08/14/2018, 06:43 PM
(This post was last modified: 08/14/2018, 10:33 PM by sheldonison.)

(08/14/2018, 05:51 PM)Chenjesu Wrote: It doesn't quite look like it's arguing for an interpolation of any fractional heights, it looks simply like it's toying around with complex *bases*. I see an interesting looking limit within a paper on the site for what appears to be an inverse relationship for the recursive relation for tetration of a given base, but I don't see a solution in the form of ^{n}b. So, what exactly does that paper accomplish? I also am not concearned with constant basis, I want to know what works for a variable base, like ^{a}x = x^(x^(x^(...))) "a" times.

You might also want to see Henryk Trapmann's paper, Uniqueness of Holomorphic Abel functions at a Complex Fixed Point Pair

At first, I was going to limit my discussion to published references given the Ops initial statement, "if it can't be attributed with a reputable source (which neither wikipedia nor a random forum on the internet is), then it can't be cited for anything".

But, yes, there is an accepted unique extension of the function to arbitrary real and complex valued heights. . This is for real bases>exp(1/e), and even for complex bases. I have written programs like fatou.gp which is available on this website that will calculate it.

Hopefully, the Op will catch up on the mathematical background, perhaps here on this website, or using some of the published links. The mathematics to understand Kneser's analytic tetration solution is pretty high level. You need a comfort level with both Complex Dynamics and Complex Analysis.

Edit: One more published link. I would use the graduate level book by Lennart Carleson and Tehodore Gamelin, "Complex Dynamics", instead of James Nixon's paper since it contains the generic solution for iterating any real valued function with a real valued fixed point and a positive multiplier at the fixed point. Nixon's paper applies only to bounded tetration with real valued bases, 1<b<exp(1/e). I would use Kneser's solution for bases b>exp(1/e) which is a much more difficult problem.

- Sheldon