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

[sheldonison]

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).

Just a quick observation, to make sure we are speaking of the same thing. When converting from base eta to base e, I find that 5.0179 goes to 0, not 1, and which corresponds with sexp(-1), not sexp(0).

I get confused sometimes when I work with exp(z)-1 instead of actually working with the iterations of eta^z, and sometimes I forget to switch properly between the two. Perhaps you are making the same mistake? A quick sanity check in SAGE is:

Code:

eta = RR(e**(1/e));
print log(log(log(log(log(eta**(eta**(eta**(eta**(eta**5.0179)))))))));

You should get something like -0.000053416. I try to remember that 5.0179 goes to 0, and 6.3344 goes to 1, and then always double check my results inside my iterating functions. If not, it's usually because I am still working in a logarithmic or double logarithmic system.

It's a moot point, for the most part, because it's a trivial shift in the sexp function, but I wanted to make sure it got caught early before you start getting results that don't match mine, and we can't figure out why!

Yes, 5.0179 goes to zero, my mistake, and the strip of interest is 5.0179 to 6.3344. I'm trying to formulating a theory that the singularities go along with a change in the windings, where the windings suddenly increment, so it wouldn't be possible to smooth out the singularity. It seems that the graphs are smooth if a real zero crossing corresponds to an imaginary maximal with a phase of 2*(n+1)*e*pi, then the graph will be smooth, whereas if it corresponds to an imaginary maximum with a phase of 2*n*e*pi, then there will be a singularity nearby. So far I haven't gotten to demonstrating this with the real cheta function, or with the critical strip. I'm on vacation, plus my brain operates slowly when contemplating cheta and base conversions....