08/15/2011, 02:59 PM
(08/15/2011, 03:44 AM)mike3 Wrote: Hi.
It's been a while but I figured I could post my attempt to tetrate complex bases with the "Kneser"/Riemann mapping algorithm.
This is the code I've got, which will tetrate a complex base, here base \( 3 + i \):
....
It does not work for all complex bases, though. It seems to work only for those relatively near the real axis. Increasing the ImagOffset (how far off the real axis we sample the Fourier integrals) appears to extend the range, however. When it fails, it looks like the iteration used to analytically continue the Schroder function gets sucked into the wrong fixed point. Any way to resolve this? The aforementioned increasing of the ImagOffset causes losses in accuracy and efficiency, and is not a cure-all that allows for tetrating all complex bases outside the STR with the possible exception of the possibly singular base 0.
Mike,
Very nice!
For base=3+i, I can calculate two repelling fixed points, each with different periods. Presumably, the upper half of the complex plane converges towards the first fixed point, and the lower half of the complex plane converges towards the second fixed point, while the merged function allows you to maintaining the definition of sexp(0)=1, and sexp(1)=base.
L1=0.324536386411256 + 1.00148180609593*I
L2=-0.0976763924712228 - 1.39768129114470*I
I assume the algorithm works when both bases are repelling?
- Sheldon
