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

[jaydfox]

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!