The "cheta" function

[jaydfox]

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.

Walker constructs an (infinitely differentiable) auxilliary function
\( h(x)=\lim_{n\to\infty} l^{[n]}(\exp^{[n]}(x)) \), where \( l(x)=\ln(x+1) \).
That satisfies
\( h(e^x)=e^{h(x)}-1 \)

Then he composes this function with the/one regular Abel function \( g \) of \( e^x-1 \), \( g(e^x-1)=g(x)+1 \).

The resulting function \( G(x)=g(h(x)) \) satisfies
\( G(e^x)=g(h(e^x))=g(e^{h(x)}-1)=g(h(x))+1=G(x)+1 \)
i.e. is an Abel function of e^x.

So where is it similar to your change of base?

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. Please review this post:

http://math.eretrandre.org/tetrationforu...306#pid306

It's post #36 in that thread, if the link doesn't take you right to it. I remembered last night that I had discussed the double logarithmic approach before, so this morning I searched until I found it.

As you'll see at the bottom of that post, I had previously made the connection between cheta and the iteration of e^x-1.

I want to work out the maths a little, so that I post something fairly coherent, but essentially, I believe, after reviewing things a bit, that my cheta with base-change is equivalent to the inverse of Walker's approach (i.e., equivalent to \( \hat{G}^{-1}(x) \).