The "cheta" function

[jaydfox]

My change of base formula relies on the following:
\( \lim_{k \to \infty} \log_a^{\circ k} \left( \exp_b^{\circ k} (x) \right) \).

I had found that it was computationally more accurate to work with the double logarithm of x.....

I've been playing around with this for a few months, on and off. I believe this limit does not converge in the complex plane.

Here's an overview of my attempt to explain why the f(x) limit equation may not converge in the complex plane. First, show the f(x) limit converges nicely for some particular real value of x. Second, show that f(x+imaginary) converges to a very different number. Third, show that no matter how small the imaginary component is, f(x+imaginary) converges to a very different number than f(x). Then the slope in the complex plane is discontinuous, and perhaps the convergence radius is zero, or f(x) is not analytic. There is one other possibility that I cannot rule out, and that is that the multi-valued logarithm allows for a solution that does converge.

I'm not sure if the limit exists in the normal sense (i.e., it's well-defined for real numbers, but as you say, not for non-real numbers).

However, rather than two distinct points x and x+d*i (where d goes to 0), consider a line segment L defined between those two points. As exp_b(x) is entire, the image exp_b(L) will be continuous, as will be exp_b(exp_b(L)), etc. Thus, no matter how many times this curve wraps around the origin in some bizarre fractal nature, there is always a well-defined "path" back to the real line.

We can then iteratively perform the logarithm log_a(x), which has branches. We start at the real endpoint of exp_b^[ok](L) as we're performing the logarithms, so that when we wrap around the origin, we will always "know" which branch of the logarithm to use.

When all is said and done, we will arrive at the correct location, with no ambiguity. However, without using this "trick" to determine which branch to use, it does seem that this limit does not converge properly for non-real x. (And at any rate, for base b>eta, it definitely does not converge on the way "up", even if by some miracle it manages to converge on the way back "down").

(And yes, I ignored the change of base "constant" for simplicity.)