Cheta with base-change: preliminary results

[jaydfox]

Unfortunately, this means my observation about the apparent radius of convergence was correct. It wasn't a fluke of the calculations. I'll try to identify a potential singularity, but assuming I can't (only 128 terms, after all), I'll try to get more terms.

It seems that 128 terms is enough to at least get a rough idea of where the singularities are (a conjugate pair). Here are graphs I computed, using z with a fixed magnitude and varying the argument from -pi to pi (one full circle of fixed radius in the complex plane).

Note that I added each new graph to the previous, so you can see that it's fairly stable around most of the circle.

First up is sexp(z), using the first 128 terms, where the magnitude of z is 0.455, and the argument is varied through -pi to pi:
   

Next up is the same graph, plus the graph of sexp(z), with mag(z) equal to 0.460:
   

Next I add the graph with with mag(z) equal to 0.465. Note that it starts to get a little "bumpier":
   

With this next one, the bumpiness is much more obvious:
   

Finally, the bumpiness turns into near chaos:

   

It just get worse after that, but all over the place, so it's not insightful to go much further than 0.475. With more terms to the power series, hopefully the picture will be a little clearer.