03/08/2016, 10:18 AM
(This post was last modified: 03/08/2016, 10:50 AM by sheldonison.)

(03/08/2016, 03:58 AM)Daniel Wrote: Greetings, the Tetration Forum has been online for almost ten years now. My own tetration website tetration.org is almost thirteen years old and I am reviewing making the site more readable. I haven't closely followed the conversations here for at least several years. What are the central concepts regarding tetration that I can read about here?

Hi Daniel,

Welcome back.

Probably the two biggest ones are that Kneser's solution has been "rediscovered", as the preferred solution for extending tetration to the real and complex numbers. In addition, Henryk Trapmann has published a uniqueness proof for Kneser's solution. I have at least two versions of pari-gp code that implement Kneser's solution, written in pari-gp. The first implements tetration for real bases greater than ; http://math.eretrandre.org/tetrationforu...hp?tid=486; convergence is not proven, but assuming convergence, it can be shown to be equal to Kneser's solution. The second program use both fixed points to generate tetration for complex bases as well, http://math.eretrandre.org/tetrationforu...p?tid=1017; again, convergence is not proven.

This second complex tetration base program generates the Abel function on a sickle, exactly meeting the uniqueness criteria. It works by solving the problem for iterating instead of where , so k=0 (which is parabolic; with a formal asymptotic series for the two solutions) corresponds to base . There is a linear transformation, so the two problems; iterating in y or z, are congruent. This is what they call the perturbed fatou coordinate in complex dynamics, which is equivalent to an slog/abel function starting by perturbing bases . Henryk Trapmann occasionally bugs me to publish my work, but so far, they are only here on his tetration forum.

There are other solutions for tetration as well, Kouznetsov's solution, and Andrew's slog using simultaneous equations, which appear to give the same solution as Kneser's solution, though there is no proof. Andrew's slog converges too slowly to give results of reasonable accuracy, though Jay has an accelerated solution that works up to a little better than double precision, given enough computer time and memory. There is also Walker's solution, generated from the upper solution for , that is infinitely differentiable, but is conjectured to be nowhere analytic! . I can find a link if you like; Jay and I have both done computations for this real valued solution, after "rediscovering" Walker's solution.

Feel free to ask questions, and generate discussion. There is a lot of neat stuff posted here on the forums, though it is sometimes hard to find stuff, especially at first.

- Sheldon Levenstein

- Sheldon