10/23/2016, 02:50 PM
(This post was last modified: 10/23/2016, 04:43 PM by sheldonison.)
(10/22/2016, 10:29 PM)Vladimir Reshetnikov Wrote: I found an analytic function given by an explicit series, that generalizes the tetration in the sense that it reproduces its values for all positive integer heights and satisfies the same functional relation for all arguments in its domain of analyticity. Numerically it seems to match Kneser's solution, a PARI/GP program for which was posted on this forum. Any questions and critique are welcome. A short note explaining my solution is attached and is also available at https://tinyurl.com/tetration
Strictly speaking, Kneser's solution only applies to real bases greater than exp(1/e) and my inclusion of the Schröder based solution for real bases<exp(1/e) is a mistake that has caused a lot of needless confusion.
More recently, I posted a pari-gp program called fatou.gp which generates the Abel solution (or slog) in a method equivalent to Kneser's solution. fatou.gp works for complex bases (as well as complex heights). For bases<exp(1/e), the analytically continued solution is no longer real valued, due to a combination of the singularity at exp(1/e) and the usage of both fixed points. The singularity at exp(1/e) is surprisingly mild, and the effect is very small if the base is too close to exp(1/e), but sexpinit(1.3) shows that sexp(z) is no longer real valued. The Schröder solution is not the analytic continuation of Kneser.
Nonetheless Schröder's functional equation is the preferred solution for tetration for bases<exp(1/e), and your equations should match that if they are correct. See https://en.wikipedia.org/wiki/Schr%C3%B6...s_equation
- Sheldon