User:Dmitrii Kouznetsov/Analytic Tetration: Difference between revisions

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[[Image:AnalyticTetrtionBase2figure0.jpg|200px|right|thumb|Fig.0. Graphic of analytic tetration <math>f=F_2(x)=\exp_2^x(1)</math> at versus <math>x</math>.]]
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{{Image|AnalyticTetrtionBase2figure0.jpg|right|200px|Fig.0. Graphic of analytic tetration <math>f=F_2(x)=\exp_2^x(1)</math> versus <math>x</math>.}}
<!--
<!--
[[Image:ExampleEquationLog01.png|300px|right|thumb|FIg.1. Example of graphic solution of equation <math>L=\log_b(L)</math> for
{{Image|ExampleEquationLog01.png|right|300px|FIg.1. Example of graphic solution of equation <math>L=\log_b(L)</math> for
<math>b=\sqrt{2}</math> (two real solutions, <math>L=2</math> and <math>L=4</math>),
<math>b=\sqrt{2}</math> (two real solutions, <math>L=2</math> and <math>L=4</math>),
<math>b=\exp(1/\rm e)</math> (one real solution <math>L=\rm e</math>)
<math>b=\exp(1/\rm e)</math> (one real solution <math>L=\rm e</math>)
<math>b=2</math> (no real solutions).]]!-->
<math>b=2</math> (no real solutions).}}!-->


{{under construction}}
In this page, I collect pieces and figures, which can be used to build-up [[tetration]].
==Abstract==
==Abstract==


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  (1) <math>F_b(z)=\exp_b^z(1)</math>
  (1) <math>F_b(z)=\exp_b^z(1)</math>


is combination of <math>x</math> exponentials on base <math>b</math>. For example,
is combination of <math>z</math> exponentials on base <math>b</math>. For example,
: <math>~ F_b(0)=\exp_b^0(1)=1</math>
: <math>~ F_b(0)=\exp_b^0(1)=1</math>
: <math>~ F_b(1)=\exp_b(1)=b</math>
: <math>~ F_b(1)=\exp_b(1)=b</math>
: <math>~ F_b(2)=\exp_b^2(1)=\exp_b(\exp_b(1))=b^b</math>
: <math>~ F_b(2)=\exp_b^2(1)=\exp_b(\exp_b(1))=b^b</math>
: <math>~ F_b(3)=\exp_b^3(1)=\exp_b\Big(\exp_b\big(\exp_b(1)\big)\Big)=b^{b^b}</math>
: <math>~ F_b(3)=\exp_b^3(1)=\exp_b\Big(\exp_b\big(\exp_b(1)\big)\Big)=b^{b^b}</math>
and so on. However, such definition is good only for positive integer values of <math>x</math>. In general, the superexponential can be defined through the [[Abel equation]]
and so on. However, such definition is good only for positive integer values of <math>z</math>. In general, the superexponential can be defined through the [[Abel equation]]
  (2) <math>~\exp_b\Big(F_b(z)\Big)=F_b(z+1) </math>
  (2) <math>~\exp_b\Big(F_b(z)\Big)=F_b(z+1) </math>
with additional condition that  
with additional condition that  
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In this paper, the way to define tetration for non-integer argument is described. For real values of the argument, at <math>~b=2</math>, such a tetration is plotted on figure 0. In the following sections,
In this paper, the way to define tetration for non-integer argument is described. For real values of the argument, at <math>~b=2</math>, 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.
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.
===Table of superfunctions===
There exist many basefuncitons such that the supercunctions can be expressed in the closed form.
For some function <math>F</math> such that both <math>F</math> and <math>F^{-1}</math>
can be expressed with simple and explicit formulas, the basefunciton
:<math>H(z)=F(1+H^{-1}(z))</math>.
In such a way, the table below can be constructed.


===Additional argument===
===Additional argument===
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<math>\exp_b^m(F_b(z))=F_b(z+m)</math>
<math>\exp_b^m(F_b(z))=F_b(z+m)</math>
The generalization to non-integer values of <math>m</math> is possible.
While writing formal expressions (and assuming existence of the inverse function <math>F^{-1}</math>), the
specific preperties of exponential function are not used; and the
generalization is possible. Such generalization is described in article [[User:Dmitrii Kouznetsov/loginal]].


===Inverse function and group properties===
===Inverse function and group properties===
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and
and
<math>~\exp(z)</math> instead of <math>~\exp_b(z)</math>;
<math>~\exp(z)</math> instead of <math>~\exp_b(z)</math>;
onitting indices. However, you may recover them at any moment.  
omitting indices. However, you may recover them at any moment.  


(I am not sure which notation is best. D.)
(I am not sure which notation is best. D.)
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===Ambiguity of the real-analytic extension===
===Ambiguity of the real-analytic extension===


==Asymptotic==
==[[Abel equation]]==
Assume, the tetration <math>F</math> is defined with the [[Abel equation]]
Assume, the tetration <math>F</math> is defined with the [[Abel equation]]


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  (11) <math> F(0)=1</math>
  (11) <math> F(0)=1</math>
===Asymptotic===
==Asymptotic==
The analytic extension of tetration <math>~F(z)</math> is supposed to grow
The analytic extension of tetration <math>~F(z)</math> is supposed to grow
fast along the real axis of the complex <math>z</math>-plane,
fast along the real axis of the complex <math>z</math>-plane,
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  (14) <math> x=\log_{b}(x)</math>.
  (14) <math> x=\log_{b}(x)</math>.


[[Image:ExampleEquationLog01.png|300px|right|thumb|FIg.1. Example of graphic solution of equation  
{{Image|ExampleEquationLog01.png|right|300px|FIg.1. Example of graphic solution of equation  
<math>x=\log_b(x)</math> for
<math>x=\log_b(x)</math> for
<math>b=\sqrt{2}</math> (two real solutions, <math>x=2</math> and <math>x=4</math>),
<math>b=\sqrt{2}</math> (two real solutions, <math>x=2</math> and <math>x=4</math>),
<math>b=\exp(1/\rm e)</math> (one real solution <math>x=\rm e</math>)
<math>b=\exp(1/\rm e)</math> (one real solution <math>x=\rm e</math>)
<math>b=2</math> (no real solutions).]]
<math>b=2</math> (no real solutions).}}


Solutions of equation (14) are called [[fixed point]]s of logarithm.
Three examples of graphical solution of equation (14) are shown in figure 1 for  
Three examples of graphical solution of equation (14) are shown in figure 1 for  
<math>b=\sqrt{2}</math>,  
<math>b=\sqrt{2}</math>,  
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At  
At  
<math>b=\exp(1/\rm e)</math> there esist one solution  
<math>b=\exp(1/\rm e)</math> there exist one solution  
<math>x=\rm e</math>.
<math>x=\rm e</math>.


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<math>x=L_{\rm e} \approx 0.318131505204764135312654+1.33723570143068940890116 \!~\rm i</math>
<math>x=L_{\rm e} \approx 0.318131505204764135312654+1.33723570143068940890116 \!~\rm i</math>
and<br>
and<br>
<math>x=L_{\rm e}^* \approx 0.318131505204764135312654+1.33723570143068940890116 \!~\rm i</math>.
<math>x=L_{\rm e}^* \approx 0.318131505204764135312654-1.33723570143068940890116 \!~\rm i</math>.


Few hundred straightforward iterations of equation (14) are sufficient to get the error smaller than the last decimal digit in the approximations above.
Few hundred straightforward iterations of equation (14) are sufficient to get the error smaller than the last decimal digit in the approximations above.
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<math>L_1=L_2      </math> takes place.
<math>L_1=L_2      </math> takes place.


[[Image:TetrationAsymptoticParameters00.jpg|700px|right|thumb|FIg.2. parameters of asymptotic of tetration versus logarithm of the base]]
{{Image|TetrationAsymptoticParameters01.jpg|right|700px|FIg.2. parameters of asymptotic of tetration versus logarithm of the base}}


The thin black solid curve at  
The thin black solid curve at  
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===Increment and periodicity===
===Increment and periodicity===
Parameter  
Parameter  
<math> Q </math> in equaiton (13) has sense of asymptotic increment. Consider possible values of  
<math> Q </math> in equaiton (13) has sense of asymptotic increment. Consider possible values of <math>Q</math>.
<math>Q</math>.


Substitution of expression (13) into the eq. (12) gives<br>
Substitution of expression (13) into the eq. (12) gives<br>
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In particular, at <math>b=\sqrt{2}</math>, the periods are
In particular, at <math>b=\sqrt{2}</math>, the periods are


<math> T_{\rm b,1} = \frac{2\pi \rmi}{Q_{b,1}}\approx -17.143148179354847104 \!~\rm i]
<math> T_{\rm b,1} = \frac{2\pi \rm i}{Q_{b,1}}\approx -17.143148179354847104 \!~\rm i
</math><br>
</math><br>
<math> T_{\rm b,2} = \frac{2\pi \rmi}{Q_{b,2}}\approx 19.236149042042854710 \!~\rm i]
<math> T_{\rm b,2} = \frac{2\pi \rm i}{Q_{b,2}}\approx ~ 19.236149042042854710 \!~\rm i
</math><br>
</math><br>


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Various functions may satisfy equations (2),(3).
Various functions may satisfy equations (2),(3).
For an abstract excersise in comlex functional analysis, [[all animals are created equal]]. However, for the applicaitons in the computational mathematics, [[some of them are more equal than other]]
For an abstract excersise in comlex functional analysis, [[all animals are created equal]]. However, for the applicaitons in the computational mathematics, [[some of them are more equal than other]]
<ref name="orwell">'''All animals are created equal''' and that with addition '''BUT SOME ANIMALS ARE MORE EQUAL THAN OTHERS''' are slogans of animals in the novel [[Animal farm]] by [[George Orwell]]. (Animals gained the superior power at the farm, bushing out the farmer). First slogan was used to gain the power. The second part appeared when pigs begun to justify their privilegies.</ref>.
<ref name="orwell">'''All animals are created equal''' and that with addition '''BUT SOME ANIMALS ARE MORE EQUAL THAN OTHERS''' are slogans of animals in the novel [[Animal farm]] by George Orwell. (Animals gained the superior power at the farm, bushing out the farmer). First slogan was used to gain the power. The second part appeared when pigs begun to justify their privilegies.</ref>.
As such a ''more equal animal'' we should choose the solution with simplest behavior, with minimum of singulatities, and easiest for the evaluation.
As such a ''more equal animal'' we should choose the solution with simplest behavior, with minimum of singulatities, and easiest for the evaluation.


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At <math>~b> \exp(1/{\rm e})</math> values of period <math>T</math> are complex; and  
At <math>~b> \exp(1/{\rm e})</math> values of period <math>T</math> are complex; and  
<math>\Re(T)>0</math>. This corresponds to the exponential growth of the asymptotic solution in the direction of the real axis; at large positive values of <math>\Re(Qz)</math> the asymptotic does not approximate the function which grows faster than any exponential.
<math>\Re(T)>0</math>. This corresponds to the exponential growth of the asymptotic solution in the direction of the real axis; at large positive values of <math>\Re(Qz)</math> the asymptotic does not approximate the function which grows faster than any exponential.
==Examples of tetration at different bases==
===Small base. Base <math>b=\sqrt{2}</math>===
===Small base. Base <math>b=\sqrt{2}</math>===
[[Image:AnalyticTetrationBaseSqrt2v00.jpg|800px|right|thumb|Fig.3. Tetration <math>f=F(z)</math> at the complex <math>z</math>-plane.
{{Image|AnalyticTetrationBaseSqrt2v00.jpg|right|220px|Fig.3a. Tetration <math>f=F(z)</math> at the complex <math>z</math>-plane.
Levels of integer <math>\Re(f)</math> and levels of integer <math>\Im(f)</math> are shown with thick lines.
Levels of integer <math>\Re(f)</math> and levels of integer <math>\Im(f)</math> are shown with thick lines.
Grid covers with unit step range  
<!--Grid covers with unit step range  
(<math>-10\le \Re(z) \le 10 </math>,
(<math>-10\le \Re(z) \le 10 </math>,
<math>-10\le \Im(z) \le 10 </math>).]]
<math>-10\le \Im(z) \le 10 </math>).!-->}}


For evaluation of Tetration we need to assume that it exists, and has asymptotic (30).
For evaluation of Tetration we need to assume that it exists, and has asymptotic (30).
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As all other tetrations, it has singularity at <math>z=-2</math>,  
As all other tetrations, it has singularity at <math>z=-2</math>,  
<math>~F(-1)=0</math> and <math>~F(0)=1</math>.
<math>~F(-1)=0</math> and <math>~F(0)=1</math>.
===Base <math>b=\sqrt{2}</math>, alternative tetration <math>G</math>===
{{Image|AnalyticTetrationBaseSqrt2u00.png|left|240px|Fig.3b. Tetration <math>f=G(z)</math> at the complex plane <math>z=x+{\rm i}y</math>.
Levels of integer <math>\Re(f)</math> and levels of integer <math>\Im(f)</math> are shown with thick lines.
<!--Grid covers with unit step range
(<math>-10\le \Re(z) \le 10 </math>,
<math>-10\le \Im(z) \le 10 </math>).!-->}}
Along the real axis, such a solution grows up faster than any exponential; it is always larger than 4 and therefore cannot be equal to unity; so, it is not possible to satisfy condition <math>G(0)=0</math> just shifting its argument for some constant. The behavior of tetration <math>G_{\sqrt{2}}</math> along the real axis makes it similar to that of tetrations with <math>b > \exp(1/\rm e)</math>; although, the larger
<math>b</math>, the larger is the growth.
===More alternative tetrations===
Alternative tetration <math> G </math> as solution of equation (1) can be calculated if we withdraw condition (2), and substitute it to
<math>
G(z)=L_{b,2}+\exp(Q_2 z)+o( \exp(Q_2 z))
</math>
Such a solution is shown in Figure 3b.


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


=== Base 2. Ackermann function===
=== Base 2. Ackermann function===
[[Image:Analytic4thAckermannFunction00.jpg|600px|right|thumb| FIg.4.
{{Image|Analytic4thAckermannFunction00.jpg|right|600px| FIg.4.
Ackermann function <math>f=A(4,z)=F_2(z+3)-3</math> at the complex <math>z</math> plane.
Ackermann function <math>f=A(4,z)=F_2(z+3)-3</math> at the complex <math>z</math> plane.
Grid covers the range (
Grid covers the range (
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-10 \le \Re(z) \le 10 ~,~
-10 \le \Re(z) \le 10 ~,~
  -4 \le \Im(z) \le  4 </math> with unit step.
  -4 \le \Im(z) \le  4 </math> with unit step.
Levels of integer values of the real part and the same for imaginary part are shown with thick lines.]]
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]]
Tertration at base 2 can be expressed through the 4th [[Ackermann function]]
<math>F_2(z)=A(4,z-3)+3</math>
<math>F_2(z)=A(4,z-3)+3</math>
This Ackermann function is plotted in fig.4. As base  
This Ackermann function is plotted in fig.4. As base  
<math>b > \exp(1/\rm e)</math>, the function is not periodic. However, there is asymptotic periods
<math>b > \exp(1/\rm e)</math>, the function is not periodic. However, there are asymptotic periods
<math>L</math> and <math>L^*</math>
<math>L</math> and <math>L^*</math>
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 singularities of the Ackermann function are at the negative integer values smaller than <math>-4</math>.
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 singularities of the Ackermann function are at the negative integer values smaller than <math>-4</math>.
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At the translation for inity along the real axis,
At the translation for inity along the real axis,


===Superlogarithm===
===Base e. Superlogarithm===
[[Image:Slogez01.jpg|300px|right|thumb|Fig.7. Inverse of the analytic tetration  
{{Image|Slogez01.jpg|left|300px|Fig.7. Inverse of the analytic tetration  
<math>f=F_{\rm e}^{-1}(z)</math> at the complex <math>z</math>-plane.]]
<math>f=F_{\rm e}^{-1}(z)</math> at the complex <math>z</math>-plane.}}


Inverse of the natural tetration can be considered as superlogarithm.
Inverse of the natural tetration can be considered as superlogarithm.
Line 389: Line 421:
<references/>
<references/>
See also the discussion at http://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3
See also the discussion at http://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3
[[Category:Mathematics]]
 
[[Category:Mathematical analysis]]
===Table of superfunctions===
[[Category:Funcitons]]
There exist many basefuncitons such that the supercunctions can be expressed in the closed form.
[[Category:tetration]]
For some function <math>F</math> such that both <math>F</math> and <math>F^{-1}</math>
can be expressed with simple and explicit formulas, the basefunciton
:<math>H(z)=F(1+F^{-1}(z))</math>.
In such a way, the table below can be constructed.
{|class="wikitable"
! <math>H(z)</math>
! <math>F(z)</math>
! <math>F^{-1}(z)</math>
! comment
|-
|<math>z+\!\!+</math>||<math>b+z</math>||<math>z-b</math> ||
|-
|<math>z+b</math>||<math>bz+c</math> || <math>(z-c)/b</math> ||
|-
|<math>cz+c</math>||<math>(b^z+\frac{c}{1-b})d</math>||<math>\log_b\frac{z-\frac{cd}{1-b}}{d}</math>||
|-
|<math>b^z</math> || <math>\mathrm{tet}_b(z)</math> || <math>\mathrm{arctet}_b(z)</math>||[[tetration]]
|-
|<math>z^b</math>|| <math>\exp(b^z)</math> || <math>\log_b(\ln(z))</math>||
|-
|<math>\big(a^b+z^b\big)^{1/b}</math>||<math>az^{1/b}</math>||<math>(z/a)^b</math>||
|-
|<math>\ln(a+\mathrm{e}^z)</math>||<math>\ln(az)</math>||<math>\exp(z)/a</math>||
|-
|<math>az/(z+a)</math>||<math>a/z</math>||<math>a/z</math>||amaising, <math>F=F^{-1}</math>
|-
|<math>2z^2-1</math>||<math>\cos(\pi 2^z)</math>||<math>\log_2(\arccos(z)/\pi)</math>||
|-
|<math>H(z)</math>||<math>F(z)</math>||<math>F^{-1}(z)</math>||
|-
|<math>P(H(Q(z)))</math>||<math>P(F(z))</math>||<math>F^{-1}(Q(z))</math>|| P(Q(z))=z
|-|}

Latest revision as of 10:20, 14 June 2024


The account of this former contributor was not re-activated after the server upgrade of March 2022.


Fig.0. Graphic of analytic tetration versus .

In this page, I collect pieces and figures, which can be used to build-up tetration.

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 [1], that I want to give them opportunity to correct them in real time.

Especially I invite

  • Arthur Knoebel
  • Henryk Trappmann
  • 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)

is combination of exponentials on base . For example,

and so on. However, such definition is good only for positive integer values of . In general, the superexponential can be defined through the Abel equation 

(2)

with additional condition that 

(3)

Then, at least for positive values of and positive integer values of , 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 , 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.

Table of superfunctions

There exist many basefuncitons such that the supercunctions can be expressed in the closed form. For some function such that both and can be expressed with simple and explicit formulas, the basefunciton

.

In such a way, the table below can be constructed.

Additional argument

One can consider to add the additional argument, replacing to . This may have sense, while is allowed to have only integer values. However, at the implementation of "good" tetration, the "argument" can be considered as inverse superexponential of some argument, ; then, ; 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 .

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

The generalization to non-integer values of is possible. While writing formal expressions (and assuming existence of the inverse 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}} ), the specific preperties of exponential function are not used; and the generalization is possible. Such generalization is described in article User:Dmitrii Kouznetsov/loginal.

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 ~F_b} and 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 ~\exp_b(z)} ; omitting 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 .

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 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(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 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 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} 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 . 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 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} 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....

Ackermann functions

Ambiguity of the real-analytic extension

Abel equation

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)

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}

Asymptotic

The analytic extension of 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(z)} is supposed to grow fast along the real axis of 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, at least for some values of 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} . However, it has no need to grow infinitely in the direction of imaginary axis. For mathematics of computation, it would be better, if 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(x+{\rm i} y)} remains bounded 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 y\rightarrow +\infty} . Consider this possibility.

Assume, there exist 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 F(z)} of equations (10), (11), analytic in 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 \Re(z) \ge 0} , with, probably, countable number of cuts and singularities at , with exponential asymptotic behavior 

(12)

within some range, 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  \varepsilon(z)=\exp(Qz+r) }
,

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} 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} are fixed complex numbers, 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} is eigenvalue of logarithm, solution of equation 

(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  x=\log_{b}(x)}
.
© Image: Dmitrii Kouznetsov
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 x=\log_b(x)} 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 x=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 x=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 x=\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).

Solutions of equation (14) are called fixed points of logarithm. Three examples of graphical solution of equation (14) are shown in figure 1 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}} , 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)} , 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=2} .

The black line shows 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 y=x} in 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 x,y} plane. The colored curves show 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 y=\log_b(x)} for cases 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}} (red), 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)} (green), 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=2} (blue).

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=\sqrt{2}} , there exist 2 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 x=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 x=4} .

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)} there exist one 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 x=\rm e} .

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=2} , there are no real solutions.

In general,

  • 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)} there are two real solutions
  • 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)} , there is one soluition, and
  • 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)} there esist two solutions, but they are complex.

In particular, 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=\sqrt{2}} , the solutions are
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=L_{\sqrt{2},1}=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 x=L_{\sqrt{2},2}=4 }
.

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} , the solutions are
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=L_2 \approx 0.824678546142074222314065+1.56743212384964786105857 \!~\rm i } 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 x=L_2^*\approx 0.824678546142074222314065-1.56743212384964786105857 \!~\rm i } .

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=\rm e} , the solutions are
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=L_{\rm e} \approx 0.318131505204764135312654+1.33723570143068940890116 \!~\rm i} 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 x=L_{\rm e}^* \approx 0.318131505204764135312654-1.33723570143068940890116 \!~\rm i} .

Few hundred straightforward iterations of equation (14) are sufficient to get the error smaller than the last decimal digit in the approximations above.

The 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 x=L_1 } 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 x=L_2 } of equation (14) are plotted in figure 2 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 \beta=\ln(b)} with thin black lines. 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 L_1<L_2 } , and only 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 \ln(b)=1/e } , the equality 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_1=L_2 } takes place.

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

The thin black solid curve 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 \beta \ge 1/\rm e} represents the real part of the 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 } 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^* } of (14); the thin black dashed curve represents the two options for the imaginary part; the two solutions are complex conjugaitons of each other. 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 \Re(L)>0} .

Increment and periodicity

Parameter 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 } in equaiton (13) has sense of asymptotic increment. Consider 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} .

Substitution of expression (13) into the eq. (12) gives
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+\varepsilon(z) {\rm e}^Q+o(\varepsilon(z))=\exp\!\Big(\ln(b)(L+\varepsilon(z)+o(\varepsilon(z))\Big) } Using equation (14) gives

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+\varepsilon(z){\rm e}^Q+o(\varepsilon(z)) = L~\Big(1+ \ln(b)\varepsilon(z) +o(\varepsilon(z))\Big) }

Then 

(16) 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=\ln(\ln(b))+\ln(L) }
.

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 \ln(b)<1/\rm e } , the two real increments correspond the the two real eigenvalues of logarithm. 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 Q_1=\ln(\ln(b))+\ln(L_1) } 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 Q_2=\ln(\ln(b))+\ln(L_2) } . Both 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_1 } 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 Q_2 } are plotted in Figure 2 with thick solid 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 Q_1\le 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 Q_2\ge 0 } , and only 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 \ln(b)=1/\rm e } the equality 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_1= Q_2= 0 } holds.

In such a way, 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 \ln(b)<1/\rm e} , there exist positive increment, which corresponds 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 L_2} and means, that the perturbation 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 \epsilon(z)} of the stationary solution grows up in the diredtion of real axis; there exist negative increment (decrement, if you like), which corresponds 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 L_1} and indicates, that the perturbation grows up in the opposite direction. 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 0<\ln(b)<2/\rm e} , there may exist 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(z)} with two different asumptotics. 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(z)} it decays to 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 L_1} and At large negative 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(z)} it decays to 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 L_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 \ln(b)\ge 1/\rm e } , there exist two possible increments, 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} 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 Q^*} . The real part 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} is plotted with thick solid line, and the imaginary part is shown with thick dashed line. The real part is positive; id est, the perturbation 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 \varepsilon(z)} grows up in the direction of real axis.

For computational mathematics, it would be good ot construct the solution, that decays both in the direction of the imaginary asis and in the opposite direction. Such a solution should have asymptotics (12),(13) at large positive values of the imaginary part of the argument and its conjugation at large negative imaginary part of the argument.

The exponential asymptotics implies the asumptotic perioditity. The 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=\frac{2 \pi{\rm i}}{Q} }

is shown in FIgure 2 with dotted lines; for the case when period is negative, the minus period is plotted. It is amaising, that 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 \ln(b)>1/\rm e} , the imajinary part is slightly larger than unity and remains almost constant.

In particular, 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=\sqrt{2}} , the periods are

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_{\rm b,1} = \frac{2\pi \rm i}{Q_{b,1}}\approx -17.143148179354847104 \!~\rm i }
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_{\rm b,2} = \frac{2\pi \rm i}{Q_{b,2}}\approx ~ 19.236149042042854710 \!~\rm i }


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} , the asymptotic 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= 5.58414243554338946020010 + 1.05421836033693734654000\!~ \rm i} ;
and 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=\rm e} , the asymptotic 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= 4.44695072006700782711227 + 1.05793999115693918376341 \!~ \rm i} .

Uniqueness

While one deal with solutions of system (10), (11) at the real axis, one imagine some real-analytic extension shown in Figure 0 and consider also 

(100) 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)=F(z+k(z))}

where 

(101) 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 soluiton of the Abel equation; at 

(102) 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 

(103) 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 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} is real and passes through the same points as </math>~F</math> at integer values of the argument.

and at small values of coefficients , 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 [2].

However, the difference between functions 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} becomes 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 some wide range, 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

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 \approx {\rm MIN}_n ~ \frac{ \ln(2/|\alpha_n|)}{2\pi n}}

At larger values of the imaginary part of the argument, the periodic function 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.

Various functions may satisfy equations (2),(3). For an abstract excersise in comlex functional analysis, all animals are created equal. However, for the applicaitons in the computational mathematics, some of them are more equal than other [3]. As such a more equal animal we should choose the solution with simplest behavior, with minimum of singulatities, and easiest for the evaluation.


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.

Examples of tetration at different bases

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.3a. 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.

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} .

The 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 .

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} .

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}} , alternative 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 G}

Fig.3b. 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=G(z)} at the complex plane 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=x+{\rm i}y} . 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.

Along the real axis, such a solution grows up faster than any exponential; it is always larger than 4 and therefore cannot be equal to unity; so, it is not possible to satisfy condition 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(0)=0} just shifting its argument for some constant. The behavior of 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 G_{\sqrt{2}}} along the real axis makes it similar to that of tetrations 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 > \exp(1/\rm e)} ; although, the larger 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} , the larger is the growth.

More alternative tetrations

Alternative 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 G } as solution of equation (1) can be calculated if we withdraw condition (2), and substitute it 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 G(z)=L_{b,2}+\exp(Q_2 z)+o( \exp(Q_2 z)) } Such a solution is shown in Figure 3b.

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 are 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 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 singularities of the Ackermann function are at the negative integer values smaller than 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} .

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,

Base e. 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}} , , 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.

Conclusions

references

  1. ↑ Publications (Those about tetrations are at the top) http://www.ils.uec.ac.jp/~dima/PAPERS
  2. ↑ Hoos
  3. ↑ All animals are created equal and that with addition BUT SOME ANIMALS ARE MORE EQUAL THAN OTHERS are slogans of animals in the novel Animal farm by George Orwell. (Animals gained the superior power at the farm, bushing out the farmer). First slogan was used to gain the power. The second part appeared when pigs begun to justify their privilegies.

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

Table of superfunctions

There exist many basefuncitons such that the supercunctions can be expressed in the closed form. For some 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} such that both 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 F^{-1}} can be expressed with simple and explicit formulas, the basefunciton

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 H(z)=F(1+F^{-1}(z))} .

In such a way, the table below can be constructed.

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 H(z)} 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)} 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}(z)} comment
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+\!\!+} 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+z} 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-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 z+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 bz+c} 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-c)/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 cz+c} 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\frac{z-\frac{cd}{1-b}}{d}}
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^z} 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 \mathrm{tet}_b(z)} 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 \mathrm{arctet}_b(z)} 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 z^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^z)} 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(\ln(z))}
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 \big(a^b+z^b\big)^{1/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 az^{1/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 (z/a)^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 \ln(a+\mathrm{e}^z)} 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(az)} 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)/a}
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 az/(z+a)} 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/z} 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/z} amaising, 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^{-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 \cos(\pi 2^z)} 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_2(\arccos(z)/\pi)}
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 H(z)} 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)} 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}(z)}
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 P(H(Q(z)))} 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 P(F(z))} 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}(Q(z))} P(Q(z))=z
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