Real and complex behaviour of the base change function (was: The "cheta" function)

[sheldonison]

Actually I am somewhat confused about the continuation of
\( f_n = \log_a^{[n]}\circ \exp_b^{[n]} \)
for some \( n \) from the real axis to the complex plane (via the method described by Jay some posts ago, i.e. considering a/the path \( \gamma \) from a point on the real axis to \( z \), looking at its image under \( \exp_b^{[n]} \) and determine the value of the subsequently following logarithms by their continuation along the image of that path).

Has anyone a clear view to which points \( f_n \) can be continued, is it even entire?

Me too, I'm making progress on n=5,6,7 ... I think the answer will eventually come out and it will be very interesting, but I don't have enough nearly enough time to do the problem proper diligence this week. Perhaps f will even converge as n increases to infinity, just as the function converges on the real number line....
- Shel