An explanation for this?

[mike3]

What is this proven algorithm

Michael Yampolski referred me to one of his articles [1].
I think you need a really good background in holomorphic dynamics to understand the article. Before that you should understand the construction of Shishikura [2] of the Fatou coordinates/Abel function where he uses the measurable Riemann theorem. Because Gaidashev gives a *constructive* measurable Riemann theorem, i.e. one where there is an algorithm to calculate it, and uses it to calculate the Fatou coordinates, if I understood that properly.

Ah. But I guess my thoughts that I've had that the Kneser, etc. tetrational function is the "best" tetrational function are likely correct:

The Kneser construction is very specific, it constructs the (i.e. the unique) Fatou coordinates for exp (which I showed in [3]) with the (plain) Riemann theorem. Then Jaydfox used an algorithm for the Riemann mapping theorem to compare Kneser's with Andrew's solution.

So would this mean that Andrew's algorithm and Kneser's construction yield the same result?

Just wanted to show the similarity and that it seems we can not circumvent the (measurable) Riemann mapping theorem, which gives no closed form coefficients; which was regarding your question:

, and does it provide any clues to finding the explicit (or close enough, e.g. if in terms of, say, infinite sums) coefficients of the Taylor series

However, I didn't think Riemann's theorem had something to do with whether or not one could give a formula of some kind for the mapping or a function related to the mapping, like \( \mathrm{tet}(z) \). Just that a mapping exists. It is true that pretty much all Riemann mappings cannot be expressed with "elementary" functions, however. Also, I thought that a "closed form" always meant a finitary expression in terms of some "standard" set of operators, e.g. the elementary functions (i.e. functions formed by finitary composition on the first three Ackermann operators and their inverses), and non-elementary standard special functions (hypergeometric, gamma, zeta, etc.). So an infinite sum, product, etc. expansion of the Taylor coefficients would not be "closed form".

But if such an expansion truly does not exist, this would seem to seriously hamper the analysis and even the usefulness of the tetrational function \( \mathrm{tet}(z) \) (note how, e.g. complex factorial is very "useful" and has "deep" connections to other areas of mathematics -- but the complex tetrational function may simply be too bizarre to be either of those). Or perhaps the formula exists, but it requires new kinds of primitive functions.

At the very least, it seems to be pointing at the idea that the tetration is a kind of function that really has no precedent, an exotic, "alien" kind of function that is really a whole lot different from the conventionally accepted set of special functions. Is this a fair assessment?