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 Definition Holomorphic function characterized in that at integer values of its argument it can be interpreted as iterated exponent. [d] [e]
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This article significantly differs from (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. Dmitrii Kouznetsov 01:00, 5 November 2008 (UTC)

How new

Hi Dmitrii, I know nothing about math but I could try and copy edit some parts if it is needed. Just how new is what you have added here? From you text below it sounds like it is almost all your own ideas, and if so how much is original thought? For me, I can't place it in the big picture of math. Chris Day 15:10, 7 November 2008 (UTC) (I write this before reading the whle article, this is based on you comments on the talk page)
Hi Chris. Thank you for your help. I make this section for your questions; and I answer them below, indicating the place of tetration in the big picture of math.
> Just how new is what you have added here?

It is completely new. There is only one paper accepted in Mathematics of Computation about this. Currently, Henryk Trappmann and I work on the detailed mathematical proof of the uniqueness.

> how much is original thought?

The problem originates, roughly, in 1950, when Kneser constructed the holomorphic generalization of exponentials, and, in particilar, . Such generalization can be based on tetration. Since that time, there were many publicaitons; they exressed doubts in uniqueness of analytic extension of tetration, but no advances in construction of the unique extension.

Now about the big picture of math you mentioned. Even at the strong zoom-out, the picture is sitll big; so, I show only the part called Mathematical analysis. I use the mathematical notations:

has only one argument;

and so on. Operation ++ may be called zeration, addition (or simmation) may be called unation, multiplication may be called duation, exponentiation may be called trination. The following operations are tetration, pentation and so on. Manipulation with the holomorphic extensions and the inverses of summation, multiplication, exponentiation form the core of the mathematical analysis.

Now you see the place of tetration in the big picture of math. It is up-to-last raw in the table above. Thank you, Chris; your questions are important. Dmitrii Kouznetsov 09:12, 8 November 2008 (UTC)

P.S. Chris Day had revealed the missed section in the article. I fixed this, using the text above. Dmitrii Kouznetsov 11:32, 8 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 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)


It seems, we can remove the condition

bounded in the range ,

from the definition. I typed it because the proof of uniqueness by Henryk Trappmann uses it. I am preparing the proof of uniqueness of tetration, that does not use this boundness; the holomorphism is sufficient. Roughly: if we slightly deform the tetration along the real axis, then, the holomorphic extension has singularities, which limit the range of the holomorphizm. In vicinity of these singulatities, the "deformed tetration" has values with ampllitude larger than any given real number. If you can advance with this faster than I do, go ahead: type the proof and remove the extra condition from the definiiton. Dmitrii Kouznetsov 11:49, 8 November 2008 (UTC)

cut for several articles?

I added the beautiful pics, and the article become slow to load. I would divide it to several articles. I prepare the update at Dmitrii Kouznetsov 07:59, 4 June 2011 (UTC)