User:Dmitrii Kouznetsov/Analytic Tetration

Fig.0. Graphic of analytic tetration $f={\rm {sex}}_{2}(x)=\exp _{2}^{x}(1)$ at versus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} .

Abstract

Analytic tetration is defined as mathematical function that coincides witht the tetration at integer values of the argument and is analytic outside the negative part of the real axis. Existence of such a function is postulated; and arguments in favor of uniqueness of such a function are considered. The algorithm of evaluation is suggested. Examples of evaluation, pictures and tables are supplied. The application and the generalization is discussed.

Preface

The colleagues indicated so many misprints in my papers about tetration, posted at my homepage , that I want to give them opportunity to correct them in real time.

Especially I invite

• Arthur Knoebel
• Henryk Trappman
• Andrew Robbins

to edit this file.

I consider the topic very important and urgent. The analytic tetration should be investigated and discussed right now; overvice, the non-analytic extension may become an ugly standard in mathematics of computation; the implementation of hige numbers with non-analytic tetration would make difficult realization of arithmetic operations and cause a lot of incompatibilities.

This is my apology for posting this research now, while the rigorous proof of existence and uniqueness of the analytic tetration is not yet found. My believe is based on the numerical check of the hypothesis of the existence and uniqueness, on smallness of the residual at the substitution of the function to the tetration equation and beauty of the resulting pictures. I cannot imagine that the agreement with 14 decimal digits occurs just by occasion without deep mathematical meaning.

In such a way I apologize for postulating of statements which should be prooven by the rigorous mathematical deduction.

Introduction

Quick start

Roughly, super-exponential

(1) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle  {\rm sex}_b(z)=\exp_b^z(1)}

is combination of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} exponentials on base Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} . Foe example,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~ {\rm sex}_b(0)=\exp_b^0(1)=1}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~ {\rm sex}_b(1)=\exp_b(1)=b}

$~{\rm {sex}}_{b}(2)=\exp _{b}^{2}(1)=\exp _{b}(\exp _{b}(1))=b^{b}$

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~ {\rm sex}_b(3)=\exp_b^3(1)=\exp_b\Big(\exp_b\big(\exp_b(1)\big)\Big)=b^{b^b}}

and so on. However, such definition is good only for positive integer values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} . In general, the superexponential can be defined through the Abel equation

(2) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\exp_b\Big({\rm sex_b}(z)\Big)={\rm sex}_b(z+1) }

with additional condition that

(3) $~{\rm {sex_{b}}}(0)=1$

Then, at least for positive values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~b} and positive integer values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~z} , such a definition can be used for the evaluation of tetration.

In this paper, the way to define tetration for non-integer argument is described. For real values of the argument, at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~b=2} , such a tetration is plotted on figure 0. In the following sections, I describe, why is it so important, how to define the tetration for non-integer values of the argument, how can it be evaluated with high precision and why it is the only correct way to define analytic tetration.

Additional argument

One can consider to add the additional argument, replacing $~\exp _{b}(1)$ to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\exp_b(a)} . This may have sense, while Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~z} is allowed to have only integer values. However, at the implementation of "good" tetration, the "argument" Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~a} can be considered as inverse superexponential of some argument, $~a={\rm {sex}}_{b}(\alpha )$; then, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\exp_b^z(a)={\rm sex}_b\Big({\rm sex}_b(\alpha) \Big)} ; in the way, similar to that of convential logarithms: it is sufficient to investigate properties of natural logarithm ln; then, any other can be expressed as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \log_b(z)=\ln(z)/\ln(b)} .

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle  \exp_b^n(\exp_b^m(z))=\exp_b^{m+n}(z) }

The exponentiation of tetration is equivalent to increment of its argument. While summaton operation forms the group, exponentiation does too.

$\exp _{b}^{m}({\rm {sex}}_{b}(z))={\rm {sex}}_{b}(z+m)$

Inverse function and group properties

In this section, I write Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~F} instead of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~{\rm sex}_b} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\exp(z)} instead of $~\exp _{b}(z)$; onitting indices. However, you may recover them at any moment.

(I am not sure which notation is best. D.)

The speculation of the previous subseciton can be written shorter.

Assume there exist function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~F^{-1}} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~ F\Big(F^{-1}(z)\Big)=z} .

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~s=\exp^u(t)=F\Big(u+F^{-1}(t)\big)} and $~q=\exp ^{v}(s)=F{\Big (}v+F^{-1}(s){\big )}$ .

Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~q =\exp^v\Big( \exp^u(t) \Big) =F\Big(v+F^{-1}\Big(F(u+F^{-1}(t))\Big)\Big) =F\Big(v+u+F^{-1}(t)\Big) } .

You can put subscript Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~_b} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~F} and $~\exp$ in the defuction above, and it will be seen, that we have no need to deal with funciton of 2 variables, considering Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exp_b^z(x)} ; it can be expressed in terms of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_b(z)} . However, we need to specify, what set shold be Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~t} and $~s$ from in the deduction above: must they be positive integer, or they can be real, of they can be also complex numbers.

History of tetration and huge numbers

Perhaps, every researcher used to see diagnostivs "floating overflow" at the evlauation of an expression with huge numbers....

Asymptotic

Assume, the tetration Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} is defined with the Abel equation

(10) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle  \exp_b(F(z))=F(z+1)}

and assume the condition

(11) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle  F(0)=1}

While one deal with real axis, one imagine some real-analytic extension shown in Figure 0 and consider also

(12) $G(z)=F(z+k(z))$

where

(13) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle  k(z)=\sum_{n=-\infty}^{\infty} \alpha_n \exp(2\pi {\rm i} n z)}

Such a funciton is also coluiton of the Abel equation; at

(14) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle  \alpha_{-n}=(\alpha_n)^* }

and

(15) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle  \alpha_0=-\sum_{n\ne 0} \alpha_n}

function $~G$ is real and passes through the same points as </math>~F</math> at integer values of the argument.

and at small values of coefficients Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} , function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~G} looks smooth, and it is difficult to guess, which of them is "true". For this reason, for the standard mathematical representation, the non-analytic stepwice funciton uxp was suggested .

However, the difference between functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~F} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~G} nefomes seen, if one of them, for example, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~F} , is analytic and regular in the upper and lower half-planes; function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~G(z)} will be analytic only within the strip Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~|z|<y_0} ; order of magnitude of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~y_0} can be estimated with

$y_{0}\approx {\rm {MIN}}_{n}~{\frac {\ln(2/|\alpha _{n}|)}{2\pi n}}$

At larger values of the imaginary part of the arguemnt, the periodic function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} takes huge values, including various negative integers. Namely at these values, function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~F} has singulatities.

For abstract excersize in comlex functional analysis,all animals are equal. However, for the applicaitons in the computaional mathematics, some of them are more equal than other. As such a more equal animal we should choose the function with simplest behavior, with minimum of singulatities, and easiest for the evaluation.

Eigenvalues of logarithm

FIg.1. Example of graphic solution of equation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L=\log_b(L)} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=\sqrt{2}} (two real solutions, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L=2} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L=4} ), Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=\exp(1/\rm e)} (one real solution Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L=\rm e} ) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=2} (no real solutions).

Function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} is supposed to grow up (faster than any exponential) in the direction of the real axis. However, it has no need to grow in other directions; neither in the opposite direction, nor in the direction of the imaginary axis.

For computational mathematics, it would be better, if it does not grow at all, approaching some limiting value. Also, for computation, it would be better, if all the signulatities of function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~F} are at the real axis; at values Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2} and less (more negative).

There is not so much opportunity in choise of this limiting value. From the equaiton of Abel, it follows, that the asymptotics should be eigenvalue of logarithm, i.e., solution Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~L } of equation

(20) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle  L=\log_b (L) }

Figure 1 shows the graphic of the left-hand (blask) and the of this equation (black line) and 3 examples of the right-hand side.

Two real solutions exist at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~b< \exp(1/\rm e)} . These solutions are plotted versus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\ln(b) } in Figure 2 with thin black solid line.

One real solution Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~L=\rm e} exists at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~b= \exp(1/\rm e)} .

Two complex solutions exist at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~b> \exp(1/\rm e)} . These solutions are mutially conjugated. The real part is plotted with solid line, and two options for the imaginary part are plotted with dashed line.

FIg.2. parameters of asymptotic of tetration versus logarithm of the base

Exponential asymptotic

Consider tetration with following asymptotic behavior:

(30) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle  F(z)=L+\exp(Qz+r) + o\Big(\exp(Qz+r) \Big)}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~Q } has sense of increment and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~r } determines the amplutude and phase of the wave, exponentially growing from the small values.

Possible values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~Q} are plotted versus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\ln(b)} in FIgure 2 with thick lines.

Quasi-period Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T=2\pi/Q} is plotted with dotted lines.

At small base, both values of quasiperiod are pure imaginary. This periodic or quasi-periodic behavior is similar to that of the conventional exponential, but function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~F} does not grow to infinity in the direction of the real axis, aproaching eigenvalue Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} of logarithm. One example of such a behavior Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~b=\sqrt{2} } is shown in Figure 3 for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~b=\sqrt{2}} . The following section describes, how it was ploted.

At Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~b> \exp(1/{\rm e})} values of period Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} are complex; and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Re(T)>0} . This corresponds to the exponential growth of the asymptotic solution in the direction of the real axis; at large positive values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Re(Qz)} the asymptotic does not approximate the function which grows faster than any exponential.

Small base. Base Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=\sqrt{2}}

Fig.3. Tetration Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f=F(z)} at the complex Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} -plane. Levels of integer Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Re(f)} and levels of integer Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Im(f)} are shown with thick lines. Grid covers with unit step range (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -10\le \Re(z) \le 10 } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -10\le \Im(z) \le 10 } ).

For evaluation of Tetration we need to assume that it exists, and has asymptotic (30).

Consider the case of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 < b < \exp(1/\rm e)} . at fixed values of real part of the argument, the function has periodic behavior in the direction of imaginary axis, as it is shown in Fig. 3. In this figure, the example with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=\sqrt{2}} is used. Function is periodic; period Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T\approx 17.1431~\rm i} .

As all analytic tetrations, function has limiting values;

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle  \lim_{x \rightarrow \infty} F_\sqrt{2}(x+{\rm i}y)=2}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle  \lim_{x \rightarrow -\infty} F_\sqrt{2}(x+{\rm i}y)=4}

At non-zero values of the imaginary part of the argument, the function decays to these asymptotiv values.

At the real axis, the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~F(z)} has cut at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z\le -2} .

Due to the periodicity, the function has cuts also at

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z+T n \le -2} for integer Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~n} .

As all other tetrations, it has singularity at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z=-2} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~F(-1)=0} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~F(0)=1} .

Other, more singular tetrations can be obtained by light periodic deformation of the argument.

Base 2. Ackermann function

FIg.4. Ackermann function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f=A(4,z)=F_2(z+3)-3} at the complex Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} plane. Grid covers the range ( Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -10 \le \Re(z) \le 10 ~,~ -4 \le \Im(z) \le 4 } with unit step. Levels of integer values of the real part and the same for imaginary part are shown with thick lines.

Tertration at base 2 can be expressed through the 4th Ackermann function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_2(z)=A(4,z-3)+3} This Ackermann function is plotted in fig.4. As base Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b > \exp(1/\rm e)} , the function is not periodic. However, there is asymptotic periods Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L^*} in the upper and lower half-planes. In vicinity of positive part of the real axis, the function shows rapid growth, and it is not possible to draw the levels there. All the eingularities are at the negative integer values smaller than -1.

Base e. Natural tetration

Fig.5.Analytic tetration at the base e , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f=F_{\rm e}(z) } in the complex Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} plane. The grid covers the range (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -9 \le \Re(z) \le 9} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -4 \le \Im(z) \le 4} ). Levels Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |f|=\exp(0), \exp(\pm 1), \exp(\pm 2),.. } and levels Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\rm Arg}(f)=0, \pm 1, \pm 2, \pm3 } are shown with thick lines.

Base Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b-\rm e} is the most natural choise. In this case, the increment Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q} is equal to the asymtotic value Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} .

At the translation for inity along the real axis,

Superlogarithm

Fig.7. Inverse of the analytic tetration Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f=F_{\rm e}^{-1}(z)} at the complex Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} -plane.

Inverse of the natural tetration can be considered as superlogarithm. Equilines of funciton Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f=F_{\rm e}^{-1}(z)} are shown in Figure 7. Levels of integer real part and those of integer imaginary part are shown with thick lines. This function has two branchpoints at eigenvalues of logarithm. In the figure, the cuts are placed along the lines Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Re(f)=-2} ; then, the function is regular in vicinity of the real axis, approaching the limitig value Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2} at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\infty} and slowly growing up at positive values of the argument.

Discussion

Analytic tetrations are shown for base Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=\sqrt{2}} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=2} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=\rm e} . In the similar way, the tetration on other bases (for example, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=10} ) can be plotted.

Knowledge of the asymptotic behavior gives the key to the efficient evaluation.

references

See also the discussion at http://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3