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

[bo198214]

My next question is, how large does k have to get before we encounter singularities?
...
As a simplification, we look at this equation, between z=4.38 and z=5.02
\( f(z) =
\lim_{k \to \infty} \log_e^{\circ k} \left( \exp_\eta^{\circ k} (z)
\right) \)

For \( k=3 \) singularities are already there (but I use \( n \) for the iteration count). The formula of the k-th singularity of \( f_n \) is:

\( \log_\eta^{[n-3]}(e(1+2\pi i k)) \)

We see that the original singularities of \( f_3 \) are situated on a vertical line through e. Thatswhy instead of depicting each single singularity I show the deformation of this line under repeated \( \log_\eta \).

The picture shows from top to bottom 1 till 5 applications of \( \log_\eta \) to this vertical line. This corresponds to n=4 till n=8. The vertical line goes from k=0 to k=5, i.e. there are the first 6 singularities of \( f_n \) on each of these lines.

   

If one mentally prolonges these lines to the right it appears to be clear that the singularities converge to the real axis. More precisely it appears that
every point on the real line in the domain of definition of the base change has in each neighborhood a singularity of \( f_n \) for some \( n \).