Iterability of exp(x)-1

[jaydfox]

Inspired by JaydFox mentioning P. L. Walker (which I didnt have heard of before) I just read the abstract of his paper [1]:

The author considers the Abel functional equation \( g(\varphi(x))=g(z)+1 \), where \( \varphi \) is a given entire function and \( g \) is an unknown entire function to be found. The inverse function \( f=g^{-1} \) (if one exists) must satisfy (1) \( f(w+1)=\varphi(f(w)) \). The purpose of this paper is to show that for a wide class of entire functions, which includes \( \varphi(z)=e^z-1 \), equation (1) has a nonconstant entire solution.

So I guess some results of I. N. Baker are indeed errournous.

[1] P. L. Walker, A class of functional equations which have entire solutions, Bull. Austral. Math. Soc. 38 (198, no. 3, 351-356

I think that settles it. If \( \varphi(z)=e^z-1 \) has an entire solution, then the cheta function should define a unique solution for tetration. And with my exact formula for base conversion, we can solve all bases.

I'll put together a consolidated post with all the necessarily formulae. Currently they are spread across separate posts (which is good, as this allows feedback on each portion of the solution).