Talk:Tetration

This article significantly differs from http://en.wikipedia.org/wiki/Tetration (Even fig.1 was not accepted there). I should greatly appreciate indication and/or correction of any misprints, miswordings and errors (if any) in the text below. Dmitrii Kouznetsov 01:00, 5 November 2008 (UTC)

About creation of this article

I feel, I should type some apology about creation and editing of this article.

The intent of this page was to collect efforts of several researchers in creation of the complete and rigorous deduction of the holomorphic extension of tetration. I planned my own role as an artist, the illustrator, and the applier of this operation to the quantum mechanics, and, in particular, theory of lasers and the fiber optics.

The main content of this article was supposed to be detailed mathematical proof of the existence and the uniqueness of the holomorphic extension of tetration. Henryk Trappmann helped me to formulate the most important part of the article: the definition of the tetration, which allows such a generalization.

Now it happens, that the mathematical proof of the existence is not yet ready (although Henryk and I currently work on this proof), but I already have the precize (14 correct decimal digits) and realtively fast implementation for the tetration and its derivative and its inverse (at least for b=2 and b=e), and I already have generated many pictures for the tetration and the related functions. I consider these pictures as very beautiful, and some of my colleagues have the same opinion. Therefore I post the most important of them with short description as the article. With my algorithms, I already have answered all the questions I had about tetration at the beginning of this activity, and I have no doubts in the existence and uniqueness of this function. I hope, soon we'll be able to present also the formal proof.

I encourage the creators and the developers of mathematical software (Mathematica, Maple (software), Matlab, C and C++ and Fortran compilers, etc., to consider implementation of tetration in their packets in two independent ways:

• 1. As a function which deserves to become not only a special function, but elementaty function in the same way, as summation, multiplication and exponentiation are. Tetration should be considered as fourth among the basic arithmetic operations.
• 2. For implementation of really HUGE real numbers. The presentation of a huge number in the form $\mathrm {HUGE} =\mathrm {tet} (x)$ may avoid "floating overflow" in the numerical analysis. I understand, that the precision of a number stored in such a way will not be able to compete with that of the conventional floating point (mantissa, logarithm) representation, but this should be excellent tool for debugging of the alforithms for the combinatorics, the theory of computability and, I hope, the quantum mechanics. The idea of beeing able to count the Feynman trajectories is really attractive.

I believe, that this artilce opens the new branch of mathematical analysis, that allows the unambiguous holomorphic extension of solutions of various recursive equations, and, in particular, those of the Abel equation. For this reason, I had to type and to edit this article, and I cannot act in a different way. I hope, you understand and accept my apology.

Dmitrii Kouznetsov 08:58, 5 November 2008 (UTC)