Extension of tetration operator
😃: The tetration operator is defined by repeating the exponentiation operation. When a, b are positive integers, a tetrated to b equals to a^a^...^a with a total of b a's, evaluated from right to left. Please provide an extension of the tetration operator to positive real numbers, so that the extended function is continuous and smooth at integer points.
🤖: To extend the tetration operator to positive real numbers, we need a function that agrees with tetration at integer points and is both continuous and smooth (i.e., infinitely differentiable) everywhere. This is a non-trivial problem, and there are several approaches in the literature. Here, I will describe one such approach using the concept of iterated exponentials and analytic continuation.
Problem Analysis
Tetration for positive integers is straightforward:
- ( {}^{1}a = a )
- ( {}^{2}a = a^a )
- ( {}^{3}a = a^{a^a} )
- and so on.
For real numbers ( x > 0 ), we need to define ( {}^{x}a ) such that:
- ( {}^{x}a ) is continuous and smooth in ( x ),
- ( {}^{n}a ) for ( n \in \mathbb{N} ) matches the integer tetration,
- ( {}^{x+1}a = a^{({}^{x}a)} ) for all ( x > 0 ).
Approach: Iterated Exponentials
One way to extend tetrati...