06/17/2014, 09:48 PM
(06/17/2014, 06:16 PM)sheldonison Wrote:(11/21/2011, 11:19 PM)in Nov 2011, Sheldon Wrote: ...The algorithm I used to generate a seed value was to start with sexp(z) from the primary fixed point, and use \( \text{sexp}(z-\sin(\frac{2\pi z} {2\pi})) \)... Then this initial approximation required an additional 42 iterations, generating ... the sexp(z) approximation around z=-1. This gave results accurate to ~32 decimal digits.
That's from the alternate fixed point post I made, #19. \( \theta(z)=\text{slog}(\text{tet}_{\text{alt}}(z))-z \) is a 1-cyclic analytic function at the real axis, even though such an alternate fixed point tet(z) doesn't have an analytic inverse at integers.... And, as you might expect, theta(z) is not entire since the fixed points are completely different in the complex plane. \( \theta(z) \) is analytic if \( |\Im(z)|\approx<0.38008 \), since tet_alt(0.38008-0.51390)~=0.3181+1.337, which is the fixed point for the Kneser primary tetration function. And at the theta(z) singularity, tet_alt(z)=tet_primary(z+theta(z)) is analytic. Should I post a graph of theta(z) near the singularity? I could also generate and post the Fourier series for \( \theta(z) \). This is a fascinating function, z+theta(z)=slog(tet_alt(z)). It circles around the integer values of tet(z) as you pass by at constant Im(z)>0. If z=-2, this means circling around a logarithmic singularity...
Yes, it would be interesting to see a graph.
