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

[bo198214]

Given a starting and ending point, it is possible to create a path between them that starts and ends in the principal branch, even if it must sometimes leave the principal branch. I'm working on pictures of what I mean, so in the meantime, hopefully you can picture what I mean.

Normally not, but in this case, yes

This has me concerned that the base-change formula is truly undefined for non-real values,

I was thinking about a different aproach: I would say all singularities of all \( f_n \) are bounded. So out there in the complex plane are points \( p \) which are distant from all singularities of all \( f_n \). Perhaps one can define the value of each \( f_n \) at \( p \) by using a path that has all singularities to the right side, respectively.
Then one would show that \( f_n \) converges uniformly in a neighborhood of \( p \). The limit function \( f \) in the neighborhood of \( p \) is necesarily again holomorphic there.
From *there* we continue the function \( f \) to the real line. I guess \( f \) has only singularities on the real line and at the fixed points of \( \exp_b \).
Then it would turn out that \( f \) has an *asymptotic* power series development (for a certain sector of approach) at all points of the (upper) real axis. Which though of course has zero convergence radius.