0)
+    >>> digamma(z, evaluate=False)
+    digamma(z)
+
+    See Also
+    ========
+
+    gamma: Gamma function.
+    lowergamma: Lower incomplete gamma function.
+    uppergamma: Upper incomplete gamma function.
+    polygamma: Polygamma function.
+    loggamma: Log Gamma function.
+    trigamma: Trigamma function.
+    beta: Euler Beta function.
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Digamma_function
+    .. [2] https://mathworld.wolfram.com/DigammaFunction.html
+    .. [3] https://functions.wolfram.com/GammaBetaErf/PolyGamma2/
+
+    """
+    def _eval_evalf(self, prec):
+        z = self.args[0]
+        nprec = prec_to_dps(prec)
+        return polygamma(0, z).evalf(n=nprec)
+
+    def fdiff(self, argindex=1):
+        z = self.args[0]
+        return polygamma(0, z).fdiff()
+
+    def _eval_is_real(self):
+        z = self.args[0]
+        return polygamma(0, z).is_real
+
+    def _eval_is_positive(self):
+        z = self.args[0]
+        return polygamma(0, z).is_positive
+
+    def _eval_is_negative(self):
+        z = self.args[0]
+        return polygamma(0, z).is_negative
+
+    def _eval_aseries(self, n, args0, x, logx):
+        as_polygamma = self.rewrite(polygamma)
+        args0 = [S.Zero,] + args0
+        return as_polygamma._eval_aseries(n, args0, x, logx)
+
+    @classmethod
+    def eval(cls, z):
+        return polygamma(0, z)
+
+    def _eval_expand_func(self, **hints):
+        z = self.args[0]
+        return polygamma(0, z).expand(func=True)
+
+    def _eval_rewrite_as_harmonic(self, z, **kwargs):
+        return harmonic(z - 1) - S.EulerGamma
+
+    def _eval_rewrite_as_polygamma(self, z, **kwargs):
+        return polygamma(0, z)
+
+    def _eval_as_leading_term(self, x, logx=None, cdir=0):
+        z = self.args[0]
+        return polygamma(0, z).as_leading_term(x)
+
+
+
+class trigamma(Function):
+    r"""
+    The ``trigamma`` function is the second derivative of the ``loggamma``
+    function
+
+    .. math::
+        \psi^{(1)}(z) := \frac{\mathrm{d}^{2}}{\mathrm{d} z^{2}} \log\Gamma(z).
+
+    In this case, ``trigamma(z) = polygamma(1, z)``.
+
+    Examples
+    ========
+
+    >>> from sympy import trigamma
+    >>> trigamma(0)
+    zoo
+    >>> from sympy import Symbol
+    >>> z = Symbol('z')
+    >>> trigamma(z)
+    polygamma(1, z)
+
+    To retain ``trigamma`` as it is:
+
+    >>> trigamma(0, evaluate=False)
+    trigamma(0)
+    >>> trigamma(z, evaluate=False)
+    trigamma(z)
+
+
+    See Also
+    ========
+
+    gamma: Gamma function.
+    lowergamma: Lower incomplete gamma function.
+    uppergamma: Upper incomplete gamma function.
+    polygamma: Polygamma function.
+    loggamma: Log Gamma function.
+    digamma: Digamma function.
+    beta: Euler Beta function.
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Trigamma_function
+    .. [2] https://mathworld.wolfram.com/TrigammaFunction.html
+    .. [3] https://functions.wolfram.com/GammaBetaErf/PolyGamma2/
+
+    """
+    def _eval_evalf(self, prec):
+        z = self.args[0]
+        nprec = prec_to_dps(prec)
+        return polygamma(1, z).evalf(n=nprec)
+
+    def fdiff(self, argindex=1):
+        z = self.args[0]
+        return polygamma(1, z).fdiff()
+
+    def _eval_is_real(self):
+        z = self.args[0]
+        return polygamma(1, z).is_real
+
+    def _eval_is_positive(self):
+        z = self.args[0]
+        return polygamma(1, z).is_positive
+
+    def _eval_is_negative(self):
+        z = self.args[0]
+        return polygamma(1, z).is_negative
+
+    def _eval_aseries(self, n, args0, x, logx):
+        as_polygamma = self.rewrite(polygamma)
+        args0 = [S.One,] + args0
+        return as_polygamma._eval_aseries(n, args0, x, logx)
+
+    @classmethod
+    def eval(cls, z):
+        return polygamma(1, z)
+
+    def _eval_expand_func(self, **hints):
+        z = self.args[0]
+        return polygamma(1, z).expand(func=True)
+
+    def _eval_rewrite_as_zeta(self, z, **kwargs):
+        return zeta(2, z)
+
+    def _eval_rewrite_as_polygamma(self, z, **kwargs):
+        return polygamma(1, z)
+
+    def _eval_rewrite_as_harmonic(self, z, **kwargs):
+        return -harmonic(z - 1, 2) + pi**2 / 6
+
+    def _eval_as_leading_term(self, x, logx=None, cdir=0):
+        z = self.args[0]
+        return polygamma(1, z).as_leading_term(x)
+
+
+###############################################################################
+##################### COMPLETE MULTIVARIATE GAMMA FUNCTION ####################
+###############################################################################
+
+
+class multigamma(Function):
+    r"""
+    The multivariate gamma function is a generalization of the gamma function
+
+    .. math::
+        \Gamma_p(z) = \pi^{p(p-1)/4}\prod_{k=1}^p \Gamma[z + (1 - k)/2].
+
+    In a special case, ``multigamma(x, 1) = gamma(x)``.
+
+    Examples
+    ========
+
+    >>> from sympy import S, multigamma
+    >>> from sympy import Symbol
+    >>> x = Symbol('x')
+    >>> p = Symbol('p', positive=True, integer=True)
+
+    >>> multigamma(x, p)
+    pi**(p*(p - 1)/4)*Product(gamma(-_k/2 + x + 1/2), (_k, 1, p))
+
+    Several special values are known:
+
+    >>> multigamma(1, 1)
+    1
+    >>> multigamma(4, 1)
+    6
+    >>> multigamma(S(3)/2, 1)
+    sqrt(pi)/2
+
+    Writing ``multigamma`` in terms of the ``gamma`` function:
+
+    >>> multigamma(x, 1)
+    gamma(x)
+
+    >>> multigamma(x, 2)
+    sqrt(pi)*gamma(x)*gamma(x - 1/2)
+
+    >>> multigamma(x, 3)
+    pi**(3/2)*gamma(x)*gamma(x - 1)*gamma(x - 1/2)
+
+    Parameters
+    ==========
+
+    p : order or dimension of the multivariate gamma function
+
+    See Also
+    ========
+
+    gamma, lowergamma, uppergamma, polygamma, loggamma, digamma, trigamma,
+    beta
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Multivariate_gamma_function
+
+    """
+    unbranched = True
+
+    def fdiff(self, argindex=2):
+        from sympy.concrete.summations import Sum
+        if argindex == 2:
+            x, p = self.args
+            k = Dummy("k")
+            return self.func(x, p)*Sum(polygamma(0, x + (1 - k)/2), (k, 1, p))
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    @classmethod
+    def eval(cls, x, p):
+        from sympy.concrete.products import Product
+        if p.is_positive is False or p.is_integer is False:
+            raise ValueError('Order parameter p must be positive integer.')
+        k = Dummy("k")
+        return (pi**(p*(p - 1)/4)*Product(gamma(x + (1 - k)/2),
+                                          (k, 1, p))).doit()
+
+    def _eval_conjugate(self):
+        x, p = self.args
+        return self.func(x.conjugate(), p)
+
+    def _eval_is_real(self):
+        x, p = self.args
+        y = 2*x
+        if y.is_integer and (y <= (p - 1)) is True:
+            return False
+        if intlike(y) and (y <= (p - 1)):
+            return False
+        if y > (p - 1) or y.is_noninteger:
+            return True

llmeval-env/lib/python3.10/site-packages/sympy/functions/special/hyper.py

@@ -0,0 +1,1152 @@
+"""Hypergeometric and Meijer G-functions"""
+from functools import reduce
+
+from sympy.core import S, ilcm, Mod
+from sympy.core.add import Add
+from sympy.core.expr import Expr
+from sympy.core.function import Function, Derivative, ArgumentIndexError
+
+from sympy.core.containers import Tuple
+from sympy.core.mul import Mul
+from sympy.core.numbers import I, pi, oo, zoo
+from sympy.core.relational import Ne
+from sympy.core.sorting import default_sort_key
+from sympy.core.symbol import Dummy
+
+from sympy.functions import (sqrt, exp, log, sin, cos, asin, atan,
+        sinh, cosh, asinh, acosh, atanh, acoth)
+from sympy.functions import factorial, RisingFactorial
+from sympy.functions.elementary.complexes import Abs, re, unpolarify
+from sympy.functions.elementary.exponential import exp_polar
+from sympy.functions.elementary.integers import ceiling
+from sympy.functions.elementary.piecewise import Piecewise
+from sympy.logic.boolalg import (And, Or)
+
+class TupleArg(Tuple):
+    def limit(self, x, xlim, dir='+'):
+        """ Compute limit x->xlim.
+        """
+        from sympy.series.limits import limit
+        return TupleArg(*[limit(f, x, xlim, dir) for f in self.args])
+
+
+# TODO should __new__ accept **options?
+# TODO should constructors should check if parameters are sensible?
+
+
+def _prep_tuple(v):
+    """
+    Turn an iterable argument *v* into a tuple and unpolarify, since both
+    hypergeometric and meijer g-functions are unbranched in their parameters.
+
+    Examples
+    ========
+
+    >>> from sympy.functions.special.hyper import _prep_tuple
+    >>> _prep_tuple([1, 2, 3])
+    (1, 2, 3)
+    >>> _prep_tuple((4, 5))
+    (4, 5)
+    >>> _prep_tuple((7, 8, 9))
+    (7, 8, 9)
+
+    """
+    return TupleArg(*[unpolarify(x) for x in v])
+
+
+class TupleParametersBase(Function):
+    """ Base class that takes care of differentiation, when some of
+        the arguments are actually tuples. """
+    # This is not deduced automatically since there are Tuples as arguments.
+    is_commutative = True
+
+    def _eval_derivative(self, s):
+        try:
+            res = 0
+            if self.args[0].has(s) or self.args[1].has(s):
+                for i, p in enumerate(self._diffargs):
+                    m = self._diffargs[i].diff(s)
+                    if m != 0:
+                        res += self.fdiff((1, i))*m
+            return res + self.fdiff(3)*self.args[2].diff(s)
+        except (ArgumentIndexError, NotImplementedError):
+            return Derivative(self, s)
+
+
+class hyper(TupleParametersBase):
+    r"""
+    The generalized hypergeometric function is defined by a series where
+    the ratios of successive terms are a rational function of the summation
+    index. When convergent, it is continued analytically to the largest
+    possible domain.
+
+    Explanation
+    ===========
+
+    The hypergeometric function depends on two vectors of parameters, called
+    the numerator parameters $a_p$, and the denominator parameters
+    $b_q$. It also has an argument $z$. The series definition is
+
+    .. math ::
+        {}_pF_q\left(\begin{matrix} a_1, \cdots, a_p \\ b_1, \cdots, b_q \end{matrix}
+                     \middle| z \right)
+        = \sum_{n=0}^\infty \frac{(a_1)_n \cdots (a_p)_n}{(b_1)_n \cdots (b_q)_n}
+                            \frac{z^n}{n!},
+
+    where $(a)_n = (a)(a+1)\cdots(a+n-1)$ denotes the rising factorial.
+
+    If one of the $b_q$ is a non-positive integer then the series is
+    undefined unless one of the $a_p$ is a larger (i.e., smaller in
+    magnitude) non-positive integer. If none of the $b_q$ is a
+    non-positive integer and one of the $a_p$ is a non-positive
+    integer, then the series reduces to a polynomial. To simplify the
+    following discussion, we assume that none of the $a_p$ or
+    $b_q$ is a non-positive integer. For more details, see the
+    references.
+
+    The series converges for all $z$ if $p \le q$, and thus
+    defines an entire single-valued function in this case. If $p =
+    q+1$ the series converges for $|z| < 1$, and can be continued
+    analytically into a half-plane. If $p > q+1$ the series is
+    divergent for all $z$.
+
+    Please note the hypergeometric function constructor currently does *not*
+    check if the parameters actually yield a well-defined function.
+
+    Examples
+    ========
+
+    The parameters $a_p$ and $b_q$ can be passed as arbitrary
+    iterables, for example:
+
+    >>> from sympy import hyper
+    >>> from sympy.abc import x, n, a
+    >>> hyper((1, 2, 3), [3, 4], x)
+    hyper((1, 2, 3), (3, 4), x)
+
+    There is also pretty printing (it looks better using Unicode):
+
+    >>> from sympy import pprint
+    >>> pprint(hyper((1, 2, 3), [3, 4], x), use_unicode=False)
+      _
+     |_  /1, 2, 3 |  \
+     |   |        | x|
+    3  2 \  3, 4  |  /
+
+    The parameters must always be iterables, even if they are vectors of
+    length one or zero:
+
+    >>> hyper((1, ), [], x)
+    hyper((1,), (), x)
+
+    But of course they may be variables (but if they depend on $x$ then you
+    should not expect much implemented functionality):
+
+    >>> hyper((n, a), (n**2,), x)
+    hyper((n, a), (n**2,), x)
+
+    The hypergeometric function generalizes many named special functions.
+    The function ``hyperexpand()`` tries to express a hypergeometric function
+    using named special functions. For example:
+
+    >>> from sympy import hyperexpand
+    >>> hyperexpand(hyper([], [], x))
+    exp(x)
+
+    You can also use ``expand_func()``:
+
+    >>> from sympy import expand_func
+    >>> expand_func(x*hyper([1, 1], [2], -x))
+    log(x + 1)
+
+    More examples:
+
+    >>> from sympy import S
+    >>> hyperexpand(hyper([], [S(1)/2], -x**2/4))
+    cos(x)
+    >>> hyperexpand(x*hyper([S(1)/2, S(1)/2], [S(3)/2], x**2))
+    asin(x)
+
+    We can also sometimes ``hyperexpand()`` parametric functions:
+
+    >>> from sympy.abc import a
+    >>> hyperexpand(hyper([-a], [], x))
+    (1 - x)**a
+
+    See Also
+    ========
+
+    sympy.simplify.hyperexpand
+    gamma
+    meijerg
+
+    References
+    ==========
+
+    .. [1] Luke, Y. L. (1969), The Special Functions and Their Approximations,
+           Volume 1
+    .. [2] https://en.wikipedia.org/wiki/Generalized_hypergeometric_function
+
+    """
+
+
+    def __new__(cls, ap, bq, z, **kwargs):
+        # TODO should we check convergence conditions?
+        return Function.__new__(cls, _prep_tuple(ap), _prep_tuple(bq), z, **kwargs)
+
+    @classmethod
+    def eval(cls, ap, bq, z):
+        if len(ap) <= len(bq) or (len(ap) == len(bq) + 1 and (Abs(z) <= 1) == True):
+            nz = unpolarify(z)
+            if z != nz:
+                return hyper(ap, bq, nz)
+
+    def fdiff(self, argindex=3):
+        if argindex != 3:
+            raise ArgumentIndexError(self, argindex)
+        nap = Tuple(*[a + 1 for a in self.ap])
+        nbq = Tuple(*[b + 1 for b in self.bq])
+        fac = Mul(*self.ap)/Mul(*self.bq)
+        return fac*hyper(nap, nbq, self.argument)
+
+    def _eval_expand_func(self, **hints):
+        from sympy.functions.special.gamma_functions import gamma
+        from sympy.simplify.hyperexpand import hyperexpand
+        if len(self.ap) == 2 and len(self.bq) == 1 and self.argument == 1:
+            a, b = self.ap
+            c = self.bq[0]
+            return gamma(c)*gamma(c - a - b)/gamma(c - a)/gamma(c - b)
+        return hyperexpand(self)
+
+    def _eval_rewrite_as_Sum(self, ap, bq, z, **kwargs):
+        from sympy.concrete.summations import Sum
+        n = Dummy("n", integer=True)
+        rfap = [RisingFactorial(a, n) for a in ap]
+        rfbq = [RisingFactorial(b, n) for b in bq]
+        coeff = Mul(*rfap) / Mul(*rfbq)
+        return Piecewise((Sum(coeff * z**n / factorial(n), (n, 0, oo)),
+                         self.convergence_statement), (self, True))
+
+    def _eval_as_leading_term(self, x, logx=None, cdir=0):
+        arg = self.args[2]
+        x0 = arg.subs(x, 0)
+        if x0 is S.NaN:
+            x0 = arg.limit(x, 0, dir='-' if re(cdir).is_negative else '+')
+
+        if x0 is S.Zero:
+            return S.One
+        return super()._eval_as_leading_term(x, logx=logx, cdir=cdir)
+
+    def _eval_nseries(self, x, n, logx, cdir=0):
+
+        from sympy.series.order import Order
+
+        arg = self.args[2]
+        x0 = arg.limit(x, 0)
+        ap = self.args[0]
+        bq = self.args[1]
+
+        if x0 != 0:
+            return super()._eval_nseries(x, n, logx)
+
+        terms = []
+
+        for i in range(n):
+            num = Mul(*[RisingFactorial(a, i) for a in ap])
+            den = Mul(*[RisingFactorial(b, i) for b in bq])
+            terms.append(((num/den) * (arg**i)) / factorial(i))
+
+        return (Add(*terms) + Order(x**n,x))
+
+    @property
+    def argument(self):
+        """ Argument of the hypergeometric function. """
+        return self.args[2]
+
+    @property
+    def ap(self):
+        """ Numerator parameters of the hypergeometric function. """
+        return Tuple(*self.args[0])
+
+    @property
+    def bq(self):
+        """ Denominator parameters of the hypergeometric function. """
+        return Tuple(*self.args[1])
+
+    @property
+    def _diffargs(self):
+        return self.ap + self.bq
+
+    @property
+    def eta(self):
+        """ A quantity related to the convergence of the series. """
+        return sum(self.ap) - sum(self.bq)
+
+    @property
+    def radius_of_convergence(self):
+        """
+        Compute the radius of convergence of the defining series.
+
+        Explanation
+        ===========
+
+        Note that even if this is not ``oo``, the function may still be
+        evaluated outside of the radius of convergence by analytic
+        continuation. But if this is zero, then the function is not actually
+        defined anywhere else.
+
+        Examples
+        ========
+
+        >>> from sympy import hyper
+        >>> from sympy.abc import z
+        >>> hyper((1, 2), [3], z).radius_of_convergence
+        1
+        >>> hyper((1, 2, 3), [4], z).radius_of_convergence
+        0
+        >>> hyper((1, 2), (3, 4), z).radius_of_convergence
+        oo
+
+        """
+        if any(a.is_integer and (a <= 0) == True for a in self.ap + self.bq):
+            aints = [a for a in self.ap if a.is_Integer and (a <= 0) == True]
+            bints = [a for a in self.bq if a.is_Integer and (a <= 0) == True]
+            if len(aints) < len(bints):
+                return S.Zero
+            popped = False
+            for b in bints:
+                cancelled = False
+                while aints:
+                    a = aints.pop()
+                    if a >= b:
+                        cancelled = True
+                        break
+                    popped = True
+                if not cancelled:
+                    return S.Zero
+            if aints or popped:
+                # There are still non-positive numerator parameters.
+                # This is a polynomial.
+                return oo
+        if len(self.ap) == len(self.bq) + 1:
+            return S.One
+        elif len(self.ap) <= len(self.bq):
+            return oo
+        else:
+            return S.Zero
+
+    @property
+    def convergence_statement(self):
+        """ Return a condition on z under which the series converges. """
+        R = self.radius_of_convergence
+        if R == 0:
+            return False
+        if R == oo:
+            return True
+        # The special functions and their approximations, page 44
+        e = self.eta
+        z = self.argument
+        c1 = And(re(e) < 0, abs(z) <= 1)
+        c2 = And(0 <= re(e), re(e) < 1, abs(z) <= 1, Ne(z, 1))
+        c3 = And(re(e) >= 1, abs(z) < 1)
+        return Or(c1, c2, c3)
+
+    def _eval_simplify(self, **kwargs):
+        from sympy.simplify.hyperexpand import hyperexpand
+        return hyperexpand(self)
+
+
+class meijerg(TupleParametersBase):
+    r"""
+    The Meijer G-function is defined by a Mellin-Barnes type integral that
+    resembles an inverse Mellin transform. It generalizes the hypergeometric
+    functions.
+
+    Explanation
+    ===========
+
+    The Meijer G-function depends on four sets of parameters. There are
+    "*numerator parameters*"
+    $a_1, \ldots, a_n$ and $a_{n+1}, \ldots, a_p$, and there are
+    "*denominator parameters*"
+    $b_1, \ldots, b_m$ and $b_{m+1}, \ldots, b_q$.
+    Confusingly, it is traditionally denoted as follows (note the position
+    of $m$, $n$, $p$, $q$, and how they relate to the lengths of the four
+    parameter vectors):
+
+    .. math ::
+        G_{p,q}^{m,n} \left(\begin{matrix}a_1, \cdots, a_n & a_{n+1}, \cdots, a_p \\
+                                        b_1, \cdots, b_m & b_{m+1}, \cdots, b_q
+                          \end{matrix} \middle| z \right).
+
+    However, in SymPy the four parameter vectors are always available
+    separately (see examples), so that there is no need to keep track of the
+    decorating sub- and super-scripts on the G symbol.
+
+    The G function is defined as the following integral:
+
+    .. math ::
+         \frac{1}{2 \pi i} \int_L \frac{\prod_{j=1}^m \Gamma(b_j - s)
+         \prod_{j=1}^n \Gamma(1 - a_j + s)}{\prod_{j=m+1}^q \Gamma(1- b_j +s)
+         \prod_{j=n+1}^p \Gamma(a_j - s)} z^s \mathrm{d}s,
+
+    where $\Gamma(z)$ is the gamma function. There are three possible
+    contours which we will not describe in detail here (see the references).
+    If the integral converges along more than one of them, the definitions
+    agree. The contours all separate the poles of $\Gamma(1-a_j+s)$
+    from the poles of $\Gamma(b_k-s)$, so in particular the G function
+    is undefined if $a_j - b_k \in \mathbb{Z}_{>0}$ for some
+    $j \le n$ and $k \le m$.
+
+    The conditions under which one of the contours yields a convergent integral
+    are complicated and we do not state them here, see the references.
+
+    Please note currently the Meijer G-function constructor does *not* check any
+    convergence conditions.
+
+    Examples
+    ========
+
+    You can pass the parameters either as four separate vectors:
+
+    >>> from sympy import meijerg, Tuple, pprint
+    >>> from sympy.abc import x, a
+    >>> pprint(meijerg((1, 2), (a, 4), (5,), [], x), use_unicode=False)
+     __1, 2 /1, 2  a, 4 |  \
+    /__     |           | x|
+    \_|4, 1 \ 5         |  /
+
+    Or as two nested vectors:
+
+    >>> pprint(meijerg([(1, 2), (3, 4)], ([5], Tuple()), x), use_unicode=False)
+     __1, 2 /1, 2  3, 4 |  \
+    /__     |           | x|
+    \_|4, 1 \ 5         |  /
+
+    As with the hypergeometric function, the parameters may be passed as
+    arbitrary iterables. Vectors of length zero and one also have to be
+    passed as iterables. The parameters need not be constants, but if they
+    depend on the argument then not much implemented functionality should be
+    expected.
+
+    All the subvectors of parameters are available:
+
+    >>> from sympy import pprint
+    >>> g = meijerg([1], [2], [3], [4], x)
+    >>> pprint(g, use_unicode=False)
+     __1, 1 /1  2 |  \
+    /__     |     | x|
+    \_|2, 2 \3  4 |  /
+    >>> g.an
+    (1,)
+    >>> g.ap
+    (1, 2)
+    >>> g.aother
+    (2,)
+    >>> g.bm
+    (3,)
+    >>> g.bq
+    (3, 4)
+    >>> g.bother
+    (4,)
+
+    The Meijer G-function generalizes the hypergeometric functions.
+    In some cases it can be expressed in terms of hypergeometric functions,
+    using Slater's theorem. For example:
+
+    >>> from sympy import hyperexpand
+    >>> from sympy.abc import a, b, c
+    >>> hyperexpand(meijerg([a], [], [c], [b], x), allow_hyper=True)
+    x**c*gamma(-a + c + 1)*hyper((-a + c + 1,),
+                                 (-b + c + 1,), -x)/gamma(-b + c + 1)
+
+    Thus the Meijer G-function also subsumes many named functions as special
+    cases. You can use ``expand_func()`` or ``hyperexpand()`` to (try to)
+    rewrite a Meijer G-function in terms of named special functions. For
+    example:
+
+    >>> from sympy import expand_func, S
+    >>> expand_func(meijerg([[],[]], [[0],[]], -x))
+    exp(x)
+    >>> hyperexpand(meijerg([[],[]], [[S(1)/2],[0]], (x/2)**2))
+    sin(x)/sqrt(pi)
+
+    See Also
+    ========
+
+    hyper
+    sympy.simplify.hyperexpand
+
+    References
+    ==========
+
+    .. [1] Luke, Y. L. (1969), The Special Functions and Their Approximations,
+           Volume 1
+    .. [2] https://en.wikipedia.org/wiki/Meijer_G-function
+
+    """
+
+
+    def __new__(cls, *args, **kwargs):
+        if len(args) == 5:
+            args = [(args[0], args[1]), (args[2], args[3]), args[4]]
+        if len(args) != 3:
+            raise TypeError("args must be either as, as', bs, bs', z or "
+                            "as, bs, z")
+
+        def tr(p):
+            if len(p) != 2:
+                raise TypeError("wrong argument")
+            return TupleArg(_prep_tuple(p[0]), _prep_tuple(p[1]))
+
+        arg0, arg1 = tr(args[0]), tr(args[1])
+        if Tuple(arg0, arg1).has(oo, zoo, -oo):
+            raise ValueError("G-function parameters must be finite")
+        if any((a - b).is_Integer and a - b > 0
+               for a in arg0[0] for b in arg1[0]):
+            raise ValueError("no parameter a1, ..., an may differ from "
+                         "any b1, ..., bm by a positive integer")
+
+        # TODO should we check convergence conditions?
+        return Function.__new__(cls, arg0, arg1, args[2], **kwargs)
+
+    def fdiff(self, argindex=3):
+        if argindex != 3:
+            return self._diff_wrt_parameter(argindex[1])
+        if len(self.an) >= 1:
+            a = list(self.an)
+            a[0] -= 1
+            G = meijerg(a, self.aother, self.bm, self.bother, self.argument)
+            return 1/self.argument * ((self.an[0] - 1)*self + G)
+        elif len(self.bm) >= 1:
+            b = list(self.bm)
+            b[0] += 1
+            G = meijerg(self.an, self.aother, b, self.bother, self.argument)
+            return 1/self.argument * (self.bm[0]*self - G)
+        else:
+            return S.Zero
+
+    def _diff_wrt_parameter(self, idx):
+        # Differentiation wrt a parameter can only be done in very special
+        # cases. In particular, if we want to differentiate with respect to
+        # `a`, all other gamma factors have to reduce to rational functions.
+        #
+        # Let MT denote mellin transform. Suppose T(-s) is the gamma factor
+        # appearing in the definition of G. Then
+        #
+        #   MT(log(z)G(z)) = d/ds T(s) = d/da T(s) + ...
+        #
+        # Thus d/da G(z) = log(z)G(z) - ...
+        # The ... can be evaluated as a G function under the above conditions,
+        # the formula being most easily derived by using
+        #
+        # d  Gamma(s + n)    Gamma(s + n) / 1    1                1     \
+        # -- ------------ =  ------------ | - + ----  + ... + --------- |
+        # ds Gamma(s)        Gamma(s)     \ s   s + 1         s + n - 1 /
+        #
+        # which follows from the difference equation of the digamma function.
+        # (There is a similar equation for -n instead of +n).
+
+        # We first figure out how to pair the parameters.
+        an = list(self.an)
+        ap = list(self.aother)
+        bm = list(self.bm)
+        bq = list(self.bother)
+        if idx < len(an):
+            an.pop(idx)
+        else:
+            idx -= len(an)
+            if idx < len(ap):
+                ap.pop(idx)
+            else:
+                idx -= len(ap)
+                if idx < len(bm):
+                    bm.pop(idx)
+                else:
+                    bq.pop(idx - len(bm))
+        pairs1 = []
+        pairs2 = []
+        for l1, l2, pairs in [(an, bq, pairs1), (ap, bm, pairs2)]:
+            while l1:
+                x = l1.pop()
+                found = None
+                for i, y in enumerate(l2):
+                    if not Mod((x - y).simplify(), 1):
+                        found = i
+                        break
+                if found is None:
+                    raise NotImplementedError('Derivative not expressible '
+                                              'as G-function?')
+                y = l2[i]
+                l2.pop(i)
+                pairs.append((x, y))
+
+        # Now build the result.
+        res = log(self.argument)*self
+
+        for a, b in pairs1:
+            sign = 1
+            n = a - b
+            base = b
+            if n < 0:
+                sign = -1
+                n = b - a
+                base = a
+            for k in range(n):
+                res -= sign*meijerg(self.an + (base + k + 1,), self.aother,
+                                    self.bm, self.bother + (base + k + 0,),
+                                    self.argument)
+
+        for a, b in pairs2:
+            sign = 1
+            n = b - a
+            base = a
+            if n < 0:
+                sign = -1
+                n = a - b
+                base = b
+            for k in range(n):
+                res -= sign*meijerg(self.an, self.aother + (base + k + 1,),
+                                    self.bm + (base + k + 0,), self.bother,
+                                    self.argument)
+
+        return res
+
+    def get_period(self):
+        """
+        Return a number $P$ such that $G(x*exp(I*P)) == G(x)$.
+
+        Examples
+        ========
+
+        >>> from sympy import meijerg, pi, S
+        >>> from sympy.abc import z
+
+        >>> meijerg([1], [], [], [], z).get_period()
+        2*pi
+        >>> meijerg([pi], [], [], [], z).get_period()
+        oo
+        >>> meijerg([1, 2], [], [], [], z).get_period()
+        oo
+        >>> meijerg([1,1], [2], [1, S(1)/2, S(1)/3], [1], z).get_period()
+        12*pi
+
+        """
+        # This follows from slater's theorem.
+        def compute(l):
+            # first check that no two differ by an integer
+            for i, b in enumerate(l):
+                if not b.is_Rational:
+                    return oo
+                for j in range(i + 1, len(l)):
+                    if not Mod((b - l[j]).simplify(), 1):
+                        return oo
+            return reduce(ilcm, (x.q for x in l), 1)
+        beta = compute(self.bm)
+        alpha = compute(self.an)
+        p, q = len(self.ap), len(self.bq)
+        if p == q:
+            if oo in (alpha, beta):
+                return oo
+            return 2*pi*ilcm(alpha, beta)
+        elif p < q:
+            return 2*pi*beta
+        else:
+            return 2*pi*alpha
+
+    def _eval_expand_func(self, **hints):
+        from sympy.simplify.hyperexpand import hyperexpand
+        return hyperexpand(self)
+
+    def _eval_evalf(self, prec):
+        # The default code is insufficient for polar arguments.
+        # mpmath provides an optional argument "r", which evaluates
+        # G(z**(1/r)). I am not sure what its intended use is, but we hijack it
+        # here in the following way: to evaluate at a number z of |argument|
+        # less than (say) n*pi, we put r=1/n, compute z' = root(z, n)
+        # (carefully so as not to loose the branch information), and evaluate
+        # G(z'**(1/r)) = G(z'**n) = G(z).
+        import mpmath
+        znum = self.argument._eval_evalf(prec)
+        if znum.has(exp_polar):
+            znum, branch = znum.as_coeff_mul(exp_polar)
+            if len(branch) != 1:
+                return
+            branch = branch[0].args[0]/I
+        else:
+            branch = S.Zero
+        n = ceiling(abs(branch/pi)) + 1
+        znum = znum**(S.One/n)*exp(I*branch / n)
+
+        # Convert all args to mpf or mpc
+        try:
+            [z, r, ap, bq] = [arg._to_mpmath(prec)
+                    for arg in [znum, 1/n, self.args[0], self.args[1]]]
+        except ValueError:
+            return
+
+        with mpmath.workprec(prec):
+            v = mpmath.meijerg(ap, bq, z, r)
+
+        return Expr._from_mpmath(v, prec)
+
+    def _eval_as_leading_term(self, x, logx=None, cdir=0):
+        from sympy.simplify.hyperexpand import hyperexpand
+        return hyperexpand(self).as_leading_term(x, logx=logx, cdir=cdir)
+
+    def integrand(self, s):
+        """ Get the defining integrand D(s). """
+        from sympy.functions.special.gamma_functions import gamma
+        return self.argument**s \
+            * Mul(*(gamma(b - s) for b in self.bm)) \
+            * Mul(*(gamma(1 - a + s) for a in self.an)) \
+            / Mul(*(gamma(1 - b + s) for b in self.bother)) \
+            / Mul(*(gamma(a - s) for a in self.aother))
+
+    @property
+    def argument(self):
+        """ Argument of the Meijer G-function. """
+        return self.args[2]
+
+    @property
+    def an(self):
+        """ First set of numerator parameters. """
+        return Tuple(*self.args[0][0])
+
+    @property
+    def ap(self):
+        """ Combined numerator parameters. """
+        return Tuple(*(self.args[0][0] + self.args[0][1]))
+
+    @property
+    def aother(self):
+        """ Second set of numerator parameters. """
+        return Tuple(*self.args[0][1])
+
+    @property
+    def bm(self):
+        """ First set of denominator parameters. """
+        return Tuple(*self.args[1][0])
+
+    @property
+    def bq(self):
+        """ Combined denominator parameters. """
+        return Tuple(*(self.args[1][0] + self.args[1][1]))
+
+    @property
+    def bother(self):
+        """ Second set of denominator parameters. """
+        return Tuple(*self.args[1][1])
+
+    @property
+    def _diffargs(self):
+        return self.ap + self.bq
+
+    @property
+    def nu(self):
+        """ A quantity related to the convergence region of the integral,
+            c.f. references. """
+        return sum(self.bq) - sum(self.ap)
+
+    @property
+    def delta(self):
+        """ A quantity related to the convergence region of the integral,
+            c.f. references. """
+        return len(self.bm) + len(self.an) - S(len(self.ap) + len(self.bq))/2
+
+    @property
+    def is_number(self):
+        """ Returns true if expression has numeric data only. """
+        return not self.free_symbols
+
+
+class HyperRep(Function):
+    """
+    A base class for "hyper representation functions".
+
+    This is used exclusively in ``hyperexpand()``, but fits more logically here.
+
+    pFq is branched at 1 if p == q+1. For use with slater-expansion, we want
+    define an "analytic continuation" to all polar numbers, which is
+    continuous on circles and on the ray t*exp_polar(I*pi). Moreover, we want
+    a "nice" expression for the various cases.
+
+    This base class contains the core logic, concrete derived classes only
+    supply the actual functions.
+
+    """
+
+
+    @classmethod
+    def eval(cls, *args):
+        newargs = tuple(map(unpolarify, args[:-1])) + args[-1:]
+        if args != newargs:
+            return cls(*newargs)
+
+    @classmethod
+    def _expr_small(cls, x):
+        """ An expression for F(x) which holds for |x| < 1. """
+        raise NotImplementedError
+
+    @classmethod
+    def _expr_small_minus(cls, x):
+        """ An expression for F(-x) which holds for |x| < 1. """
+        raise NotImplementedError
+
+    @classmethod
+    def _expr_big(cls, x, n):
+        """ An expression for F(exp_polar(2*I*pi*n)*x), |x| > 1. """
+        raise NotImplementedError
+
+    @classmethod
+    def _expr_big_minus(cls, x, n):
+        """ An expression for F(exp_polar(2*I*pi*n + pi*I)*x), |x| > 1. """
+        raise NotImplementedError
+
+    def _eval_rewrite_as_nonrep(self, *args, **kwargs):
+        x, n = self.args[-1].extract_branch_factor(allow_half=True)
+        minus = False
+        newargs = self.args[:-1] + (x,)
+        if not n.is_Integer:
+            minus = True
+            n -= S.Half
+        newerargs = newargs + (n,)
+        if minus:
+            small = self._expr_small_minus(*newargs)
+            big = self._expr_big_minus(*newerargs)
+        else:
+            small = self._expr_small(*newargs)
+            big = self._expr_big(*newerargs)
+
+        if big == small:
+            return small
+        return Piecewise((big, abs(x) > 1), (small, True))
+
+    def _eval_rewrite_as_nonrepsmall(self, *args, **kwargs):
+        x, n = self.args[-1].extract_branch_factor(allow_half=True)
+        args = self.args[:-1] + (x,)
+        if not n.is_Integer:
+            return self._expr_small_minus(*args)
+        return self._expr_small(*args)
+
+
+class HyperRep_power1(HyperRep):
+    """ Return a representative for hyper([-a], [], z) == (1 - z)**a. """
+
+    @classmethod
+    def _expr_small(cls, a, x):
+        return (1 - x)**a
+
+    @classmethod
+    def _expr_small_minus(cls, a, x):
+        return (1 + x)**a
+
+    @classmethod
+    def _expr_big(cls, a, x, n):
+        if a.is_integer:
+            return cls._expr_small(a, x)
+        return (x - 1)**a*exp((2*n - 1)*pi*I*a)
+
+    @classmethod
+    def _expr_big_minus(cls, a, x, n):
+        if a.is_integer:
+            return cls._expr_small_minus(a, x)
+        return (1 + x)**a*exp(2*n*pi*I*a)
+
+
+class HyperRep_power2(HyperRep):
+    """ Return a representative for hyper([a, a - 1/2], [2*a], z). """
+
+    @classmethod
+    def _expr_small(cls, a, x):
+        return 2**(2*a - 1)*(1 + sqrt(1 - x))**(1 - 2*a)
+
+    @classmethod
+    def _expr_small_minus(cls, a, x):
+        return 2**(2*a - 1)*(1 + sqrt(1 + x))**(1 - 2*a)
+
+    @classmethod
+    def _expr_big(cls, a, x, n):
+        sgn = -1
+        if n.is_odd:
+            sgn = 1
+            n -= 1
+        return 2**(2*a - 1)*(1 + sgn*I*sqrt(x - 1))**(1 - 2*a) \
+            *exp(-2*n*pi*I*a)
+
+    @classmethod
+    def _expr_big_minus(cls, a, x, n):
+        sgn = 1
+        if n.is_odd:
+            sgn = -1
+        return sgn*2**(2*a - 1)*(sqrt(1 + x) + sgn)**(1 - 2*a)*exp(-2*pi*I*a*n)
+
+
+class HyperRep_log1(HyperRep):
+    """ Represent -z*hyper([1, 1], [2], z) == log(1 - z). """
+    @classmethod
+    def _expr_small(cls, x):
+        return log(1 - x)
+
+    @classmethod
+    def _expr_small_minus(cls, x):
+        return log(1 + x)
+
+    @classmethod
+    def _expr_big(cls, x, n):
+        return log(x - 1) + (2*n - 1)*pi*I
+
+    @classmethod
+    def _expr_big_minus(cls, x, n):
+        return log(1 + x) + 2*n*pi*I
+
+
+class HyperRep_atanh(HyperRep):
+    """ Represent hyper([1/2, 1], [3/2], z) == atanh(sqrt(z))/sqrt(z). """
+    @classmethod
+    def _expr_small(cls, x):
+        return atanh(sqrt(x))/sqrt(x)
+
+    def _expr_small_minus(cls, x):
+        return atan(sqrt(x))/sqrt(x)
+
+    def _expr_big(cls, x, n):
+        if n.is_even:
+            return (acoth(sqrt(x)) + I*pi/2)/sqrt(x)
+        else:
+            return (acoth(sqrt(x)) - I*pi/2)/sqrt(x)
+
+    def _expr_big_minus(cls, x, n):
+        if n.is_even:
+            return atan(sqrt(x))/sqrt(x)
+        else:
+            return (atan(sqrt(x)) - pi)/sqrt(x)
+
+
+class HyperRep_asin1(HyperRep):
+    """ Represent hyper([1/2, 1/2], [3/2], z) == asin(sqrt(z))/sqrt(z). """
+    @classmethod
+    def _expr_small(cls, z):
+        return asin(sqrt(z))/sqrt(z)
+
+    @classmethod
+    def _expr_small_minus(cls, z):
+        return asinh(sqrt(z))/sqrt(z)
+
+    @classmethod
+    def _expr_big(cls, z, n):
+        return S.NegativeOne**n*((S.Half - n)*pi/sqrt(z) + I*acosh(sqrt(z))/sqrt(z))
+
+    @classmethod
+    def _expr_big_minus(cls, z, n):
+        return S.NegativeOne**n*(asinh(sqrt(z))/sqrt(z) + n*pi*I/sqrt(z))
+
+
+class HyperRep_asin2(HyperRep):
+    """ Represent hyper([1, 1], [3/2], z) == asin(sqrt(z))/sqrt(z)/sqrt(1-z). """
+    # TODO this can be nicer
+    @classmethod
+    def _expr_small(cls, z):
+        return HyperRep_asin1._expr_small(z) \
+            /HyperRep_power1._expr_small(S.Half, z)
+
+    @classmethod
+    def _expr_small_minus(cls, z):
+        return HyperRep_asin1._expr_small_minus(z) \
+            /HyperRep_power1._expr_small_minus(S.Half, z)
+
+    @classmethod
+    def _expr_big(cls, z, n):
+        return HyperRep_asin1._expr_big(z, n) \
+            /HyperRep_power1._expr_big(S.Half, z, n)
+
+    @classmethod
+    def _expr_big_minus(cls, z, n):
+        return HyperRep_asin1._expr_big_minus(z, n) \
+            /HyperRep_power1._expr_big_minus(S.Half, z, n)
+
+
+class HyperRep_sqrts1(HyperRep):
+    """ Return a representative for hyper([-a, 1/2 - a], [1/2], z). """
+
+    @classmethod
+    def _expr_small(cls, a, z):
+        return ((1 - sqrt(z))**(2*a) + (1 + sqrt(z))**(2*a))/2
+
+    @classmethod
+    def _expr_small_minus(cls, a, z):
+        return (1 + z)**a*cos(2*a*atan(sqrt(z)))
+
+    @classmethod
+    def _expr_big(cls, a, z, n):
+        if n.is_even:
+            return ((sqrt(z) + 1)**(2*a)*exp(2*pi*I*n*a) +
+                    (sqrt(z) - 1)**(2*a)*exp(2*pi*I*(n - 1)*a))/2
+        else:
+            n -= 1
+            return ((sqrt(z) - 1)**(2*a)*exp(2*pi*I*a*(n + 1)) +
+                    (sqrt(z) + 1)**(2*a)*exp(2*pi*I*a*n))/2
+
+    @classmethod
+    def _expr_big_minus(cls, a, z, n):
+        if n.is_even:
+            return (1 + z)**a*exp(2*pi*I*n*a)*cos(2*a*atan(sqrt(z)))
+        else:
+            return (1 + z)**a*exp(2*pi*I*n*a)*cos(2*a*atan(sqrt(z)) - 2*pi*a)
+
+
+class HyperRep_sqrts2(HyperRep):
+    """ Return a representative for
+          sqrt(z)/2*[(1-sqrt(z))**2a - (1 + sqrt(z))**2a]
+          == -2*z/(2*a+1) d/dz hyper([-a - 1/2, -a], [1/2], z)"""
+
+    @classmethod
+    def _expr_small(cls, a, z):
+        return sqrt(z)*((1 - sqrt(z))**(2*a) - (1 + sqrt(z))**(2*a))/2
+
+    @classmethod
+    def _expr_small_minus(cls, a, z):
+        return sqrt(z)*(1 + z)**a*sin(2*a*atan(sqrt(z)))
+
+    @classmethod
+    def _expr_big(cls, a, z, n):
+        if n.is_even:
+            return sqrt(z)/2*((sqrt(z) - 1)**(2*a)*exp(2*pi*I*a*(n - 1)) -
+                              (sqrt(z) + 1)**(2*a)*exp(2*pi*I*a*n))
+        else:
+            n -= 1
+            return sqrt(z)/2*((sqrt(z) - 1)**(2*a)*exp(2*pi*I*a*(n + 1)) -
+                              (sqrt(z) + 1)**(2*a)*exp(2*pi*I*a*n))
+
+    def _expr_big_minus(cls, a, z, n):
+        if n.is_even:
+            return (1 + z)**a*exp(2*pi*I*n*a)*sqrt(z)*sin(2*a*atan(sqrt(z)))
+        else:
+            return (1 + z)**a*exp(2*pi*I*n*a)*sqrt(z) \
+                *sin(2*a*atan(sqrt(z)) - 2*pi*a)
+
+
+class HyperRep_log2(HyperRep):
+    """ Represent log(1/2 + sqrt(1 - z)/2) == -z/4*hyper([3/2, 1, 1], [2, 2], z) """
+
+    @classmethod
+    def _expr_small(cls, z):
+        return log(S.Half + sqrt(1 - z)/2)
+
+    @classmethod
+    def _expr_small_minus(cls, z):
+        return log(S.Half + sqrt(1 + z)/2)
+
+    @classmethod
+    def _expr_big(cls, z, n):
+        if n.is_even:
+            return (n - S.Half)*pi*I + log(sqrt(z)/2) + I*asin(1/sqrt(z))
+        else:
+            return (n - S.Half)*pi*I + log(sqrt(z)/2) - I*asin(1/sqrt(z))
+
+    def _expr_big_minus(cls, z, n):
+        if n.is_even:
+            return pi*I*n + log(S.Half + sqrt(1 + z)/2)
+        else:
+            return pi*I*n + log(sqrt(1 + z)/2 - S.Half)
+
+
+class HyperRep_cosasin(HyperRep):
+    """ Represent hyper([a, -a], [1/2], z) == cos(2*a*asin(sqrt(z))). """
+    # Note there are many alternative expressions, e.g. as powers of a sum of
+    # square roots.
+
+    @classmethod
+    def _expr_small(cls, a, z):
+        return cos(2*a*asin(sqrt(z)))
+
+    @classmethod
+    def _expr_small_minus(cls, a, z):
+        return cosh(2*a*asinh(sqrt(z)))
+
+    @classmethod
+    def _expr_big(cls, a, z, n):
+        return cosh(2*a*acosh(sqrt(z)) + a*pi*I*(2*n - 1))
+
+    @classmethod
+    def _expr_big_minus(cls, a, z, n):
+        return cosh(2*a*asinh(sqrt(z)) + 2*a*pi*I*n)
+
+
+class HyperRep_sinasin(HyperRep):
+    """ Represent 2*a*z*hyper([1 - a, 1 + a], [3/2], z)
+        == sqrt(z)/sqrt(1-z)*sin(2*a*asin(sqrt(z))) """
+
+    @classmethod
+    def _expr_small(cls, a, z):
+        return sqrt(z)/sqrt(1 - z)*sin(2*a*asin(sqrt(z)))
+
+    @classmethod
+    def _expr_small_minus(cls, a, z):
+        return -sqrt(z)/sqrt(1 + z)*sinh(2*a*asinh(sqrt(z)))
+
+    @classmethod
+    def _expr_big(cls, a, z, n):
+        return -1/sqrt(1 - 1/z)*sinh(2*a*acosh(sqrt(z)) + a*pi*I*(2*n - 1))
+
+    @classmethod
+    def _expr_big_minus(cls, a, z, n):
+        return -1/sqrt(1 + 1/z)*sinh(2*a*asinh(sqrt(z)) + 2*a*pi*I*n)
+
+class appellf1(Function):
+    r"""
+    This is the Appell hypergeometric function of two variables as:
+
+    .. math ::
+        F_1(a,b_1,b_2,c,x,y) = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty}
+        \frac{(a)_{m+n} (b_1)_m (b_2)_n}{(c)_{m+n}}
+        \frac{x^m y^n}{m! n!}.
+
+    Examples
+    ========
+
+    >>> from sympy import appellf1, symbols
+    >>> x, y, a, b1, b2, c = symbols('x y a b1 b2 c')
+    >>> appellf1(2., 1., 6., 4., 5., 6.)
+    0.0063339426292673
+    >>> appellf1(12., 12., 6., 4., 0.5, 0.12)
+    172870711.659936
+    >>> appellf1(40, 2, 6, 4, 15, 60)
+    appellf1(40, 2, 6, 4, 15, 60)
+    >>> appellf1(20., 12., 10., 3., 0.5, 0.12)
+    15605338197184.4
+    >>> appellf1(40, 2, 6, 4, x, y)
+    appellf1(40, 2, 6, 4, x, y)
+    >>> appellf1(a, b1, b2, c, x, y)
+    appellf1(a, b1, b2, c, x, y)
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Appell_series
+    .. [2] https://functions.wolfram.com/HypergeometricFunctions/AppellF1/
+
+    """
+
+    @classmethod
+    def eval(cls, a, b1, b2, c, x, y):
+        if default_sort_key(b1) > default_sort_key(b2):
+            b1, b2 = b2, b1
+            x, y = y, x
+            return cls(a, b1, b2, c, x, y)
+        elif b1 == b2 and default_sort_key(x) > default_sort_key(y):
+            x, y = y, x
+            return cls(a, b1, b2, c, x, y)
+        if x == 0 and y == 0:
+            return S.One
+
+    def fdiff(self, argindex=5):
+        a, b1, b2, c, x, y = self.args
+        if argindex == 5:
+            return (a*b1/c)*appellf1(a + 1, b1 + 1, b2, c + 1, x, y)
+        elif argindex == 6:
+            return (a*b2/c)*appellf1(a + 1, b1, b2 + 1, c + 1, x, y)
+        elif argindex in (1, 2, 3, 4):
+            return Derivative(self, self.args[argindex-1])
+        else:
+            raise ArgumentIndexError(self, argindex)

llmeval-env/lib/python3.10/site-packages/sympy/functions/special/mathieu_functions.py

@@ -0,0 +1,269 @@
+""" This module contains the Mathieu functions.
+"""
+
+from sympy.core.function import Function, ArgumentIndexError
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.trigonometric import sin, cos
+
+
+class MathieuBase(Function):
+    """
+    Abstract base class for Mathieu functions.
+
+    This class is meant to reduce code duplication.
+
+    """
+
+    unbranched = True
+
+    def _eval_conjugate(self):
+        a, q, z = self.args
+        return self.func(a.conjugate(), q.conjugate(), z.conjugate())
+
+
+class mathieus(MathieuBase):
+    r"""
+    The Mathieu Sine function $S(a,q,z)$.
+
+    Explanation
+    ===========
+
+    This function is one solution of the Mathieu differential equation:
+
+    .. math ::
+        y(x)^{\prime\prime} + (a - 2 q \cos(2 x)) y(x) = 0
+
+    The other solution is the Mathieu Cosine function.
+
+    Examples
+    ========
+
+    >>> from sympy import diff, mathieus
+    >>> from sympy.abc import a, q, z
+
+    >>> mathieus(a, q, z)
+    mathieus(a, q, z)
+
+    >>> mathieus(a, 0, z)
+    sin(sqrt(a)*z)
+
+    >>> diff(mathieus(a, q, z), z)
+    mathieusprime(a, q, z)
+
+    See Also
+    ========
+
+    mathieuc: Mathieu cosine function.
+    mathieusprime: Derivative of Mathieu sine function.
+    mathieucprime: Derivative of Mathieu cosine function.
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Mathieu_function
+    .. [2] https://dlmf.nist.gov/28
+    .. [3] https://mathworld.wolfram.com/MathieuFunction.html
+    .. [4] https://functions.wolfram.com/MathieuandSpheroidalFunctions/MathieuS/
+
+    """
+
+    def fdiff(self, argindex=1):
+        if argindex == 3:
+            a, q, z = self.args
+            return mathieusprime(a, q, z)
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    @classmethod
+    def eval(cls, a, q, z):
+        if q.is_Number and q.is_zero:
+            return sin(sqrt(a)*z)
+        # Try to pull out factors of -1
+        if z.could_extract_minus_sign():
+            return -cls(a, q, -z)
+
+
+class mathieuc(MathieuBase):
+    r"""
+    The Mathieu Cosine function $C(a,q,z)$.
+
+    Explanation
+    ===========
+
+    This function is one solution of the Mathieu differential equation:
+
+    .. math ::
+        y(x)^{\prime\prime} + (a - 2 q \cos(2 x)) y(x) = 0
+
+    The other solution is the Mathieu Sine function.
+
+    Examples
+    ========
+
+    >>> from sympy import diff, mathieuc
+    >>> from sympy.abc import a, q, z
+
+    >>> mathieuc(a, q, z)
+    mathieuc(a, q, z)
+
+    >>> mathieuc(a, 0, z)
+    cos(sqrt(a)*z)
+
+    >>> diff(mathieuc(a, q, z), z)
+    mathieucprime(a, q, z)
+
+    See Also
+    ========
+
+    mathieus: Mathieu sine function
+    mathieusprime: Derivative of Mathieu sine function
+    mathieucprime: Derivative of Mathieu cosine function
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Mathieu_function
+    .. [2] https://dlmf.nist.gov/28
+    .. [3] https://mathworld.wolfram.com/MathieuFunction.html
+    .. [4] https://functions.wolfram.com/MathieuandSpheroidalFunctions/MathieuC/
+
+    """
+
+    def fdiff(self, argindex=1):
+        if argindex == 3:
+            a, q, z = self.args
+            return mathieucprime(a, q, z)
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    @classmethod
+    def eval(cls, a, q, z):
+        if q.is_Number and q.is_zero:
+            return cos(sqrt(a)*z)
+        # Try to pull out factors of -1
+        if z.could_extract_minus_sign():
+            return cls(a, q, -z)
+
+
+class mathieusprime(MathieuBase):
+    r"""
+    The derivative $S^{\prime}(a,q,z)$ of the Mathieu Sine function.
+
+    Explanation
+    ===========
+
+    This function is one solution of the Mathieu differential equation:
+
+    .. math ::
+        y(x)^{\prime\prime} + (a - 2 q \cos(2 x)) y(x) = 0
+
+    The other solution is the Mathieu Cosine function.
+
+    Examples
+    ========
+
+    >>> from sympy import diff, mathieusprime
+    >>> from sympy.abc import a, q, z
+
+    >>> mathieusprime(a, q, z)
+    mathieusprime(a, q, z)
+
+    >>> mathieusprime(a, 0, z)
+    sqrt(a)*cos(sqrt(a)*z)
+
+    >>> diff(mathieusprime(a, q, z), z)
+    (-a + 2*q*cos(2*z))*mathieus(a, q, z)
+
+    See Also
+    ========
+
+    mathieus: Mathieu sine function
+    mathieuc: Mathieu cosine function
+    mathieucprime: Derivative of Mathieu cosine function
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Mathieu_function
+    .. [2] https://dlmf.nist.gov/28
+    .. [3] https://mathworld.wolfram.com/MathieuFunction.html
+    .. [4] https://functions.wolfram.com/MathieuandSpheroidalFunctions/MathieuSPrime/
+
+    """
+
+    def fdiff(self, argindex=1):
+        if argindex == 3:
+            a, q, z = self.args
+            return (2*q*cos(2*z) - a)*mathieus(a, q, z)
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    @classmethod
+    def eval(cls, a, q, z):
+        if q.is_Number and q.is_zero:
+            return sqrt(a)*cos(sqrt(a)*z)
+        # Try to pull out factors of -1
+        if z.could_extract_minus_sign():
+            return cls(a, q, -z)
+
+
+class mathieucprime(MathieuBase):
+    r"""
+    The derivative $C^{\prime}(a,q,z)$ of the Mathieu Cosine function.
+
+    Explanation
+    ===========
+
+    This function is one solution of the Mathieu differential equation:
+
+    .. math ::
+        y(x)^{\prime\prime} + (a - 2 q \cos(2 x)) y(x) = 0
+
+    The other solution is the Mathieu Sine function.
+
+    Examples
+    ========
+
+    >>> from sympy import diff, mathieucprime
+    >>> from sympy.abc import a, q, z
+
+    >>> mathieucprime(a, q, z)
+    mathieucprime(a, q, z)
+
+    >>> mathieucprime(a, 0, z)
+    -sqrt(a)*sin(sqrt(a)*z)
+
+    >>> diff(mathieucprime(a, q, z), z)
+    (-a + 2*q*cos(2*z))*mathieuc(a, q, z)
+
+    See Also
+    ========
+
+    mathieus: Mathieu sine function
+    mathieuc: Mathieu cosine function
+    mathieusprime: Derivative of Mathieu sine function
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Mathieu_function
+    .. [2] https://dlmf.nist.gov/28
+    .. [3] https://mathworld.wolfram.com/MathieuFunction.html
+    .. [4] https://functions.wolfram.com/MathieuandSpheroidalFunctions/MathieuCPrime/
+
+    """
+
+    def fdiff(self, argindex=1):
+        if argindex == 3:
+            a, q, z = self.args
+            return (2*q*cos(2*z) - a)*mathieuc(a, q, z)
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    @classmethod
+    def eval(cls, a, q, z):
+        if q.is_Number and q.is_zero:
+            return -sqrt(a)*sin(sqrt(a)*z)
+        # Try to pull out factors of -1
+        if z.could_extract_minus_sign():
+            return -cls(a, q, -z)

llmeval-env/lib/python3.10/site-packages/sympy/functions/special/polynomials.py

@@ -0,0 +1,1447 @@
+"""
+This module mainly implements special orthogonal polynomials.
+
+See also functions.combinatorial.numbers which contains some
+combinatorial polynomials.
+
+"""
+
+from sympy.core import Rational
+from sympy.core.function import Function, ArgumentIndexError
+from sympy.core.singleton import S
+from sympy.core.symbol import Dummy
+from sympy.functions.combinatorial.factorials import binomial, factorial, RisingFactorial
+from sympy.functions.elementary.complexes import re
+from sympy.functions.elementary.exponential import exp
+from sympy.functions.elementary.integers import floor
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.trigonometric import cos, sec
+from sympy.functions.special.gamma_functions import gamma
+from sympy.functions.special.hyper import hyper
+from sympy.polys.orthopolys import (chebyshevt_poly, chebyshevu_poly,
+                                    gegenbauer_poly, hermite_poly, hermite_prob_poly,
+                                    jacobi_poly, laguerre_poly, legendre_poly)
+
+_x = Dummy('x')
+
+
+class OrthogonalPolynomial(Function):
+    """Base class for orthogonal polynomials.
+    """
+
+    @classmethod
+    def _eval_at_order(cls, n, x):
+        if n.is_integer and n >= 0:
+            return cls._ortho_poly(int(n), _x).subs(_x, x)
+
+    def _eval_conjugate(self):
+        return self.func(self.args[0], self.args[1].conjugate())
+
+#----------------------------------------------------------------------------
+# Jacobi polynomials
+#
+
+
+class jacobi(OrthogonalPolynomial):
+    r"""
+    Jacobi polynomial $P_n^{\left(\alpha, \beta\right)}(x)$.
+
+    Explanation
+    ===========
+
+    ``jacobi(n, alpha, beta, x)`` gives the $n$th Jacobi polynomial
+    in $x$, $P_n^{\left(\alpha, \beta\right)}(x)$.
+
+    The Jacobi polynomials are orthogonal on $[-1, 1]$ with respect
+    to the weight $\left(1-x\right)^\alpha \left(1+x\right)^\beta$.
+
+    Examples
+    ========
+
+    >>> from sympy import jacobi, S, conjugate, diff
+    >>> from sympy.abc import a, b, n, x
+
+    >>> jacobi(0, a, b, x)
+    1
+    >>> jacobi(1, a, b, x)
+    a/2 - b/2 + x*(a/2 + b/2 + 1)
+    >>> jacobi(2, a, b, x)
+    a**2/8 - a*b/4 - a/8 + b**2/8 - b/8 + x**2*(a**2/8 + a*b/4 + 7*a/8 + b**2/8 + 7*b/8 + 3/2) + x*(a**2/4 + 3*a/4 - b**2/4 - 3*b/4) - 1/2
+
+    >>> jacobi(n, a, b, x)
+    jacobi(n, a, b, x)
+
+    >>> jacobi(n, a, a, x)
+    RisingFactorial(a + 1, n)*gegenbauer(n,
+        a + 1/2, x)/RisingFactorial(2*a + 1, n)
+
+    >>> jacobi(n, 0, 0, x)
+    legendre(n, x)
+
+    >>> jacobi(n, S(1)/2, S(1)/2, x)
+    RisingFactorial(3/2, n)*chebyshevu(n, x)/factorial(n + 1)
+
+    >>> jacobi(n, -S(1)/2, -S(1)/2, x)
+    RisingFactorial(1/2, n)*chebyshevt(n, x)/factorial(n)
+
+    >>> jacobi(n, a, b, -x)
+    (-1)**n*jacobi(n, b, a, x)
+
+    >>> jacobi(n, a, b, 0)
+    gamma(a + n + 1)*hyper((-b - n, -n), (a + 1,), -1)/(2**n*factorial(n)*gamma(a + 1))
+    >>> jacobi(n, a, b, 1)
+    RisingFactorial(a + 1, n)/factorial(n)
+
+    >>> conjugate(jacobi(n, a, b, x))
+    jacobi(n, conjugate(a), conjugate(b), conjugate(x))
+
+    >>> diff(jacobi(n,a,b,x), x)
+    (a/2 + b/2 + n/2 + 1/2)*jacobi(n - 1, a + 1, b + 1, x)
+
+    See Also
+    ========
+
+    gegenbauer,
+    chebyshevt_root, chebyshevu, chebyshevu_root,
+    legendre, assoc_legendre,
+    hermite, hermite_prob,
+    laguerre, assoc_laguerre,
+    sympy.polys.orthopolys.jacobi_poly,
+    sympy.polys.orthopolys.gegenbauer_poly
+    sympy.polys.orthopolys.chebyshevt_poly
+    sympy.polys.orthopolys.chebyshevu_poly
+    sympy.polys.orthopolys.hermite_poly
+    sympy.polys.orthopolys.legendre_poly
+    sympy.polys.orthopolys.laguerre_poly
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Jacobi_polynomials
+    .. [2] https://mathworld.wolfram.com/JacobiPolynomial.html
+    .. [3] https://functions.wolfram.com/Polynomials/JacobiP/
+
+    """
+
+    @classmethod
+    def eval(cls, n, a, b, x):
+        # Simplify to other polynomials
+        # P^{a, a}_n(x)
+        if a == b:
+            if a == Rational(-1, 2):
+                return RisingFactorial(S.Half, n) / factorial(n) * chebyshevt(n, x)
+            elif a.is_zero:
+                return legendre(n, x)
+            elif a == S.Half:
+                return RisingFactorial(3*S.Half, n) / factorial(n + 1) * chebyshevu(n, x)
+            else:
+                return RisingFactorial(a + 1, n) / RisingFactorial(2*a + 1, n) * gegenbauer(n, a + S.Half, x)
+        elif b == -a:
+            # P^{a, -a}_n(x)
+            return gamma(n + a + 1) / gamma(n + 1) * (1 + x)**(a/2) / (1 - x)**(a/2) * assoc_legendre(n, -a, x)
+
+        if not n.is_Number:
+            # Symbolic result P^{a,b}_n(x)
+            # P^{a,b}_n(-x)  --->  (-1)**n * P^{b,a}_n(-x)
+            if x.could_extract_minus_sign():
+                return S.NegativeOne**n * jacobi(n, b, a, -x)
+            # We can evaluate for some special values of x
+            if x.is_zero:
+                return (2**(-n) * gamma(a + n + 1) / (gamma(a + 1) * factorial(n)) *
+                        hyper([-b - n, -n], [a + 1], -1))
+            if x == S.One:
+                return RisingFactorial(a + 1, n) / factorial(n)
+            elif x is S.Infinity:
+                if n.is_positive:
+                    # Make sure a+b+2*n \notin Z
+                    if (a + b + 2*n).is_integer:
+                        raise ValueError("Error. a + b + 2*n should not be an integer.")
+                    return RisingFactorial(a + b + n + 1, n) * S.Infinity
+        else:
+            # n is a given fixed integer, evaluate into polynomial
+            return jacobi_poly(n, a, b, x)
+
+    def fdiff(self, argindex=4):
+        from sympy.concrete.summations import Sum
+        if argindex == 1:
+            # Diff wrt n
+            raise ArgumentIndexError(self, argindex)
+        elif argindex == 2:
+            # Diff wrt a
+            n, a, b, x = self.args
+            k = Dummy("k")
+            f1 = 1 / (a + b + n + k + 1)
+            f2 = ((a + b + 2*k + 1) * RisingFactorial(b + k + 1, n - k) /
+                  ((n - k) * RisingFactorial(a + b + k + 1, n - k)))
+            return Sum(f1 * (jacobi(n, a, b, x) + f2*jacobi(k, a, b, x)), (k, 0, n - 1))
+        elif argindex == 3:
+            # Diff wrt b
+            n, a, b, x = self.args
+            k = Dummy("k")
+            f1 = 1 / (a + b + n + k + 1)
+            f2 = (-1)**(n - k) * ((a + b + 2*k + 1) * RisingFactorial(a + k + 1, n - k) /
+                  ((n - k) * RisingFactorial(a + b + k + 1, n - k)))
+            return Sum(f1 * (jacobi(n, a, b, x) + f2*jacobi(k, a, b, x)), (k, 0, n - 1))
+        elif argindex == 4:
+            # Diff wrt x
+            n, a, b, x = self.args
+            return S.Half * (a + b + n + 1) * jacobi(n - 1, a + 1, b + 1, x)
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    def _eval_rewrite_as_Sum(self, n, a, b, x, **kwargs):
+        from sympy.concrete.summations import Sum
+        # Make sure n \in N
+        if n.is_negative or n.is_integer is False:
+            raise ValueError("Error: n should be a non-negative integer.")
+        k = Dummy("k")
+        kern = (RisingFactorial(-n, k) * RisingFactorial(a + b + n + 1, k) * RisingFactorial(a + k + 1, n - k) /
+                factorial(k) * ((1 - x)/2)**k)
+        return 1 / factorial(n) * Sum(kern, (k, 0, n))
+
+    def _eval_rewrite_as_polynomial(self, n, a, b, x, **kwargs):
+        # This function is just kept for backwards compatibility
+        # but should not be used
+        return self._eval_rewrite_as_Sum(n, a, b, x, **kwargs)
+
+    def _eval_conjugate(self):
+        n, a, b, x = self.args
+        return self.func(n, a.conjugate(), b.conjugate(), x.conjugate())
+
+
+def jacobi_normalized(n, a, b, x):
+    r"""
+    Jacobi polynomial $P_n^{\left(\alpha, \beta\right)}(x)$.
+
+    Explanation
+    ===========
+
+    ``jacobi_normalized(n, alpha, beta, x)`` gives the $n$th
+    Jacobi polynomial in $x$, $P_n^{\left(\alpha, \beta\right)}(x)$.
+
+    The Jacobi polynomials are orthogonal on $[-1, 1]$ with respect
+    to the weight $\left(1-x\right)^\alpha \left(1+x\right)^\beta$.
+
+    This functions returns the polynomials normilzed:
+
+    .. math::
+
+        \int_{-1}^{1}
+          P_m^{\left(\alpha, \beta\right)}(x)
+          P_n^{\left(\alpha, \beta\right)}(x)
+          (1-x)^{\alpha} (1+x)^{\beta} \mathrm{d}x
+        = \delta_{m,n}
+
+    Examples
+    ========
+
+    >>> from sympy import jacobi_normalized
+    >>> from sympy.abc import n,a,b,x
+
+    >>> jacobi_normalized(n, a, b, x)
+    jacobi(n, a, b, x)/sqrt(2**(a + b + 1)*gamma(a + n + 1)*gamma(b + n + 1)/((a + b + 2*n + 1)*factorial(n)*gamma(a + b + n + 1)))
+
+    Parameters
+    ==========
+
+    n : integer degree of polynomial
+
+    a : alpha value
+
+    b : beta value
+
+    x : symbol
+
+    See Also
+    ========
+
+    gegenbauer,
+    chebyshevt_root, chebyshevu, chebyshevu_root,
+    legendre, assoc_legendre,
+    hermite, hermite_prob,
+    laguerre, assoc_laguerre,
+    sympy.polys.orthopolys.jacobi_poly,
+    sympy.polys.orthopolys.gegenbauer_poly
+    sympy.polys.orthopolys.chebyshevt_poly
+    sympy.polys.orthopolys.chebyshevu_poly
+    sympy.polys.orthopolys.hermite_poly
+    sympy.polys.orthopolys.legendre_poly
+    sympy.polys.orthopolys.laguerre_poly
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Jacobi_polynomials
+    .. [2] https://mathworld.wolfram.com/JacobiPolynomial.html
+    .. [3] https://functions.wolfram.com/Polynomials/JacobiP/
+
+    """
+    nfactor = (S(2)**(a + b + 1) * (gamma(n + a + 1) * gamma(n + b + 1))
+               / (2*n + a + b + 1) / (factorial(n) * gamma(n + a + b + 1)))
+
+    return jacobi(n, a, b, x) / sqrt(nfactor)
+
+
+#----------------------------------------------------------------------------
+# Gegenbauer polynomials
+#
+
+
+class gegenbauer(OrthogonalPolynomial):
+    r"""
+    Gegenbauer polynomial $C_n^{\left(\alpha\right)}(x)$.
+
+    Explanation
+    ===========
+
+    ``gegenbauer(n, alpha, x)`` gives the $n$th Gegenbauer polynomial
+    in $x$, $C_n^{\left(\alpha\right)}(x)$.
+
+    The Gegenbauer polynomials are orthogonal on $[-1, 1]$ with
+    respect to the weight $\left(1-x^2\right)^{\alpha-\frac{1}{2}}$.
+
+    Examples
+    ========
+
+    >>> from sympy import gegenbauer, conjugate, diff
+    >>> from sympy.abc import n,a,x
+    >>> gegenbauer(0, a, x)
+    1
+    >>> gegenbauer(1, a, x)
+    2*a*x
+    >>> gegenbauer(2, a, x)
+    -a + x**2*(2*a**2 + 2*a)
+    >>> gegenbauer(3, a, x)
+    x**3*(4*a**3/3 + 4*a**2 + 8*a/3) + x*(-2*a**2 - 2*a)
+
+    >>> gegenbauer(n, a, x)
+    gegenbauer(n, a, x)
+    >>> gegenbauer(n, a, -x)
+    (-1)**n*gegenbauer(n, a, x)
+
+    >>> gegenbauer(n, a, 0)
+    2**n*sqrt(pi)*gamma(a + n/2)/(gamma(a)*gamma(1/2 - n/2)*gamma(n + 1))
+    >>> gegenbauer(n, a, 1)
+    gamma(2*a + n)/(gamma(2*a)*gamma(n + 1))
+
+    >>> conjugate(gegenbauer(n, a, x))
+    gegenbauer(n, conjugate(a), conjugate(x))
+
+    >>> diff(gegenbauer(n, a, x), x)
+    2*a*gegenbauer(n - 1, a + 1, x)
+
+    See Also
+    ========
+
+    jacobi,
+    chebyshevt_root, chebyshevu, chebyshevu_root,
+    legendre, assoc_legendre,
+    hermite, hermite_prob,
+    laguerre, assoc_laguerre,
+    sympy.polys.orthopolys.jacobi_poly
+    sympy.polys.orthopolys.gegenbauer_poly
+    sympy.polys.orthopolys.chebyshevt_poly
+    sympy.polys.orthopolys.chebyshevu_poly
+    sympy.polys.orthopolys.hermite_poly
+    sympy.polys.orthopolys.hermite_prob_poly
+    sympy.polys.orthopolys.legendre_poly
+    sympy.polys.orthopolys.laguerre_poly
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Gegenbauer_polynomials
+    .. [2] https://mathworld.wolfram.com/GegenbauerPolynomial.html
+    .. [3] https://functions.wolfram.com/Polynomials/GegenbauerC3/
+
+    """
+
+    @classmethod
+    def eval(cls, n, a, x):
+        # For negative n the polynomials vanish
+        # See https://functions.wolfram.com/Polynomials/GegenbauerC3/03/01/03/0012/
+        if n.is_negative:
+            return S.Zero
+
+        # Some special values for fixed a
+        if a == S.Half:
+            return legendre(n, x)
+        elif a == S.One:
+            return chebyshevu(n, x)
+        elif a == S.NegativeOne:
+            return S.Zero
+
+        if not n.is_Number:
+            # Handle this before the general sign extraction rule
+            if x == S.NegativeOne:
+                if (re(a) > S.Half) == True:
+                    return S.ComplexInfinity
+                else:
+                    return (cos(S.Pi*(a+n)) * sec(S.Pi*a) * gamma(2*a+n) /
+                                (gamma(2*a) * gamma(n+1)))
+
+            # Symbolic result C^a_n(x)
+            # C^a_n(-x)  --->  (-1)**n * C^a_n(x)
+            if x.could_extract_minus_sign():
+                return S.NegativeOne**n * gegenbauer(n, a, -x)
+            # We can evaluate for some special values of x
+            if x.is_zero:
+                return (2**n * sqrt(S.Pi) * gamma(a + S.Half*n) /
+                        (gamma((1 - n)/2) * gamma(n + 1) * gamma(a)) )
+            if x == S.One:
+                return gamma(2*a + n) / (gamma(2*a) * gamma(n + 1))
+            elif x is S.Infinity:
+                if n.is_positive:
+                    return RisingFactorial(a, n) * S.Infinity
+        else:
+            # n is a given fixed integer, evaluate into polynomial
+            return gegenbauer_poly(n, a, x)
+
+    def fdiff(self, argindex=3):
+        from sympy.concrete.summations import Sum
+        if argindex == 1:
+            # Diff wrt n
+            raise ArgumentIndexError(self, argindex)
+        elif argindex == 2:
+            # Diff wrt a
+            n, a, x = self.args
+            k = Dummy("k")
+            factor1 = 2 * (1 + (-1)**(n - k)) * (k + a) / ((k +
+                           n + 2*a) * (n - k))
+            factor2 = 2*(k + 1) / ((k + 2*a) * (2*k + 2*a + 1)) + \
+                2 / (k + n + 2*a)
+            kern = factor1*gegenbauer(k, a, x) + factor2*gegenbauer(n, a, x)
+            return Sum(kern, (k, 0, n - 1))
+        elif argindex == 3:
+            # Diff wrt x
+            n, a, x = self.args
+            return 2*a*gegenbauer(n - 1, a + 1, x)
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    def _eval_rewrite_as_Sum(self, n, a, x, **kwargs):
+        from sympy.concrete.summations import Sum
+        k = Dummy("k")
+        kern = ((-1)**k * RisingFactorial(a, n - k) * (2*x)**(n - 2*k) /
+                (factorial(k) * factorial(n - 2*k)))
+        return Sum(kern, (k, 0, floor(n/2)))
+
+    def _eval_rewrite_as_polynomial(self, n, a, x, **kwargs):
+        # This function is just kept for backwards compatibility
+        # but should not be used
+        return self._eval_rewrite_as_Sum(n, a, x, **kwargs)
+
+    def _eval_conjugate(self):
+        n, a, x = self.args
+        return self.func(n, a.conjugate(), x.conjugate())
+
+#----------------------------------------------------------------------------
+# Chebyshev polynomials of first and second kind
+#
+
+
+class chebyshevt(OrthogonalPolynomial):
+    r"""
+    Chebyshev polynomial of the first kind, $T_n(x)$.
+
+    Explanation
+    ===========
+
+    ``chebyshevt(n, x)`` gives the $n$th Chebyshev polynomial (of the first
+    kind) in $x$, $T_n(x)$.
+
+    The Chebyshev polynomials of the first kind are orthogonal on
+    $[-1, 1]$ with respect to the weight $\frac{1}{\sqrt{1-x^2}}$.
+
+    Examples
+    ========
+
+    >>> from sympy import chebyshevt, diff
+    >>> from sympy.abc import n,x
+    >>> chebyshevt(0, x)
+    1
+    >>> chebyshevt(1, x)
+    x
+    >>> chebyshevt(2, x)
+    2*x**2 - 1
+
+    >>> chebyshevt(n, x)
+    chebyshevt(n, x)
+    >>> chebyshevt(n, -x)
+    (-1)**n*chebyshevt(n, x)
+    >>> chebyshevt(-n, x)
+    chebyshevt(n, x)
+
+    >>> chebyshevt(n, 0)
+    cos(pi*n/2)
+    >>> chebyshevt(n, -1)
+    (-1)**n
+
+    >>> diff(chebyshevt(n, x), x)
+    n*chebyshevu(n - 1, x)
+
+    See Also
+    ========
+
+    jacobi, gegenbauer,
+    chebyshevt_root, chebyshevu, chebyshevu_root,
+    legendre, assoc_legendre,
+    hermite, hermite_prob,
+    laguerre, assoc_laguerre,
+    sympy.polys.orthopolys.jacobi_poly
+    sympy.polys.orthopolys.gegenbauer_poly
+    sympy.polys.orthopolys.chebyshevt_poly
+    sympy.polys.orthopolys.chebyshevu_poly
+    sympy.polys.orthopolys.hermite_poly
+    sympy.polys.orthopolys.hermite_prob_poly
+    sympy.polys.orthopolys.legendre_poly
+    sympy.polys.orthopolys.laguerre_poly
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Chebyshev_polynomial
+    .. [2] https://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html
+    .. [3] https://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html
+    .. [4] https://functions.wolfram.com/Polynomials/ChebyshevT/
+    .. [5] https://functions.wolfram.com/Polynomials/ChebyshevU/
+
+    """
+
+    _ortho_poly = staticmethod(chebyshevt_poly)
+
+    @classmethod
+    def eval(cls, n, x):
+        if not n.is_Number:
+            # Symbolic result T_n(x)
+            # T_n(-x)  --->  (-1)**n * T_n(x)
+            if x.could_extract_minus_sign():
+                return S.NegativeOne**n * chebyshevt(n, -x)
+            # T_{-n}(x)  --->  T_n(x)
+            if n.could_extract_minus_sign():
+                return chebyshevt(-n, x)
+            # We can evaluate for some special values of x
+            if x.is_zero:
+                return cos(S.Half * S.Pi * n)
+            if x == S.One:
+                return S.One
+            elif x is S.Infinity:
+                return S.Infinity
+        else:
+            # n is a given fixed integer, evaluate into polynomial
+            if n.is_negative:
+                # T_{-n}(x) == T_n(x)
+                return cls._eval_at_order(-n, x)
+            else:
+                return cls._eval_at_order(n, x)
+
+    def fdiff(self, argindex=2):
+        if argindex == 1:
+            # Diff wrt n
+            raise ArgumentIndexError(self, argindex)
+        elif argindex == 2:
+            # Diff wrt x
+            n, x = self.args
+            return n * chebyshevu(n - 1, x)
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    def _eval_rewrite_as_Sum(self, n, x, **kwargs):
+        from sympy.concrete.summations import Sum
+        k = Dummy("k")
+        kern = binomial(n, 2*k) * (x**2 - 1)**k * x**(n - 2*k)
+        return Sum(kern, (k, 0, floor(n/2)))
+
+    def _eval_rewrite_as_polynomial(self, n, x, **kwargs):
+        # This function is just kept for backwards compatibility
+        # but should not be used
+        return self._eval_rewrite_as_Sum(n, x, **kwargs)
+
+
+class chebyshevu(OrthogonalPolynomial):
+    r"""
+    Chebyshev polynomial of the second kind, $U_n(x)$.
+
+    Explanation
+    ===========
+
+    ``chebyshevu(n, x)`` gives the $n$th Chebyshev polynomial of the second
+    kind in x, $U_n(x)$.
+
+    The Chebyshev polynomials of the second kind are orthogonal on
+    $[-1, 1]$ with respect to the weight $\sqrt{1-x^2}$.
+
+    Examples
+    ========
+
+    >>> from sympy import chebyshevu, diff
+    >>> from sympy.abc import n,x
+    >>> chebyshevu(0, x)
+    1
+    >>> chebyshevu(1, x)
+    2*x
+    >>> chebyshevu(2, x)
+    4*x**2 - 1
+
+    >>> chebyshevu(n, x)
+    chebyshevu(n, x)
+    >>> chebyshevu(n, -x)
+    (-1)**n*chebyshevu(n, x)
+    >>> chebyshevu(-n, x)
+    -chebyshevu(n - 2, x)
+
+    >>> chebyshevu(n, 0)
+    cos(pi*n/2)
+    >>> chebyshevu(n, 1)
+    n + 1
+
+    >>> diff(chebyshevu(n, x), x)
+    (-x*chebyshevu(n, x) + (n + 1)*chebyshevt(n + 1, x))/(x**2 - 1)
+
+    See Also
+    ========
+
+    jacobi, gegenbauer,
+    chebyshevt, chebyshevt_root, chebyshevu_root,
+    legendre, assoc_legendre,
+    hermite, hermite_prob,
+    laguerre, assoc_laguerre,
+    sympy.polys.orthopolys.jacobi_poly
+    sympy.polys.orthopolys.gegenbauer_poly
+    sympy.polys.orthopolys.chebyshevt_poly
+    sympy.polys.orthopolys.chebyshevu_poly
+    sympy.polys.orthopolys.hermite_poly
+    sympy.polys.orthopolys.hermite_prob_poly
+    sympy.polys.orthopolys.legendre_poly
+    sympy.polys.orthopolys.laguerre_poly
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Chebyshev_polynomial
+    .. [2] https://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html
+    .. [3] https://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html
+    .. [4] https://functions.wolfram.com/Polynomials/ChebyshevT/
+    .. [5] https://functions.wolfram.com/Polynomials/ChebyshevU/
+
+    """
+
+    _ortho_poly = staticmethod(chebyshevu_poly)
+
+    @classmethod
+    def eval(cls, n, x):
+        if not n.is_Number:
+            # Symbolic result U_n(x)
+            # U_n(-x)  --->  (-1)**n * U_n(x)
+            if x.could_extract_minus_sign():
+                return S.NegativeOne**n * chebyshevu(n, -x)
+            # U_{-n}(x)  --->  -U_{n-2}(x)
+            if n.could_extract_minus_sign():
+                if n == S.NegativeOne:
+                    # n can not be -1 here
+                    return S.Zero
+                elif not (-n - 2).could_extract_minus_sign():
+                    return -chebyshevu(-n - 2, x)
+            # We can evaluate for some special values of x
+            if x.is_zero:
+                return cos(S.Half * S.Pi * n)
+            if x == S.One:
+                return S.One + n
+            elif x is S.Infinity:
+                return S.Infinity
+        else:
+            # n is a given fixed integer, evaluate into polynomial
+            if n.is_negative:
+                # U_{-n}(x)  --->  -U_{n-2}(x)
+                if n == S.NegativeOne:
+                    return S.Zero
+                else:
+                    return -cls._eval_at_order(-n - 2, x)
+            else:
+                return cls._eval_at_order(n, x)
+
+    def fdiff(self, argindex=2):
+        if argindex == 1:
+            # Diff wrt n
+            raise ArgumentIndexError(self, argindex)
+        elif argindex == 2:
+            # Diff wrt x
+            n, x = self.args
+            return ((n + 1) * chebyshevt(n + 1, x) - x * chebyshevu(n, x)) / (x**2 - 1)
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    def _eval_rewrite_as_Sum(self, n, x, **kwargs):
+        from sympy.concrete.summations import Sum
+        k = Dummy("k")
+        kern = S.NegativeOne**k * factorial(
+            n - k) * (2*x)**(n - 2*k) / (factorial(k) * factorial(n - 2*k))
+        return Sum(kern, (k, 0, floor(n/2)))
+
+    def _eval_rewrite_as_polynomial(self, n, x, **kwargs):
+        # This function is just kept for backwards compatibility
+        # but should not be used
+        return self._eval_rewrite_as_Sum(n, x, **kwargs)
+
+
+class chebyshevt_root(Function):
+    r"""
+    ``chebyshev_root(n, k)`` returns the $k$th root (indexed from zero) of
+    the $n$th Chebyshev polynomial of the first kind; that is, if
+    $0 \le k < n$, ``chebyshevt(n, chebyshevt_root(n, k)) == 0``.
+
+    Examples
+    ========
+
+    >>> from sympy import chebyshevt, chebyshevt_root
+    >>> chebyshevt_root(3, 2)
+    -sqrt(3)/2
+    >>> chebyshevt(3, chebyshevt_root(3, 2))
+    0
+
+    See Also
+    ========
+
+    jacobi, gegenbauer,
+    chebyshevt, chebyshevu, chebyshevu_root,
+    legendre, assoc_legendre,
+    hermite, hermite_prob,
+    laguerre, assoc_laguerre,
+    sympy.polys.orthopolys.jacobi_poly
+    sympy.polys.orthopolys.gegenbauer_poly
+    sympy.polys.orthopolys.chebyshevt_poly
+    sympy.polys.orthopolys.chebyshevu_poly
+    sympy.polys.orthopolys.hermite_poly
+    sympy.polys.orthopolys.hermite_prob_poly
+    sympy.polys.orthopolys.legendre_poly
+    sympy.polys.orthopolys.laguerre_poly
+    """
+
+    @classmethod
+    def eval(cls, n, k):
+        if not ((0 <= k) and (k < n)):
+            raise ValueError("must have 0 <= k < n, "
+                "got k = %s and n = %s" % (k, n))
+        return cos(S.Pi*(2*k + 1)/(2*n))
+
+
+class chebyshevu_root(Function):
+    r"""
+    ``chebyshevu_root(n, k)`` returns the $k$th root (indexed from zero) of the
+    $n$th Chebyshev polynomial of the second kind; that is, if $0 \le k < n$,
+    ``chebyshevu(n, chebyshevu_root(n, k)) == 0``.
+
+    Examples
+    ========
+
+    >>> from sympy import chebyshevu, chebyshevu_root
+    >>> chebyshevu_root(3, 2)
+    -sqrt(2)/2
+    >>> chebyshevu(3, chebyshevu_root(3, 2))
+    0
+
+    See Also
+    ========
+
+    chebyshevt, chebyshevt_root, chebyshevu,
+    legendre, assoc_legendre,
+    hermite, hermite_prob,
+    laguerre, assoc_laguerre,
+    sympy.polys.orthopolys.jacobi_poly
+    sympy.polys.orthopolys.gegenbauer_poly
+    sympy.polys.orthopolys.chebyshevt_poly
+    sympy.polys.orthopolys.chebyshevu_poly
+    sympy.polys.orthopolys.hermite_poly
+    sympy.polys.orthopolys.hermite_prob_poly
+    sympy.polys.orthopolys.legendre_poly
+    sympy.polys.orthopolys.laguerre_poly
+    """
+
+
+    @classmethod
+    def eval(cls, n, k):
+        if not ((0 <= k) and (k < n)):
+            raise ValueError("must have 0 <= k < n, "
+                "got k = %s and n = %s" % (k, n))
+        return cos(S.Pi*(k + 1)/(n + 1))
+
+#----------------------------------------------------------------------------
+# Legendre polynomials and Associated Legendre polynomials
+#
+
+
+class legendre(OrthogonalPolynomial):
+    r"""
+    ``legendre(n, x)`` gives the $n$th Legendre polynomial of $x$, $P_n(x)$
+
+    Explanation
+    ===========
+
+    The Legendre polynomials are orthogonal on $[-1, 1]$ with respect to
+    the constant weight 1. They satisfy $P_n(1) = 1$ for all $n$; further,
+    $P_n$ is odd for odd $n$ and even for even $n$.
+
+    Examples
+    ========
+
+    >>> from sympy import legendre, diff
+    >>> from sympy.abc import x, n
+    >>> legendre(0, x)
+    1
+    >>> legendre(1, x)
+    x
+    >>> legendre(2, x)
+    3*x**2/2 - 1/2
+    >>> legendre(n, x)
+    legendre(n, x)
+    >>> diff(legendre(n,x), x)
+    n*(x*legendre(n, x) - legendre(n - 1, x))/(x**2 - 1)
+
+    See Also
+    ========
+
+    jacobi, gegenbauer,
+    chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,
+    assoc_legendre,
+    hermite, hermite_prob,
+    laguerre, assoc_laguerre,
+    sympy.polys.orthopolys.jacobi_poly
+    sympy.polys.orthopolys.gegenbauer_poly
+    sympy.polys.orthopolys.chebyshevt_poly
+    sympy.polys.orthopolys.chebyshevu_poly
+    sympy.polys.orthopolys.hermite_poly
+    sympy.polys.orthopolys.hermite_prob_poly
+    sympy.polys.orthopolys.legendre_poly
+    sympy.polys.orthopolys.laguerre_poly
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Legendre_polynomial
+    .. [2] https://mathworld.wolfram.com/LegendrePolynomial.html
+    .. [3] https://functions.wolfram.com/Polynomials/LegendreP/
+    .. [4] https://functions.wolfram.com/Polynomials/LegendreP2/
+
+    """
+
+    _ortho_poly = staticmethod(legendre_poly)
+
+    @classmethod
+    def eval(cls, n, x):
+        if not n.is_Number:
+            # Symbolic result L_n(x)
+            # L_n(-x)  --->  (-1)**n * L_n(x)
+            if x.could_extract_minus_sign():
+                return S.NegativeOne**n * legendre(n, -x)
+            # L_{-n}(x)  --->  L_{n-1}(x)
+            if n.could_extract_minus_sign() and not(-n - 1).could_extract_minus_sign():
+                return legendre(-n - S.One, x)
+            # We can evaluate for some special values of x
+            if x.is_zero:
+                return sqrt(S.Pi)/(gamma(S.Half - n/2)*gamma(S.One + n/2))
+            elif x == S.One:
+                return S.One
+            elif x is S.Infinity:
+                return S.Infinity
+        else:
+            # n is a given fixed integer, evaluate into polynomial;
+            # L_{-n}(x)  --->  L_{n-1}(x)
+            if n.is_negative:
+                n = -n - S.One
+            return cls._eval_at_order(n, x)
+
+    def fdiff(self, argindex=2):
+        if argindex == 1:
+            # Diff wrt n
+            raise ArgumentIndexError(self, argindex)
+        elif argindex == 2:
+            # Diff wrt x
+            # Find better formula, this is unsuitable for x = +/-1
+            # https://www.autodiff.org/ad16/Oral/Buecker_Legendre.pdf says
+            # at x = 1:
+            #    n*(n + 1)/2            , m = 0
+            #    oo                     , m = 1
+            #    -(n-1)*n*(n+1)*(n+2)/4 , m = 2
+            #    0                      , m = 3, 4, ..., n
+            #
+            # at x = -1
+            #    (-1)**(n+1)*n*(n + 1)/2       , m = 0
+            #    (-1)**n*oo                    , m = 1
+            #    (-1)**n*(n-1)*n*(n+1)*(n+2)/4 , m = 2
+            #    0                             , m = 3, 4, ..., n
+            n, x = self.args
+            return n/(x**2 - 1)*(x*legendre(n, x) - legendre(n - 1, x))
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    def _eval_rewrite_as_Sum(self, n, x, **kwargs):
+        from sympy.concrete.summations import Sum
+        k = Dummy("k")
+        kern = S.NegativeOne**k*binomial(n, k)**2*((1 + x)/2)**(n - k)*((1 - x)/2)**k
+        return Sum(kern, (k, 0, n))
+
+    def _eval_rewrite_as_polynomial(self, n, x, **kwargs):
+        # This function is just kept for backwards compatibility
+        # but should not be used
+        return self._eval_rewrite_as_Sum(n, x, **kwargs)
+
+
+class assoc_legendre(Function):
+    r"""
+    ``assoc_legendre(n, m, x)`` gives $P_n^m(x)$, where $n$ and $m$ are
+    the degree and order or an expression which is related to the nth
+    order Legendre polynomial, $P_n(x)$ in the following manner:
+
+    .. math::
+        P_n^m(x) = (-1)^m (1 - x^2)^{\frac{m}{2}}
+                   \frac{\mathrm{d}^m P_n(x)}{\mathrm{d} x^m}
+
+    Explanation
+    ===========
+
+    Associated Legendre polynomials are orthogonal on $[-1, 1]$ with:
+
+    - weight $= 1$            for the same $m$ and different $n$.
+    - weight $= \frac{1}{1-x^2}$   for the same $n$ and different $m$.
+
+    Examples
+    ========
+
+    >>> from sympy import assoc_legendre
+    >>> from sympy.abc import x, m, n
+    >>> assoc_legendre(0,0, x)
+    1
+    >>> assoc_legendre(1,0, x)
+    x
+    >>> assoc_legendre(1,1, x)
+    -sqrt(1 - x**2)
+    >>> assoc_legendre(n,m,x)
+    assoc_legendre(n, m, x)
+
+    See Also
+    ========
+
+    jacobi, gegenbauer,
+    chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,
+    legendre,
+    hermite, hermite_prob,
+    laguerre, assoc_laguerre,
+    sympy.polys.orthopolys.jacobi_poly
+    sympy.polys.orthopolys.gegenbauer_poly
+    sympy.polys.orthopolys.chebyshevt_poly
+    sympy.polys.orthopolys.chebyshevu_poly
+    sympy.polys.orthopolys.hermite_poly
+    sympy.polys.orthopolys.hermite_prob_poly
+    sympy.polys.orthopolys.legendre_poly
+    sympy.polys.orthopolys.laguerre_poly
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Associated_Legendre_polynomials
+    .. [2] https://mathworld.wolfram.com/LegendrePolynomial.html
+    .. [3] https://functions.wolfram.com/Polynomials/LegendreP/
+    .. [4] https://functions.wolfram.com/Polynomials/LegendreP2/
+
+    """
+
+    @classmethod
+    def _eval_at_order(cls, n, m):
+        P = legendre_poly(n, _x, polys=True).diff((_x, m))
+        return S.NegativeOne**m * (1 - _x**2)**Rational(m, 2) * P.as_expr()
+
+    @classmethod
+    def eval(cls, n, m, x):
+        if m.could_extract_minus_sign():
+            # P^{-m}_n  --->  F * P^m_n
+            return S.NegativeOne**(-m) * (factorial(m + n)/factorial(n - m)) * assoc_legendre(n, -m, x)
+        if m == 0:
+            # P^0_n  --->  L_n
+            return legendre(n, x)
+        if x == 0:
+            return 2**m*sqrt(S.Pi) / (gamma((1 - m - n)/2)*gamma(1 - (m - n)/2))
+        if n.is_Number and m.is_Number and n.is_integer and m.is_integer:
+            if n.is_negative:
+                raise ValueError("%s : 1st index must be nonnegative integer (got %r)" % (cls, n))
+            if abs(m) > n:
+                raise ValueError("%s : abs('2nd index') must be <= '1st index' (got %r, %r)" % (cls, n, m))
+            return cls._eval_at_order(int(n), abs(int(m))).subs(_x, x)
+
+    def fdiff(self, argindex=3):
+        if argindex == 1:
+            # Diff wrt n
+            raise ArgumentIndexError(self, argindex)
+        elif argindex == 2:
+            # Diff wrt m
+            raise ArgumentIndexError(self, argindex)
+        elif argindex == 3:
+            # Diff wrt x
+            # Find better formula, this is unsuitable for x = 1
+            n, m, x = self.args
+            return 1/(x**2 - 1)*(x*n*assoc_legendre(n, m, x) - (m + n)*assoc_legendre(n - 1, m, x))
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    def _eval_rewrite_as_Sum(self, n, m, x, **kwargs):
+        from sympy.concrete.summations import Sum
+        k = Dummy("k")
+        kern = factorial(2*n - 2*k)/(2**n*factorial(n - k)*factorial(
+            k)*factorial(n - 2*k - m))*S.NegativeOne**k*x**(n - m - 2*k)
+        return (1 - x**2)**(m/2) * Sum(kern, (k, 0, floor((n - m)*S.Half)))
+
+    def _eval_rewrite_as_polynomial(self, n, m, x, **kwargs):
+        # This function is just kept for backwards compatibility
+        # but should not be used
+        return self._eval_rewrite_as_Sum(n, m, x, **kwargs)
+
+    def _eval_conjugate(self):
+        n, m, x = self.args
+        return self.func(n, m.conjugate(), x.conjugate())
+
+#----------------------------------------------------------------------------
+# Hermite polynomials
+#
+
+
+class hermite(OrthogonalPolynomial):
+    r"""
+    ``hermite(n, x)`` gives the $n$th Hermite polynomial in $x$, $H_n(x)$.
+
+    Explanation
+    ===========
+
+    The Hermite polynomials are orthogonal on $(-\infty, \infty)$
+    with respect to the weight $\exp\left(-x^2\right)$.
+
+    Examples
+    ========
+
+    >>> from sympy import hermite, diff
+    >>> from sympy.abc import x, n
+    >>> hermite(0, x)
+    1
+    >>> hermite(1, x)
+    2*x
+    >>> hermite(2, x)
+    4*x**2 - 2
+    >>> hermite(n, x)
+    hermite(n, x)
+    >>> diff(hermite(n,x), x)
+    2*n*hermite(n - 1, x)
+    >>> hermite(n, -x)
+    (-1)**n*hermite(n, x)
+
+    See Also
+    ========
+
+    jacobi, gegenbauer,
+    chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,
+    legendre, assoc_legendre,
+    hermite_prob,
+    laguerre, assoc_laguerre,
+    sympy.polys.orthopolys.jacobi_poly
+    sympy.polys.orthopolys.gegenbauer_poly
+    sympy.polys.orthopolys.chebyshevt_poly
+    sympy.polys.orthopolys.chebyshevu_poly
+    sympy.polys.orthopolys.hermite_poly
+    sympy.polys.orthopolys.hermite_prob_poly
+    sympy.polys.orthopolys.legendre_poly
+    sympy.polys.orthopolys.laguerre_poly
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Hermite_polynomial
+    .. [2] https://mathworld.wolfram.com/HermitePolynomial.html
+    .. [3] https://functions.wolfram.com/Polynomials/HermiteH/
+
+    """
+
+    _ortho_poly = staticmethod(hermite_poly)
+
+    @classmethod
+    def eval(cls, n, x):
+        if not n.is_Number:
+            # Symbolic result H_n(x)
+            # H_n(-x)  --->  (-1)**n * H_n(x)
+            if x.could_extract_minus_sign():
+                return S.NegativeOne**n * hermite(n, -x)
+            # We can evaluate for some special values of x
+            if x.is_zero:
+                return 2**n * sqrt(S.Pi) / gamma((S.One - n)/2)
+            elif x is S.Infinity:
+                return S.Infinity
+        else:
+            # n is a given fixed integer, evaluate into polynomial
+            if n.is_negative:
+                raise ValueError(
+                    "The index n must be nonnegative integer (got %r)" % n)
+            else:
+                return cls._eval_at_order(n, x)
+
+    def fdiff(self, argindex=2):
+        if argindex == 1:
+            # Diff wrt n
+            raise ArgumentIndexError(self, argindex)
+        elif argindex == 2:
+            # Diff wrt x
+            n, x = self.args
+            return 2*n*hermite(n - 1, x)
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    def _eval_rewrite_as_Sum(self, n, x, **kwargs):
+        from sympy.concrete.summations import Sum
+        k = Dummy("k")
+        kern = S.NegativeOne**k / (factorial(k)*factorial(n - 2*k)) * (2*x)**(n - 2*k)
+        return factorial(n)*Sum(kern, (k, 0, floor(n/2)))
+
+    def _eval_rewrite_as_polynomial(self, n, x, **kwargs):
+        # This function is just kept for backwards compatibility
+        # but should not be used
+        return self._eval_rewrite_as_Sum(n, x, **kwargs)
+
+    def _eval_rewrite_as_hermite_prob(self, n, x, **kwargs):
+        return sqrt(2)**n * hermite_prob(n, x*sqrt(2))
+
+
+class hermite_prob(OrthogonalPolynomial):
+    r"""
+    ``hermite_prob(n, x)`` gives the $n$th probabilist's Hermite polynomial
+    in $x$, $He_n(x)$.
+
+    Explanation
+    ===========
+
+    The probabilist's Hermite polynomials are orthogonal on $(-\infty, \infty)$
+    with respect to the weight $\exp\left(-\frac{x^2}{2}\right)$. They are monic
+    polynomials, related to the plain Hermite polynomials (:py:class:`~.hermite`) by
+
+    .. math :: He_n(x) = 2^{-n/2} H_n(x/\sqrt{2})
+
+    Examples
+    ========
+
+    >>> from sympy import hermite_prob, diff, I
+    >>> from sympy.abc import x, n
+    >>> hermite_prob(1, x)
+    x
+    >>> hermite_prob(5, x)
+    x**5 - 10*x**3 + 15*x
+    >>> diff(hermite_prob(n,x), x)
+    n*hermite_prob(n - 1, x)
+    >>> hermite_prob(n, -x)
+    (-1)**n*hermite_prob(n, x)
+
+    The sum of absolute values of coefficients of $He_n(x)$ is the number of
+    matchings in the complete graph $K_n$ or telephone number, A000085 in the OEIS:
+
+    >>> [hermite_prob(n,I) / I**n for n in range(11)]
+    [1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496]
+
+    See Also
+    ========
+
+    jacobi, gegenbauer,
+    chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,
+    legendre, assoc_legendre,
+    hermite,
+    laguerre, assoc_laguerre,
+    sympy.polys.orthopolys.jacobi_poly
+    sympy.polys.orthopolys.gegenbauer_poly
+    sympy.polys.orthopolys.chebyshevt_poly
+    sympy.polys.orthopolys.chebyshevu_poly
+    sympy.polys.orthopolys.hermite_poly
+    sympy.polys.orthopolys.hermite_prob_poly
+    sympy.polys.orthopolys.legendre_poly
+    sympy.polys.orthopolys.laguerre_poly
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Hermite_polynomial
+    .. [2] https://mathworld.wolfram.com/HermitePolynomial.html
+    """
+
+    _ortho_poly = staticmethod(hermite_prob_poly)
+
+    @classmethod
+    def eval(cls, n, x):
+        if not n.is_Number:
+            if x.could_extract_minus_sign():
+                return S.NegativeOne**n * hermite_prob(n, -x)
+            if x.is_zero:
+                return sqrt(S.Pi) / gamma((S.One-n) / 2)
+            elif x is S.Infinity:
+                return S.Infinity
+        else:
+            if n.is_negative:
+                ValueError("n must be a nonnegative integer, not %r" % n)
+            else:
+                return cls._eval_at_order(n, x)
+
+    def fdiff(self, argindex=2):
+        if argindex == 2:
+            n, x = self.args
+            return n*hermite_prob(n-1, x)
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    def _eval_rewrite_as_Sum(self, n, x, **kwargs):
+        from sympy.concrete.summations import Sum
+        k = Dummy("k")
+        kern = (-S.Half)**k * x**(n-2*k) / (factorial(k) * factorial(n-2*k))
+        return factorial(n)*Sum(kern, (k, 0, floor(n/2)))
+
+    def _eval_rewrite_as_polynomial(self, n, x, **kwargs):
+        # This function is just kept for backwards compatibility
+        # but should not be used
+        return self._eval_rewrite_as_Sum(n, x, **kwargs)
+
+    def _eval_rewrite_as_hermite(self, n, x, **kwargs):
+        return sqrt(2)**(-n) * hermite(n, x/sqrt(2))
+
+
+#----------------------------------------------------------------------------
+# Laguerre polynomials
+#
+
+
+class laguerre(OrthogonalPolynomial):
+    r"""
+    Returns the $n$th Laguerre polynomial in $x$, $L_n(x)$.
+
+    Examples
+    ========
+
+    >>> from sympy import laguerre, diff
+    >>> from sympy.abc import x, n
+    >>> laguerre(0, x)
+    1
+    >>> laguerre(1, x)
+    1 - x
+    >>> laguerre(2, x)
+    x**2/2 - 2*x + 1
+    >>> laguerre(3, x)
+    -x**3/6 + 3*x**2/2 - 3*x + 1
+
+    >>> laguerre(n, x)
+    laguerre(n, x)
+
+    >>> diff(laguerre(n, x), x)
+    -assoc_laguerre(n - 1, 1, x)
+
+    Parameters
+    ==========
+
+    n : int
+        Degree of Laguerre polynomial. Must be `n \ge 0`.
+
+    See Also
+    ========
+
+    jacobi, gegenbauer,
+    chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,
+    legendre, assoc_legendre,
+    hermite, hermite_prob,
+    assoc_laguerre,
+    sympy.polys.orthopolys.jacobi_poly
+    sympy.polys.orthopolys.gegenbauer_poly
+    sympy.polys.orthopolys.chebyshevt_poly
+    sympy.polys.orthopolys.chebyshevu_poly
+    sympy.polys.orthopolys.hermite_poly
+    sympy.polys.orthopolys.hermite_prob_poly
+    sympy.polys.orthopolys.legendre_poly
+    sympy.polys.orthopolys.laguerre_poly
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Laguerre_polynomial
+    .. [2] https://mathworld.wolfram.com/LaguerrePolynomial.html
+    .. [3] https://functions.wolfram.com/Polynomials/LaguerreL/
+    .. [4] https://functions.wolfram.com/Polynomials/LaguerreL3/
+
+    """
+
+    _ortho_poly = staticmethod(laguerre_poly)
+
+    @classmethod
+    def eval(cls, n, x):
+        if n.is_integer is False:
+            raise ValueError("Error: n should be an integer.")
+        if not n.is_Number:
+            # Symbolic result L_n(x)
+            # L_{n}(-x)  --->  exp(-x) * L_{-n-1}(x)
+            # L_{-n}(x)  --->  exp(x) * L_{n-1}(-x)
+            if n.could_extract_minus_sign() and not(-n - 1).could_extract_minus_sign():
+                return exp(x)*laguerre(-n - 1, -x)
+            # We can evaluate for some special values of x
+            if x.is_zero:
+                return S.One
+            elif x is S.NegativeInfinity:
+                return S.Infinity
+            elif x is S.Infinity:
+                return S.NegativeOne**n * S.Infinity
+        else:
+            if n.is_negative:
+                return exp(x)*laguerre(-n - 1, -x)
+            else:
+                return cls._eval_at_order(n, x)
+
+    def fdiff(self, argindex=2):
+        if argindex == 1:
+            # Diff wrt n
+            raise ArgumentIndexError(self, argindex)
+        elif argindex == 2:
+            # Diff wrt x
+            n, x = self.args
+            return -assoc_laguerre(n - 1, 1, x)
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    def _eval_rewrite_as_Sum(self, n, x, **kwargs):
+        from sympy.concrete.summations import Sum
+        # Make sure n \in N_0
+        if n.is_negative:
+            return exp(x) * self._eval_rewrite_as_Sum(-n - 1, -x, **kwargs)
+        if n.is_integer is False:
+            raise ValueError("Error: n should be an integer.")
+        k = Dummy("k")
+        kern = RisingFactorial(-n, k) / factorial(k)**2 * x**k
+        return Sum(kern, (k, 0, n))
+
+    def _eval_rewrite_as_polynomial(self, n, x, **kwargs):
+        # This function is just kept for backwards compatibility
+        # but should not be used
+        return self._eval_rewrite_as_Sum(n, x, **kwargs)
+
+
+class assoc_laguerre(OrthogonalPolynomial):
+    r"""
+    Returns the $n$th generalized Laguerre polynomial in $x$, $L_n(x)$.
+
+    Examples
+    ========
+
+    >>> from sympy import assoc_laguerre, diff
+    >>> from sympy.abc import x, n, a
+    >>> assoc_laguerre(0, a, x)
+    1
+    >>> assoc_laguerre(1, a, x)
+    a - x + 1
+    >>> assoc_laguerre(2, a, x)
+    a**2/2 + 3*a/2 + x**2/2 + x*(-a - 2) + 1
+    >>> assoc_laguerre(3, a, x)
+    a**3/6 + a**2 + 11*a/6 - x**3/6 + x**2*(a/2 + 3/2) +
+        x*(-a**2/2 - 5*a/2 - 3) + 1
+
+    >>> assoc_laguerre(n, a, 0)
+    binomial(a + n, a)
+
+    >>> assoc_laguerre(n, a, x)
+    assoc_laguerre(n, a, x)
+
+    >>> assoc_laguerre(n, 0, x)
+    laguerre(n, x)
+
+    >>> diff(assoc_laguerre(n, a, x), x)
+    -assoc_laguerre(n - 1, a + 1, x)
+
+    >>> diff(assoc_laguerre(n, a, x), a)
+    Sum(assoc_laguerre(_k, a, x)/(-a + n), (_k, 0, n - 1))
+
+    Parameters
+    ==========
+
+    n : int
+        Degree of Laguerre polynomial. Must be `n \ge 0`.
+
+    alpha : Expr
+        Arbitrary expression. For ``alpha=0`` regular Laguerre
+        polynomials will be generated.
+
+    See Also
+    ========
+
+    jacobi, gegenbauer,
+    chebyshevt, chebyshevt_root, chebyshevu, chebyshevu_root,
+    legendre, assoc_legendre,
+    hermite, hermite_prob,
+    laguerre,
+    sympy.polys.orthopolys.jacobi_poly
+    sympy.polys.orthopolys.gegenbauer_poly
+    sympy.polys.orthopolys.chebyshevt_poly
+    sympy.polys.orthopolys.chebyshevu_poly
+    sympy.polys.orthopolys.hermite_poly
+    sympy.polys.orthopolys.hermite_prob_poly
+    sympy.polys.orthopolys.legendre_poly
+    sympy.polys.orthopolys.laguerre_poly
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Laguerre_polynomial#Generalized_Laguerre_polynomials
+    .. [2] https://mathworld.wolfram.com/AssociatedLaguerrePolynomial.html
+    .. [3] https://functions.wolfram.com/Polynomials/LaguerreL/
+    .. [4] https://functions.wolfram.com/Polynomials/LaguerreL3/
+
+    """
+
+    @classmethod
+    def eval(cls, n, alpha, x):
+        # L_{n}^{0}(x)  --->  L_{n}(x)
+        if alpha.is_zero:
+            return laguerre(n, x)
+
+        if not n.is_Number:
+            # We can evaluate for some special values of x
+            if x.is_zero:
+                return binomial(n + alpha, alpha)
+            elif x is S.Infinity and n > 0:
+                return S.NegativeOne**n * S.Infinity
+            elif x is S.NegativeInfinity and n > 0:
+                return S.Infinity
+        else:
+            # n is a given fixed integer, evaluate into polynomial
+            if n.is_negative:
+                raise ValueError(
+                    "The index n must be nonnegative integer (got %r)" % n)
+            else:
+                return laguerre_poly(n, x, alpha)
+
+    def fdiff(self, argindex=3):
+        from sympy.concrete.summations import Sum
+        if argindex == 1:
+            # Diff wrt n
+            raise ArgumentIndexError(self, argindex)
+        elif argindex == 2:
+            # Diff wrt alpha
+            n, alpha, x = self.args
+            k = Dummy("k")
+            return Sum(assoc_laguerre(k, alpha, x) / (n - alpha), (k, 0, n - 1))
+        elif argindex == 3:
+            # Diff wrt x
+            n, alpha, x = self.args
+            return -assoc_laguerre(n - 1, alpha + 1, x)
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    def _eval_rewrite_as_Sum(self, n, alpha, x, **kwargs):
+        from sympy.concrete.summations import Sum
+        # Make sure n \in N_0
+        if n.is_negative or n.is_integer is False:
+            raise ValueError("Error: n should be a non-negative integer.")
+        k = Dummy("k")
+        kern = RisingFactorial(
+            -n, k) / (gamma(k + alpha + 1) * factorial(k)) * x**k
+        return gamma(n + alpha + 1) / factorial(n) * Sum(kern, (k, 0, n))
+
+    def _eval_rewrite_as_polynomial(self, n, alpha, x, **kwargs):
+        # This function is just kept for backwards compatibility
+        # but should not be used
+        return self._eval_rewrite_as_Sum(n, alpha, x, **kwargs)
+
+    def _eval_conjugate(self):
+        n, alpha, x = self.args
+        return self.func(n, alpha.conjugate(), x.conjugate())

llmeval-env/lib/python3.10/site-packages/sympy/functions/special/singularity_functions.py

@@ -0,0 +1,228 @@
+from sympy.core import S, oo, diff
+from sympy.core.function import Function, ArgumentIndexError
+from sympy.core.logic import fuzzy_not
+from sympy.core.relational import Eq
+from sympy.functions.elementary.complexes import im
+from sympy.functions.elementary.piecewise import Piecewise
+from sympy.functions.special.delta_functions import Heaviside
+
+###############################################################################
+############################# SINGULARITY FUNCTION ############################
+###############################################################################
+
+
+class SingularityFunction(Function):
+    r"""
+    Singularity functions are a class of discontinuous functions.
+
+    Explanation
+    ===========
+
+    Singularity functions take a variable, an offset, and an exponent as
+    arguments. These functions are represented using Macaulay brackets as:
+
+    SingularityFunction(x, a, n) := <x - a>^n
+
+    The singularity function will automatically evaluate to
+    ``Derivative(DiracDelta(x - a), x, -n - 1)`` if ``n < 0``
+    and ``(x - a)**n*Heaviside(x - a)`` if ``n >= 0``.
+
+    Examples
+    ========
+
+    >>> from sympy import SingularityFunction, diff, Piecewise, DiracDelta, Heaviside, Symbol
+    >>> from sympy.abc import x, a, n
+    >>> SingularityFunction(x, a, n)
+    SingularityFunction(x, a, n)
+    >>> y = Symbol('y', positive=True)
+    >>> n = Symbol('n', nonnegative=True)
+    >>> SingularityFunction(y, -10, n)
+    (y + 10)**n
+    >>> y = Symbol('y', negative=True)
+    >>> SingularityFunction(y, 10, n)
+    0
+    >>> SingularityFunction(x, 4, -1).subs(x, 4)
+    oo
+    >>> SingularityFunction(x, 10, -2).subs(x, 10)
+    oo
+    >>> SingularityFunction(4, 1, 5)
+    243
+    >>> diff(SingularityFunction(x, 1, 5) + SingularityFunction(x, 1, 4), x)
+    4*SingularityFunction(x, 1, 3) + 5*SingularityFunction(x, 1, 4)
+    >>> diff(SingularityFunction(x, 4, 0), x, 2)
+    SingularityFunction(x, 4, -2)
+    >>> SingularityFunction(x, 4, 5).rewrite(Piecewise)
+    Piecewise(((x - 4)**5, x > 4), (0, True))
+    >>> expr = SingularityFunction(x, a, n)
+    >>> y = Symbol('y', positive=True)
+    >>> n = Symbol('n', nonnegative=True)
+    >>> expr.subs({x: y, a: -10, n: n})
+    (y + 10)**n
+
+    The methods ``rewrite(DiracDelta)``, ``rewrite(Heaviside)``, and
+    ``rewrite('HeavisideDiracDelta')`` returns the same output. One can use any
+    of these methods according to their choice.
+
+    >>> expr = SingularityFunction(x, 4, 5) + SingularityFunction(x, -3, -1) - SingularityFunction(x, 0, -2)
+    >>> expr.rewrite(Heaviside)
+    (x - 4)**5*Heaviside(x - 4) + DiracDelta(x + 3) - DiracDelta(x, 1)
+    >>> expr.rewrite(DiracDelta)
+    (x - 4)**5*Heaviside(x - 4) + DiracDelta(x + 3) - DiracDelta(x, 1)
+    >>> expr.rewrite('HeavisideDiracDelta')
+    (x - 4)**5*Heaviside(x - 4) + DiracDelta(x + 3) - DiracDelta(x, 1)
+
+    See Also
+    ========
+
+    DiracDelta, Heaviside
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Singularity_function
+
+    """
+
+    is_real = True
+
+    def fdiff(self, argindex=1):
+        """
+        Returns the first derivative of a DiracDelta Function.
+
+        Explanation
+        ===========
+
+        The difference between ``diff()`` and ``fdiff()`` is: ``diff()`` is the
+        user-level function and ``fdiff()`` is an object method. ``fdiff()`` is
+        a convenience method available in the ``Function`` class. It returns
+        the derivative of the function without considering the chain rule.
+        ``diff(function, x)`` calls ``Function._eval_derivative`` which in turn
+        calls ``fdiff()`` internally to compute the derivative of the function.
+
+        """
+
+        if argindex == 1:
+            x, a, n = self.args
+            if n in (S.Zero, S.NegativeOne):
+                return self.func(x, a, n-1)
+            elif n.is_positive:
+                return n*self.func(x, a, n-1)
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    @classmethod
+    def eval(cls, variable, offset, exponent):
+        """
+        Returns a simplified form or a value of Singularity Function depending
+        on the argument passed by the object.
+
+        Explanation
+        ===========
+
+        The ``eval()`` method is automatically called when the
+        ``SingularityFunction`` class is about to be instantiated and it
+        returns either some simplified instance or the unevaluated instance
+        depending on the argument passed. In other words, ``eval()`` method is
+        not needed to be called explicitly, it is being called and evaluated
+        once the object is called.
+
+        Examples
+        ========
+
+        >>> from sympy import SingularityFunction, Symbol, nan
+        >>> from sympy.abc import x, a, n
+        >>> SingularityFunction(x, a, n)
+        SingularityFunction(x, a, n)
+        >>> SingularityFunction(5, 3, 2)
+        4
+        >>> SingularityFunction(x, a, nan)
+        nan
+        >>> SingularityFunction(x, 3, 0).subs(x, 3)
+        1
+        >>> SingularityFunction(4, 1, 5)
+        243
+        >>> x = Symbol('x', positive = True)
+        >>> a = Symbol('a', negative = True)
+        >>> n = Symbol('n', nonnegative = True)
+        >>> SingularityFunction(x, a, n)
+        (-a + x)**n
+        >>> x = Symbol('x', negative = True)
+        >>> a = Symbol('a', positive = True)
+        >>> SingularityFunction(x, a, n)
+        0
+
+        """
+
+        x = variable
+        a = offset
+        n = exponent
+        shift = (x - a)
+
+        if fuzzy_not(im(shift).is_zero):
+            raise ValueError("Singularity Functions are defined only for Real Numbers.")
+        if fuzzy_not(im(n).is_zero):
+            raise ValueError("Singularity Functions are not defined for imaginary exponents.")
+        if shift is S.NaN or n is S.NaN:
+            return S.NaN
+        if (n + 2).is_negative:
+            raise ValueError("Singularity Functions are not defined for exponents less than -2.")
+        if shift.is_extended_negative:
+            return S.Zero
+        if n.is_nonnegative and shift.is_extended_nonnegative:
+            return (x - a)**n
+        if n in (S.NegativeOne, -2):
+            if shift.is_negative or shift.is_extended_positive:
+                return S.Zero
+            if shift.is_zero:
+                return oo
+
+    def _eval_rewrite_as_Piecewise(self, *args, **kwargs):
+        '''
+        Converts a Singularity Function expression into its Piecewise form.
+
+        '''
+        x, a, n = self.args
+
+        if n in (S.NegativeOne, S(-2)):
+            return Piecewise((oo, Eq((x - a), 0)), (0, True))
+        elif n.is_nonnegative:
+            return Piecewise(((x - a)**n, (x - a) > 0), (0, True))
+
+    def _eval_rewrite_as_Heaviside(self, *args, **kwargs):
+        '''
+        Rewrites a Singularity Function expression using Heavisides and DiracDeltas.
+
+        '''
+        x, a, n = self.args
+
+        if n == -2:
+            return diff(Heaviside(x - a), x.free_symbols.pop(), 2)
+        if n == -1:
+            return diff(Heaviside(x - a), x.free_symbols.pop(), 1)
+        if n.is_nonnegative:
+            return (x - a)**n*Heaviside(x - a)
+
+    def _eval_as_leading_term(self, x, logx=None, cdir=0):
+        z, a, n = self.args
+        shift = (z - a).subs(x, 0)
+        if n < 0:
+            return S.Zero
+        elif n.is_zero and shift.is_zero:
+            return S.Zero if cdir == -1 else S.One
+        elif shift.is_positive:
+            return shift**n
+        return S.Zero
+
+    def _eval_nseries(self, x, n, logx=None, cdir=0):
+        z, a, n = self.args
+        shift = (z - a).subs(x, 0)
+        if n < 0:
+            return S.Zero
+        elif n.is_zero and shift.is_zero:
+            return S.Zero if cdir == -1 else S.One
+        elif shift.is_positive:
+            return ((z - a)**n)._eval_nseries(x, n, logx=logx, cdir=cdir)
+        return S.Zero
+
+    _eval_rewrite_as_DiracDelta = _eval_rewrite_as_Heaviside
+    _eval_rewrite_as_HeavisideDiracDelta = _eval_rewrite_as_Heaviside