The "cheta" function

[jaydfox]

The bright side is, this saves me the work of having to rigorously prove various properties, as they apparently have been proven for nearly 20 years. I need only work on getting good numerical approximations (several thousands of bits of accuracy), to use in my change-of-base formula.

Haha dont dare! You have to show that your algorithm indeed reproduces the regular iteration!
And whether the corresponding superfunction for base \( b>\eta \) is holomorphic is also not solved yet, we only know about infinite differentiability by Walker.
If he uses your change of base at all, but to find out is your task now.

I recall he used something similar to my change of base formula; indeed, when I first read [1], it was a comment at the bottom of page 729 (the page headed "3. Values of Generalized Logarithms"), where it mentioned the h(x) function has a sufficient approximation after at most 5 iterations.

This statement seemed odd, so I worked out what he meant. It was then that I realized that for any two bases of tetration, a and b each real and greater than eta, and a sufficiently large real x, the value \( \lim_{k \to \infty} \log_a^{\circ k} \left( \exp_b^{\circ k} (x) \right) \) is well-defined, and furthermore, relatively easy to calculate to full machine precision with, usually, a very small k (arbitrarily large precision for some math libraries, limited by hardware). (Note: by "sufficiently large real x", I mean that x must be large enough so that it does not become negative with remaining logarithms to be performed, which would lead to non-unique complex results.)

I had already deduced that tetration in base eta was solvable exactly without bizarre matrix inversions or whatever, but this would correspond to "heta", the solution to the fixed point from the negative real direction. In other words, a simple, elegant formula existed. Furthermore, the formula would be provably the unique solution, be infinitely differentiable, etc. Up to that point, everything I'd read about tetration suggested that tetration was impossible to solve uniquely or with infinite differentiabilty, but the literature I had access to was sparse and outdated. So I was excited to be able to solve a base, even if it wasn't e or 2 or 10 or something more useful.

What I needed was a way to get a solution for bases larger than eta, and armed with my change of base formula, I decided to approach the fixed point from the positive real direction, which gave me the cheta function. I actually didn't realize the connection with exp(z)-1 at the time, which is kind of a shame because I got the idea from having read Walker's paper! I didn't see the connection at first because I was approaching the problem from the point of view of "tetration", as opposed to thinking in terms of Abel functions and parabolic fixed points and what-have-you.

[1] Walker, P. (1991). Infinitely differentiable generalized logarithmic and exponential functions. Math. Comput., 57(196), 723–733.