04/17/2012, 12:48 PM
(This post was last modified: 04/17/2012, 02:28 PM by sheldonison.)
(04/17/2012, 07:51 AM)mike3 Wrote: ....
EDIT: I've continued it further, to \( e - 0.5i \) and then back up to \( e \), and noticed nothing "singularitylike". However, I did find that on returning to \( e \), I was once again at a sickel between the two principal fixed points, only now they had been swapped. Wtf? Does this mean there's a merged tetrational using the two fixed points in opposite order? That'd be very bizarre. Or, perhaps, this in fact indicates that there is a failure somewhere (probably at the STR boundary) and the continuation is in fact not possible.
edited Where both fixed points are repelling, there is only one solution, and the fixed points cannot be swapped (check your math, and/or else post of sexp(z)). Henryk has a paper with a uniquess criteria for sexp(z), and I hope to eventually show the theta(z) uniqueness criteria I posted yesterday, \( \theta(z)=\text{slog}(f(z))-z \), can be shown to be complete if both fixed points are repelling and theta(z) has a singularity.
But, inside the Shell Thron boundary, where one of the fixed points is attracting, there are two different solutions for the same base, which also makes the uniqueness criteria more complicated. Are you calculating sexp(z), or slog(z)? I've started to make plots of slog(z) as well, for the bipolar solutions, inside the Shell Thron boundary. Also, as you point out, as you approach the second Shell Thron crossing, the singularities for sexp(z) in the upper half of the complex plane (rotating around eta counterclockwise) start to bunch up and get arbitrarily close to each other and the real axis.
- Sheldon
