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

[sheldonison]

However did we silently switch from base change \( \eta\to e \) (\( b=\eta,a=e \)) to the base change \( e\to\eta \) (\( b=e,a=\eta \))?
I think we are more interested in the first one!

Well, they are both important, as singularities in either will be an issue, but I suppose that the change from base eta back to base e should be my more immediate concern.

In theory, most of the principles should be the same. The trivial singularities would be at all branched locations of log^[n-2](1) instead of of log^[n-2](-1), if I'm thinking this through correctly (which I might not be, it's 4 AM here and I just got up to give someone a lift to the airport).

My next question is, how large does k have to get before we encounter singularities? After a very hectic week at work, I'm having an equally hectic time out of town on vacation, and perhaps missed out on a lot of the fun. And still, I don't have time to do this problem proper diligence.

Jay, you confused me converting between base e to eta, instead of eta to e. I agree that the principles are the same, and the singularities seem to be fatal. Also, base eta can be represented as iterated exp(z-1), which is another potential source of confusion. I wanted to step back to the base conversion equation, in the strip where \( \text{sexp_\eta(x) \) varies from 4.38 to 5.02, which would correspond more or less to sexp_e=0 to 1. The hope was to extend such a strip to the complex plane. I wanted to share a few observations about the singularities in this strip, where z = \( \text{slog_\eta(5.02) \). Here f(z)=1, which corresponds to sexp_e(0).

\( f(z) =
\lim_{k \to \infty} \log_e^{\circ k}
\text{sexp_\eta}(z+k)) \)

As a simplification, we look at this equation, between z=4.38 and z=5.02
\( f(z) =
\lim_{k \to \infty} \log_e^{\circ k} \left( \exp_\eta^{\circ k} (z)
\right) \)

I was interested in how large k had to get before we encounter singularities. In this scenario, we first need to find the smallest value of k such that f(z)=e. and ln(ln(ln(e)))=singularity. In this strip, does f(z) for k<=4 reach a value of exactly e or exactly 1? I found some singularities for k=5.

I'm enjoying all the posts; you guys seem to be close to showing the base conversion equation has zero radius of convergence. I don't have enought time while on vacation though.....
- Shel