08/23/2018, 10:17 AM
(This post was last modified: 08/23/2018, 10:31 AM by sheldonison.)
(08/21/2018, 02:33 AM)Chenjesu Wrote: Well I appreciate you taking the time to right about it. When I look at the formula you posted again though, I can understand how it applies to complex bases, but I still don't see how it addresses complex heights. You're still only picking an integer 2 to define the tetration, but I don't see how it tells me what something like \( ^{i/3}(z) \) outside of maybe the sexp/slog formula that also doesn't seem to have an explicit representation.
Here are a few example bases with b^^(1/3) and b^^(i/3) using Kneser for real bases>exp(1/e), and Schroder for real bases<exp(1/e). I also included b=i, for both Kneser and Schroder. Schroder can only be used to calculate b^^z if there is an attracting fixed point for the limit of b^^n as n gets arbitrarily large. Kneser is not real valued at the real axis for real bases<exp(1/e). Rigorously extending Kneser to complex bases requires an application of perturbed fatou coordinates, which is the recent work of Shishikura in complex dynamics. My fatou.gp program can compute tetration for complex bases. Unfortunately, I only understand understand Shishikura's work at an high level (not in detail); whereas I do understand Kneser in detail. One could use my program fatou.gp to calculate b^^(i/3) for real and complex valued bases and it would be an analytic function in b, with a singularity at b=exp(1/e).
Code:
base algorithm b^^(1/3) b^^(i/3)
1.25 Schroder 1.12034262083907 1.03239262899284+0.143429154452345i
sqrt(2) Schroder 1.17241053375763 1.02664360767221+0.192822433840326i
2 kneser 1.30092658318488 0.998967851029509+0.292980086642819i
e kneser 1.40302138219711 0.970673924797826+0.356232199460891i
3 kneser 1.43437813759630 0.961475968170544+0.373403025656933i
i kneser 1.18928618189822+0.535109505735665i 0.719270646867979+0.215917090873838i
i Schroder 1.15607095286632+0.428615552193531i 0.712178499068194+0.193070761174410i
- Sheldon