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ckpts/universal/global_step80/zero/10.input_layernorm.weight/exp_avg_sq.pt

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ckpts/universal/global_step80/zero/10.input_layernorm.weight/fp32.pt

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ckpts/universal/global_step80/zero/4.mlp.dense_h_to_4h.weight/fp32.pt

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ckpts/universal/global_step80/zero/7.attention.query_key_value.weight/exp_avg_sq.pt

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ckpts/universal/global_step80/zero/7.attention.query_key_value.weight/fp32.pt

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

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+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

venv/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

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

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

venv/lib/python3.10/site-packages/sympy/functions/special/benchmarks/bench_special.py

@@ -0,0 +1,8 @@
+from sympy.core.symbol import symbols
+from sympy.functions.special.spherical_harmonics import Ynm
+
+x, y = symbols('x,y')
+
+
+def timeit_Ynm_xy():
+    Ynm(1, 1, x, y)

venv/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

venv/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)

venv/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)

venv/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.'''))

venv/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))

venv/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)

venv/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

venv/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)

venv/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)

venv/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())

venv/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

venv/lib/python3.10/site-packages/sympy/functions/special/spherical_harmonics.py

@@ -0,0 +1,334 @@
+from sympy.core.expr import Expr
+from sympy.core.function import Function, ArgumentIndexError
+from sympy.core.numbers import I, pi
+from sympy.core.singleton import S
+from sympy.core.symbol import Dummy
+from sympy.functions import assoc_legendre
+from sympy.functions.combinatorial.factorials import factorial
+from sympy.functions.elementary.complexes import Abs, conjugate
+from sympy.functions.elementary.exponential import exp
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.trigonometric import sin, cos, cot
+
+_x = Dummy("x")
+
+class Ynm(Function):
+    r"""
+    Spherical harmonics defined as
+
+    .. math::
+        Y_n^m(\theta, \varphi) := \sqrt{\frac{(2n+1)(n-m)!}{4\pi(n+m)!}}
+                                  \exp(i m \varphi)
+                                  \mathrm{P}_n^m\left(\cos(\theta)\right)
+
+    Explanation
+    ===========
+
+    ``Ynm()`` gives the spherical harmonic function of order $n$ and $m$
+    in $\theta$ and $\varphi$, $Y_n^m(\theta, \varphi)$. The four
+    parameters are as follows: $n \geq 0$ an integer and $m$ an integer
+    such that $-n \leq m \leq n$ holds. The two angles are real-valued
+    with $\theta \in [0, \pi]$ and $\varphi \in [0, 2\pi]$.
+
+    Examples
+    ========
+
+    >>> from sympy import Ynm, Symbol, simplify
+    >>> from sympy.abc import n,m
+    >>> theta = Symbol("theta")
+    >>> phi = Symbol("phi")
+
+    >>> Ynm(n, m, theta, phi)
+    Ynm(n, m, theta, phi)
+
+    Several symmetries are known, for the order:
+
+    >>> Ynm(n, -m, theta, phi)
+    (-1)**m*exp(-2*I*m*phi)*Ynm(n, m, theta, phi)
+
+    As well as for the angles:
+
+    >>> Ynm(n, m, -theta, phi)
+    Ynm(n, m, theta, phi)
+
+    >>> Ynm(n, m, theta, -phi)
+    exp(-2*I*m*phi)*Ynm(n, m, theta, phi)
+
+    For specific integers $n$ and $m$ we can evaluate the harmonics
+    to more useful expressions:
+
+    >>> simplify(Ynm(0, 0, theta, phi).expand(func=True))
+    1/(2*sqrt(pi))
+
+    >>> simplify(Ynm(1, -1, theta, phi).expand(func=True))
+    sqrt(6)*exp(-I*phi)*sin(theta)/(4*sqrt(pi))
+
+    >>> simplify(Ynm(1, 0, theta, phi).expand(func=True))
+    sqrt(3)*cos(theta)/(2*sqrt(pi))
+
+    >>> simplify(Ynm(1, 1, theta, phi).expand(func=True))
+    -sqrt(6)*exp(I*phi)*sin(theta)/(4*sqrt(pi))
+
+    >>> simplify(Ynm(2, -2, theta, phi).expand(func=True))
+    sqrt(30)*exp(-2*I*phi)*sin(theta)**2/(8*sqrt(pi))
+
+    >>> simplify(Ynm(2, -1, theta, phi).expand(func=True))
+    sqrt(30)*exp(-I*phi)*sin(2*theta)/(8*sqrt(pi))
+
+    >>> simplify(Ynm(2, 0, theta, phi).expand(func=True))
+    sqrt(5)*(3*cos(theta)**2 - 1)/(4*sqrt(pi))
+
+    >>> simplify(Ynm(2, 1, theta, phi).expand(func=True))
+    -sqrt(30)*exp(I*phi)*sin(2*theta)/(8*sqrt(pi))
+
+    >>> simplify(Ynm(2, 2, theta, phi).expand(func=True))
+    sqrt(30)*exp(2*I*phi)*sin(theta)**2/(8*sqrt(pi))
+
+    We can differentiate the functions with respect
+    to both angles:
+
+    >>> from sympy import Ynm, Symbol, diff
+    >>> from sympy.abc import n,m
+    >>> theta = Symbol("theta")
+    >>> phi = Symbol("phi")
+
+    >>> diff(Ynm(n, m, theta, phi), theta)
+    m*cot(theta)*Ynm(n, m, theta, phi) + sqrt((-m + n)*(m + n + 1))*exp(-I*phi)*Ynm(n, m + 1, theta, phi)
+
+    >>> diff(Ynm(n, m, theta, phi), phi)
+    I*m*Ynm(n, m, theta, phi)
+
+    Further we can compute the complex conjugation:
+
+    >>> from sympy import Ynm, Symbol, conjugate
+    >>> from sympy.abc import n,m
+    >>> theta = Symbol("theta")
+    >>> phi = Symbol("phi")
+
+    >>> conjugate(Ynm(n, m, theta, phi))
+    (-1)**(2*m)*exp(-2*I*m*phi)*Ynm(n, m, theta, phi)
+
+    To get back the well known expressions in spherical
+    coordinates, we use full expansion:
+
+    >>> from sympy import Ynm, Symbol, expand_func
+    >>> from sympy.abc import n,m
+    >>> theta = Symbol("theta")
+    >>> phi = Symbol("phi")
+
+    >>> expand_func(Ynm(n, m, theta, phi))
+    sqrt((2*n + 1)*factorial(-m + n)/factorial(m + n))*exp(I*m*phi)*assoc_legendre(n, m, cos(theta))/(2*sqrt(pi))
+
+    See Also
+    ========
+
+    Ynm_c, Znm
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Spherical_harmonics
+    .. [2] https://mathworld.wolfram.com/SphericalHarmonic.html
+    .. [3] https://functions.wolfram.com/Polynomials/SphericalHarmonicY/
+    .. [4] https://dlmf.nist.gov/14.30
+
+    """
+
+    @classmethod
+    def eval(cls, n, m, theta, phi):
+        # Handle negative index m and arguments theta, phi
+        if m.could_extract_minus_sign():
+            m = -m
+            return S.NegativeOne**m * exp(-2*I*m*phi) * Ynm(n, m, theta, phi)
+        if theta.could_extract_minus_sign():
+            theta = -theta
+            return Ynm(n, m, theta, phi)
+        if phi.could_extract_minus_sign():
+            phi = -phi
+            return exp(-2*I*m*phi) * Ynm(n, m, theta, phi)
+
+        # TODO Add more simplififcation here
+
+    def _eval_expand_func(self, **hints):
+        n, m, theta, phi = self.args
+        rv = (sqrt((2*n + 1)/(4*pi) * factorial(n - m)/factorial(n + m)) *
+                exp(I*m*phi) * assoc_legendre(n, m, cos(theta)))
+        # We can do this because of the range of theta
+        return rv.subs(sqrt(-cos(theta)**2 + 1), sin(theta))
+
+    def fdiff(self, argindex=4):
+        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 theta
+            n, m, theta, phi = self.args
+            return (m * cot(theta) * Ynm(n, m, theta, phi) +
+                    sqrt((n - m)*(n + m + 1)) * exp(-I*phi) * Ynm(n, m + 1, theta, phi))
+        elif argindex == 4:
+            # Diff wrt phi
+            n, m, theta, phi = self.args
+            return I * m * Ynm(n, m, theta, phi)
+        else:
+            raise ArgumentIndexError(self, argindex)
+
+    def _eval_rewrite_as_polynomial(self, n, m, theta, phi, **kwargs):
+        # TODO: Make sure n \in N
+        # TODO: Assert |m| <= n ortherwise we should return 0
+        return self.expand(func=True)
+
+    def _eval_rewrite_as_sin(self, n, m, theta, phi, **kwargs):
+        return self.rewrite(cos)
+
+    def _eval_rewrite_as_cos(self, n, m, theta, phi, **kwargs):
+        # This method can be expensive due to extensive use of simplification!
+        from sympy.simplify import simplify, trigsimp
+        # TODO: Make sure n \in N
+        # TODO: Assert |m| <= n ortherwise we should return 0
+        term = simplify(self.expand(func=True))
+        # We can do this because of the range of theta
+        term = term.xreplace({Abs(sin(theta)):sin(theta)})
+        return simplify(trigsimp(term))
+
+    def _eval_conjugate(self):
+        # TODO: Make sure theta \in R and phi \in R
+        n, m, theta, phi = self.args
+        return S.NegativeOne**m * self.func(n, -m, theta, phi)
+
+    def as_real_imag(self, deep=True, **hints):
+        # TODO: Handle deep and hints
+        n, m, theta, phi = self.args
+        re = (sqrt((2*n + 1)/(4*pi) * factorial(n - m)/factorial(n + m)) *
+              cos(m*phi) * assoc_legendre(n, m, cos(theta)))
+        im = (sqrt((2*n + 1)/(4*pi) * factorial(n - m)/factorial(n + m)) *
+              sin(m*phi) * assoc_legendre(n, m, cos(theta)))
+        return (re, im)
+
+    def _eval_evalf(self, prec):
+        # Note: works without this function by just calling
+        #       mpmath for Legendre polynomials. But using
+        #       the dedicated function directly is cleaner.
+        from mpmath import mp, workprec
+        n = self.args[0]._to_mpmath(prec)
+        m = self.args[1]._to_mpmath(prec)
+        theta = self.args[2]._to_mpmath(prec)
+        phi = self.args[3]._to_mpmath(prec)
+        with workprec(prec):
+            res = mp.spherharm(n, m, theta, phi)
+        return Expr._from_mpmath(res, prec)
+
+
+def Ynm_c(n, m, theta, phi):
+    r"""
+    Conjugate spherical harmonics defined as
+
+    .. math::
+        \overline{Y_n^m(\theta, \varphi)} := (-1)^m Y_n^{-m}(\theta, \varphi).
+
+    Examples
+    ========
+
+    >>> from sympy import Ynm_c, Symbol, simplify
+    >>> from sympy.abc import n,m
+    >>> theta = Symbol("theta")
+    >>> phi = Symbol("phi")
+    >>> Ynm_c(n, m, theta, phi)
+    (-1)**(2*m)*exp(-2*I*m*phi)*Ynm(n, m, theta, phi)
+    >>> Ynm_c(n, m, -theta, phi)
+    (-1)**(2*m)*exp(-2*I*m*phi)*Ynm(n, m, theta, phi)
+
+    For specific integers $n$ and $m$ we can evaluate the harmonics
+    to more useful expressions:
+
+    >>> simplify(Ynm_c(0, 0, theta, phi).expand(func=True))
+    1/(2*sqrt(pi))
+    >>> simplify(Ynm_c(1, -1, theta, phi).expand(func=True))
+    sqrt(6)*exp(I*(-phi + 2*conjugate(phi)))*sin(theta)/(4*sqrt(pi))
+
+    See Also
+    ========
+
+    Ynm, Znm
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Spherical_harmonics
+    .. [2] https://mathworld.wolfram.com/SphericalHarmonic.html
+    .. [3] https://functions.wolfram.com/Polynomials/SphericalHarmonicY/
+
+    """
+    return conjugate(Ynm(n, m, theta, phi))
+
+
+class Znm(Function):
+    r"""
+    Real spherical harmonics defined as
+
+    .. math::
+
+        Z_n^m(\theta, \varphi) :=
+        \begin{cases}
+          \frac{Y_n^m(\theta, \varphi) + \overline{Y_n^m(\theta, \varphi)}}{\sqrt{2}} &\quad m > 0 \\
+          Y_n^m(\theta, \varphi) &\quad m = 0 \\
+          \frac{Y_n^m(\theta, \varphi) - \overline{Y_n^m(\theta, \varphi)}}{i \sqrt{2}} &\quad m < 0 \\
+        \end{cases}
+
+    which gives in simplified form
+
+    .. math::
+
+        Z_n^m(\theta, \varphi) =
+        \begin{cases}
+          \frac{Y_n^m(\theta, \varphi) + (-1)^m Y_n^{-m}(\theta, \varphi)}{\sqrt{2}} &\quad m > 0 \\
+          Y_n^m(\theta, \varphi) &\quad m = 0 \\
+          \frac{Y_n^m(\theta, \varphi) - (-1)^m Y_n^{-m}(\theta, \varphi)}{i \sqrt{2}} &\quad m < 0 \\
+        \end{cases}
+
+    Examples
+    ========
+
+    >>> from sympy import Znm, Symbol, simplify
+    >>> from sympy.abc import n, m
+    >>> theta = Symbol("theta")
+    >>> phi = Symbol("phi")
+    >>> Znm(n, m, theta, phi)
+    Znm(n, m, theta, phi)
+
+    For specific integers n and m we can evaluate the harmonics
+    to more useful expressions:
+
+    >>> simplify(Znm(0, 0, theta, phi).expand(func=True))
+    1/(2*sqrt(pi))
+    >>> simplify(Znm(1, 1, theta, phi).expand(func=True))
+    -sqrt(3)*sin(theta)*cos(phi)/(2*sqrt(pi))
+    >>> simplify(Znm(2, 1, theta, phi).expand(func=True))
+    -sqrt(15)*sin(2*theta)*cos(phi)/(4*sqrt(pi))
+
+    See Also
+    ========
+
+    Ynm, Ynm_c
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Spherical_harmonics
+    .. [2] https://mathworld.wolfram.com/SphericalHarmonic.html
+    .. [3] https://functions.wolfram.com/Polynomials/SphericalHarmonicY/
+
+    """
+
+    @classmethod
+    def eval(cls, n, m, theta, phi):
+        if m.is_positive:
+            zz = (Ynm(n, m, theta, phi) + Ynm_c(n, m, theta, phi)) / sqrt(2)
+            return zz
+        elif m.is_zero:
+            return Ynm(n, m, theta, phi)
+        elif m.is_negative:
+            zz = (Ynm(n, m, theta, phi) - Ynm_c(n, m, theta, phi)) / (sqrt(2)*I)
+            return zz

venv/lib/python3.10/site-packages/sympy/functions/special/tensor_functions.py

@@ -0,0 +1,474 @@
+from math import prod
+
+from sympy.core import S, Integer
+from sympy.core.function import Function
+from sympy.core.logic import fuzzy_not
+from sympy.core.relational import Ne
+from sympy.core.sorting import default_sort_key
+from sympy.external.gmpy import SYMPY_INTS
+from sympy.functions.combinatorial.factorials import factorial
+from sympy.functions.elementary.piecewise import Piecewise
+from sympy.utilities.iterables import has_dups
+
+###############################################################################
+###################### Kronecker Delta, Levi-Civita etc. ######################
+###############################################################################
+
+
+def Eijk(*args, **kwargs):
+    """
+    Represent the Levi-Civita symbol.
+
+    This is a compatibility wrapper to ``LeviCivita()``.
+
+    See Also
+    ========
+
+    LeviCivita
+
+    """
+    return LeviCivita(*args, **kwargs)
+
+
+def eval_levicivita(*args):
+    """Evaluate Levi-Civita symbol."""
+    n = len(args)
+    return prod(
+        prod(args[j] - args[i] for j in range(i + 1, n))
+        / factorial(i) for i in range(n))
+    # converting factorial(i) to int is slightly faster
+
+
+class LeviCivita(Function):
+    """
+    Represent the Levi-Civita symbol.
+
+    Explanation
+    ===========
+
+    For even permutations of indices it returns 1, for odd permutations -1, and
+    for everything else (a repeated index) it returns 0.
+
+    Thus it represents an alternating pseudotensor.
+
+    Examples
+    ========
+
+    >>> from sympy import LeviCivita
+    >>> from sympy.abc import i, j, k
+    >>> LeviCivita(1, 2, 3)
+    1
+    >>> LeviCivita(1, 3, 2)
+    -1
+    >>> LeviCivita(1, 2, 2)
+    0
+    >>> LeviCivita(i, j, k)
+    LeviCivita(i, j, k)
+    >>> LeviCivita(i, j, i)
+    0
+
+    See Also
+    ========
+
+    Eijk
+
+    """
+
+    is_integer = True
+
+    @classmethod
+    def eval(cls, *args):
+        if all(isinstance(a, (SYMPY_INTS, Integer)) for a in args):
+            return eval_levicivita(*args)
+        if has_dups(args):
+            return S.Zero
+
+    def doit(self, **hints):
+        return eval_levicivita(*self.args)
+
+
+class KroneckerDelta(Function):
+    """
+    The discrete, or Kronecker, delta function.
+
+    Explanation
+    ===========
+
+    A function that takes in two integers $i$ and $j$. It returns $0$ if $i$
+    and $j$ are not equal, or it returns $1$ if $i$ and $j$ are equal.
+
+    Examples
+    ========
+
+    An example with integer indices:
+
+        >>> from sympy import KroneckerDelta
+        >>> KroneckerDelta(1, 2)
+        0
+        >>> KroneckerDelta(3, 3)
+        1
+
+    Symbolic indices:
+
+        >>> from sympy.abc import i, j, k
+        >>> KroneckerDelta(i, j)
+        KroneckerDelta(i, j)
+        >>> KroneckerDelta(i, i)
+        1
+        >>> KroneckerDelta(i, i + 1)
+        0
+        >>> KroneckerDelta(i, i + 1 + k)
+        KroneckerDelta(i, i + k + 1)
+
+    Parameters
+    ==========
+
+    i : Number, Symbol
+        The first index of the delta function.
+    j : Number, Symbol
+        The second index of the delta function.
+
+    See Also
+    ========
+
+    eval
+    DiracDelta
+
+    References
+    ==========
+
+    .. [1] https://en.wikipedia.org/wiki/Kronecker_delta
+
+    """
+
+    is_integer = True
+
+    @classmethod
+    def eval(cls, i, j, delta_range=None):
+        """
+        Evaluates the discrete delta function.
+
+        Examples
+        ========
+
+        >>> from sympy import KroneckerDelta
+        >>> from sympy.abc import i, j, k
+
+        >>> KroneckerDelta(i, j)
+        KroneckerDelta(i, j)
+        >>> KroneckerDelta(i, i)
+        1
+        >>> KroneckerDelta(i, i + 1)
+        0
+        >>> KroneckerDelta(i, i + 1 + k)
+        KroneckerDelta(i, i + k + 1)
+
+        # indirect doctest
+
+        """
+
+        if delta_range is not None:
+            dinf, dsup = delta_range
+            if (dinf - i > 0) == True:
+                return S.Zero
+            if (dinf - j > 0) == True:
+                return S.Zero
+            if (dsup - i < 0) == True:
+                return S.Zero
+            if (dsup - j < 0) == True:
+                return S.Zero
+
+        diff = i - j
+        if diff.is_zero:
+            return S.One
+        elif fuzzy_not(diff.is_zero):
+            return S.Zero
+
+        if i.assumptions0.get("below_fermi") and \
+                j.assumptions0.get("above_fermi"):
+            return S.Zero
+        if j.assumptions0.get("below_fermi") and \
+                i.assumptions0.get("above_fermi"):
+            return S.Zero
+        # to make KroneckerDelta canonical
+        # following lines will check if inputs are in order
+        # if not, will return KroneckerDelta with correct order
+        if i != min(i, j, key=default_sort_key):
+            if delta_range:
+                return cls(j, i, delta_range)
+            else:
+                return cls(j, i)
+
+    @property
+    def delta_range(self):
+        if len(self.args) > 2:
+            return self.args[2]
+
+    def _eval_power(self, expt):
+        if expt.is_positive:
+            return self
+        if expt.is_negative and expt is not S.NegativeOne:
+            return 1/self
+
+    @property
+    def is_above_fermi(self):
+        """
+        True if Delta can be non-zero above fermi.
+
+        Examples
+        ========
+
+        >>> from sympy import KroneckerDelta, Symbol
+        >>> a = Symbol('a', above_fermi=True)
+        >>> i = Symbol('i', below_fermi=True)
+        >>> p = Symbol('p')
+        >>> q = Symbol('q')
+        >>> KroneckerDelta(p, a).is_above_fermi
+        True
+        >>> KroneckerDelta(p, i).is_above_fermi
+        False
+        >>> KroneckerDelta(p, q).is_above_fermi
+        True
+
+        See Also
+        ========
+
+        is_below_fermi, is_only_below_fermi, is_only_above_fermi
+
+        """
+        if self.args[0].assumptions0.get("below_fermi"):
+            return False
+        if self.args[1].assumptions0.get("below_fermi"):
+            return False
+        return True
+
+    @property
+    def is_below_fermi(self):
+        """
+        True if Delta can be non-zero below fermi.
+
+        Examples
+        ========
+
+        >>> from sympy import KroneckerDelta, Symbol
+        >>> a = Symbol('a', above_fermi=True)
+        >>> i = Symbol('i', below_fermi=True)
+        >>> p = Symbol('p')
+        >>> q = Symbol('q')
+        >>> KroneckerDelta(p, a).is_below_fermi
+        False
+        >>> KroneckerDelta(p, i).is_below_fermi
+        True
+        >>> KroneckerDelta(p, q).is_below_fermi
+        True
+
+        See Also
+        ========
+
+        is_above_fermi, is_only_above_fermi, is_only_below_fermi
+
+        """
+        if self.args[0].assumptions0.get("above_fermi"):
+            return False
+        if self.args[1].assumptions0.get("above_fermi"):
+            return False
+        return True
+
+    @property
+    def is_only_above_fermi(self):
+        """
+        True if Delta is restricted to above fermi.
+
+        Examples
+        ========
+
+        >>> from sympy import KroneckerDelta, Symbol
+        >>> a = Symbol('a', above_fermi=True)
+        >>> i = Symbol('i', below_fermi=True)
+        >>> p = Symbol('p')
+        >>> q = Symbol('q')
+        >>> KroneckerDelta(p, a).is_only_above_fermi
+        True
+        >>> KroneckerDelta(p, q).is_only_above_fermi
+        False
+        >>> KroneckerDelta(p, i).is_only_above_fermi
+        False
+
+        See Also
+        ========
+
+        is_above_fermi, is_below_fermi, is_only_below_fermi
+
+        """
+        return ( self.args[0].assumptions0.get("above_fermi")
+                or
+                self.args[1].assumptions0.get("above_fermi")
+                ) or False
+
+    @property
+    def is_only_below_fermi(self):
+        """
+        True if Delta is restricted to below fermi.
+
+        Examples
+        ========
+
+        >>> from sympy import KroneckerDelta, Symbol
+        >>> a = Symbol('a', above_fermi=True)
+        >>> i = Symbol('i', below_fermi=True)
+        >>> p = Symbol('p')
+        >>> q = Symbol('q')
+        >>> KroneckerDelta(p, i).is_only_below_fermi
+        True
+        >>> KroneckerDelta(p, q).is_only_below_fermi
+        False
+        >>> KroneckerDelta(p, a).is_only_below_fermi
+        False
+
+        See Also
+        ========
+
+        is_above_fermi, is_below_fermi, is_only_above_fermi
+
+        """
+        return ( self.args[0].assumptions0.get("below_fermi")
+                or
+                self.args[1].assumptions0.get("below_fermi")
+                ) or False
+
+    @property
+    def indices_contain_equal_information(self):
+        """
+        Returns True if indices are either both above or below fermi.
+
+        Examples
+        ========
+
+        >>> from sympy import KroneckerDelta, Symbol
+        >>> a = Symbol('a', above_fermi=True)
+        >>> i = Symbol('i', below_fermi=True)
+        >>> p = Symbol('p')
+        >>> q = Symbol('q')
+        >>> KroneckerDelta(p, q).indices_contain_equal_information
+        True
+        >>> KroneckerDelta(p, q+1).indices_contain_equal_information
+        True
+        >>> KroneckerDelta(i, p).indices_contain_equal_information
+        False
+
+        """
+        if (self.args[0].assumptions0.get("below_fermi") and
+                self.args[1].assumptions0.get("below_fermi")):
+            return True
+        if (self.args[0].assumptions0.get("above_fermi")
+                and self.args[1].assumptions0.get("above_fermi")):
+            return True
+
+        # if both indices are general we are True, else false
+        return self.is_below_fermi and self.is_above_fermi
+
+    @property
+    def preferred_index(self):
+        """
+        Returns the index which is preferred to keep in the final expression.
+
+        Explanation
+        ===========
+
+        The preferred index is the index with more information regarding fermi
+        level. If indices contain the same information, 'a' is preferred before
+        'b'.
+
+        Examples
+        ========
+
+        >>> from sympy import KroneckerDelta, Symbol
+        >>> a = Symbol('a', above_fermi=True)
+        >>> i = Symbol('i', below_fermi=True)
+        >>> j = Symbol('j', below_fermi=True)
+        >>> p = Symbol('p')
+        >>> KroneckerDelta(p, i).preferred_index
+        i
+        >>> KroneckerDelta(p, a).preferred_index
+        a
+        >>> KroneckerDelta(i, j).preferred_index
+        i
+
+        See Also
+        ========
+
+        killable_index
+
+        """
+        if self._get_preferred_index():
+            return self.args[1]
+        else:
+            return self.args[0]
+
+    @property
+    def killable_index(self):
+        """
+        Returns the index which is preferred to substitute in the final
+        expression.
+
+        Explanation
+        ===========
+
+        The index to substitute is the index with less information regarding
+        fermi level. If indices contain the same information, 'a' is preferred
+        before 'b'.
+
+        Examples
+        ========
+
+        >>> from sympy import KroneckerDelta, Symbol
+        >>> a = Symbol('a', above_fermi=True)
+        >>> i = Symbol('i', below_fermi=True)
+        >>> j = Symbol('j', below_fermi=True)
+        >>> p = Symbol('p')
+        >>> KroneckerDelta(p, i).killable_index
+        p
+        >>> KroneckerDelta(p, a).killable_index
+        p
+        >>> KroneckerDelta(i, j).killable_index
+        j
+
+        See Also
+        ========
+
+        preferred_index
+
+        """
+        if self._get_preferred_index():
+            return self.args[0]
+        else:
+            return self.args[1]
+
+    def _get_preferred_index(self):
+        """
+        Returns the index which is preferred to keep in the final expression.
+
+        The preferred index is the index with more information regarding fermi
+        level. If indices contain the same information, index 0 is returned.
+
+        """
+        if not self.is_above_fermi:
+            if self.args[0].assumptions0.get("below_fermi"):
+                return 0
+            else:
+                return 1
+        elif not self.is_below_fermi:
+            if self.args[0].assumptions0.get("above_fermi"):
+                return 0
+            else:
+                return 1
+        else:
+            return 0
+
+    @property
+    def indices(self):
+        return self.args[0:2]
+
+    def _eval_rewrite_as_Piecewise(self, *args, **kwargs):
+        i, j = args
+        return Piecewise((0, Ne(i, j)), (1, True))