Taylor series of cheta

[Xorter]

I am interested in that how we could expand cheta function and its inverse to reals and complexes.
cheta(x) = e[x]e
It is important for me because of the rational and complex hyper-operators.
Please, help me.

There is a formal asymptotic series for the Abel \( \alpha(z) \) function solution for the parabolic case. Iterating \( f(z)=\exp(z)-1 \), is congruent through a simple linear transformation to iterating cheta \( f(y)=\eta^y \;\;\; \eta=\exp(\frac{1}{e}) \), by mapping \( z \mapsto \frac{y}{e}-1 \)



\(
\alpha(z) = \frac{-1}{2z} + \frac{\ln(z)}{3} - \frac{z}{36} + \frac{z^2}{540} + \frac{z^3}{7776} + \frac{-71z^4}{435456} + ....
\alpha(\exp(z)-1) = \alpha(z)+1
\)

The formal series will work in either half plane, by changing the ln(z) to ln(-z). But it will not work in both at the same time. It helps to iterate (exp(z)-1) or ln(z+1) a few times to get closer to the fixed point of zero before evaulating the asymptotic series. See G Edgar's post in mathoverflow for some theoretical background on the parabolic case. http://mathoverflow.net/questions/45608/...x-converge

Hello, Sheldon!

I tried your series for abel function. But it does not seem it would be the inverse of cheta function. Why not? Did I make a mistake or what?
(Here is the picture of the graphs.)