08/18/2009, 04:37 AM

(08/17/2009, 05:26 PM)jaydfox Wrote: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....(08/17/2009, 04:01 PM)sheldonison Wrote: I wanted to step back to the base conversion equation, in the strip where 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 = . 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:

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.Code:`eta = RR(e**(1/e));`

print log(log(log(log(log(eta**(eta**(eta**(eta**(eta**5.0179)))))))));

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!