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

[sheldonison]

Though currently I wonder whether these arbitrary close singularities indeed imply that the function is not analytic in any point.
I mean there is a theorem that if a holomorphic function sequence converges locally uniformly (i.e. for each point there is a neighborhood where it converges uniformly) then the limit is again a holomorphic function (which is not true for just differentiable functions).

However I dont think that the inverse statement is also true, that if a function sequence does not converge locally uniformly that then resulting function can not be holomorphic.

For example a sequence of non-continuous functions can have a continuous function as a limit. Also Jay showed that the singularities gets milder with increasing n. So there maybe a very little tiny hope that the resulting function is analytic despite.

Henryk, I finally caught up with your formula for the singularities -- its very helpful. I have a pretty good intuitive feel for the singularities for small values of k.
\( \log_\eta^{[n]}(e(1+2\pi i k)) \)
It appears there are about 500,000 or so n=3 singularities in the critical strip used by the base change equation (from 5.016 to 6.330). I would assume the n=4 singularities would be super-exponentially denser yet.
k=1, 4.8688+0.5713i
k=2, 5.0732+0.4586i
k=3, 5.1734+0.4068i
k=500,000 6.3301+0.0706i

In looking at the singularities for small values of k, It seems that the function becomes undefined (or multi-valued?), once passing the neighborhood of the singularity. Each singularity is associated with a particular increment of the windings. Even ignoring the singularities associated with very large values of k, values of n>4, (which approach arbitrarily close to the real axis), can we continue the function for smaller of k, where n=4 as opposed to n=infinity?

Typo correction: Actually I used n=3 in Henryk's equation, but to see the singularities in the "f" base change equation below requires using n=4.
\( f_n = \log_e^{[n]}\circ \exp_\eta^{[n]} \)

I have not yet made the leap to understanding the singularities associated with larger values of k, and how they change the behavior of f, but I hope to do so.
- Sheldon