08/04/2019, 12:55 PM
(This post was last modified: 08/04/2019, 01:18 PM by Ember Edison.)
(08/02/2019, 09:34 PM)sheldonison Wrote: ……The last edit it's sounds strange. So Will the Kneser Tetration and its inverse can exist if Schroeder function doesn't exist?
[edit1] The theta mapping for these bases greater than exp(-e) seems poorly behaved and requires a larger number of coefficients to converge than one might expect. It would be nice to better understand the theta mapping and its radius of convergence for base=0.15, and its nearest singularity. For B=0.1+I*E-30; I needed 250 terms in the theta mapping! matrix_ir(B,400,250,14/15,45/46) to get a bit over 16 decimal digits of precision as compared with 90 terms for B=0.15 matrix_ir(0.15,400,90,14/15,45/46). Either way, where is the nearest singularity in the theta function? I also don't know why I needed the small imaginary offset for B=0.1; otherwise the Schroeder function to Abel function conversion gets confused, but I don't know why.
[edit2] The Schroeder function for neutral points does not converge if the period is a rational number. So a period=4 Schroeder function doesn't exist since it doesn't converge. But if the period is a well behaved irrational number (with imaginary part=0) then the Schroeder function does converge. For example, a period=pi Schroeder function does converge. See
https://en.wikipedia.org/wiki/Brjuno_number, and http://www.scholarpedia.org/article/Sieg...egel_disks. The Kneser Tetration and its inverse are analytic even if the period is rational. The experiment is to look at all of the Taylor series coefficients of the slog and see how they vary as the base changes in the complex plane. There is an analytic function for each of these Taylor series coefficients. So from the bases around the rational base, one can construct an arbitrarily accurate series even though the fatou.gp algorithm isn't ideal for computation of such neutral rational period bases.
The base = E^-E is a singularity?
And Can you get more digital evidence to "proof" add small imaginary will not add side effects?
Ps: As long as the fatou.gp be faster, I can accept more memory usage. 2GB is good, but I can accept >10GB if you can really speed up (but I don't think you can lot).
