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)

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. The section "about creation.." below was written before I saw your comments; so, make new section for your questions; and I answer your questions below.

> Just how new is what you have added here?

I believe it is completely new. There is only one publication accepted in Mathematics of Computation, it deals about case of base $b=\mathrm {e}$. Currently, Henryk Trappmann and I work on the detailen mathematical proof of the statements declared. My believe in these statements is based mainly on the numerical analysis: I got the very small residual at the approximation of equations in definition of tetration with my numerical solution. Deviation of a funciton from this solution for $10^{-10}$ along the real axis, say, at the range between $-1$ and 0, reduces the range of holomorphism, approximately, to the strip $|\Im (z)|<4$ at the complex $z$- plane, at least in the range of the figures. It is difficult to believe that these 10 orders of magnitude are just coincidence.

> how much is original thought?

The problem originates, roughly, in 1950, when Kneser deduced that there exist holomorphic generalization of exponentials, and, in particilar, ${\sqrt {exp}}$. I cite this work. There is a lot of bzz around, see the article in Wikipedia and cites therein. From my point of view, since Kneser, almost nothing was done about holomorphic tetration.

Nobody works with a piece-vice approximation for the exponential or the Gamma function. One could approximate $\Gamma (z)$ the range $1\leq \Re (z)z\leq 2$ with constant, and extend the function for other values of $z$, using $\Gamma (z\!+\!1)=z\Gamma (z)$. For some reason, the colleagues consider in serious the same thing about tetration. They write papers about this, they accept them for publication, and they discuee them seriously.

• M.H.Hooshmand, (2006). "Ultra power and ultra exponential functions". Integral Transforms and Special Functions 17 (8): 549-558. doi:10.1080/10652460500422247.
• M.H.Hooshmand, (2008). "Infra logarithm and ultra power part functions". Integral Transforms and Special Functions 19 (7): 497-507. doi:10.1080/10652460801965555.
• http://en.wikipedia.org/wiki/Uxp , which cites both above.

Perhaps, this activity has sense, but it has nothing to do with advance of holomorphic extension of basic recursive mathematical functions. I copipast the piece I wrot in mathematical notation about most basic functions:

$++$ (which has ony one argument)

$+$

$*$

$\exp$

At least for integers, each operation, beginning with operation +, in this sequance is recurrence of operations from previous row. Most of conventional calculus is based on the operations summation, multiplication, exponentiation; and the inverse functions. These functions are holomorpically extended to the whole complex plane. As for the operation, which is supposed to be next in this sequence, the holomorphic estension is not so widely known. (Perhaps, only few colleagues know about it: you, me, Henryc Trappmann, the participants of his forum, and editors and reviewers of Mathematics of Computation, who had accepted my paper). (I shall type more, I just break for few minites) Dmitrii Kouznetsov 06:49, 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 $\mathrm {HUGE} =\mathrm {tet} (x)$ 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)