Tetration: Difference between revisions

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<math>x</math>.}}
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'''Tetration''' is a rapidly growing [[mathematical function]], which was introduced in the 20th century and proposed for the representation of huge numbers in the [[mathematics of computation]]. For positive integer values of its argument <math>x</math>, tetration <math>\mathrm{tet}_b(x)</math> on base <math>b</math> can be defined with:
'''Tetration''' is a rapidly growing [[mathematical function]], which was introduced in the 20th century and proposed for the representation of huge numbers in the Mathematics of Computation. For positive integer values of its argument <math>x</math>, tetration <math>\mathrm{tet}_b(x)</math> on base <math>b</math> can be defined with:


<math>{\ \mathrm{tet}_b(x) = \ \atop {\ }} {{\underbrace{b^{b^{\cdot^{\cdot^{b}}}}}} \atop {x}}~</math>
<math>{\ \mathrm{tet}_b(x) = \ \atop {\ }} {{\underbrace{b^{b^{\cdot^{\cdot^{b}}}}}} \atop {x}}~</math>
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|author=D.Kouznetsov.
|author=D.Kouznetsov.
|title=Solutions of <math>F(z+1)=\exp(F(z))</math> in the complex <math>z</math>plane.  
|title=Solutions of <math>F(z+1)=\exp(F(z))</math> in the complex <math>z</math>plane.  
|journal=[[Mathematics of Computation]],
|journal=Mathematics of Computation,
|year=2009
|year=2009
|volume=78
|volume=78
Line 92: Line 92:
|author= P.Walker
|author= P.Walker
|title=Infinitely differentiable generalized logarithmic and exponential functions
|title=Infinitely differentiable generalized logarithmic and exponential functions
|journal=[[Mathematics of computation]]
|journal=Mathematics of Computation
|volume=196
|volume=196
|year=1991
|year=1991
Line 125: Line 125:


===Fast growth and application===
===Fast growth and application===
For <math>~b\! >\! \exp(1\!/\!\mathrm{e})~</math> tetration <math>\mathrm{tet}_b(x)</math> grows faster than any exponential function. For this reason tetration has been proposed for the representation of huge numbers in the [[mathematics of computation]]<ref name="w">
For <math>~b\! >\! \exp(1\!/\!\mathrm{e})~</math> tetration <math>\mathrm{tet}_b(x)</math> grows faster than any exponential function. For this reason tetration has been proposed for the representation of huge numbers in the Mathematics of Computation<ref name="w">
P.Walker. Infinitely differentiable generalized logarithmic and exponential functions. Mathematics
P.Walker. Infinitely differentiable generalized logarithmic and exponential functions. Mathematics
of computation, 196 (1991), 723-733.</ref>.
of computation, 196 (1991), 723-733.</ref>.
Line 273: Line 273:
<math>\Re(z)=2~,~\Im(z)=2\pi |T| m~ \forall m \in \mathbb{N}</math>; the cut lines are
<math>\Re(z)=2~,~\Im(z)=2\pi |T| m~ \forall m \in \mathbb{N}</math>; the cut lines are
<math>\Re(z)<2~,~\Im(z)=2\pi |T| m~ \forall m \in \mathbb{N}</math>.  For this value of the base, the period
<math>\Re(z)<2~,~\Im(z)=2\pi |T| m~ \forall m \in \mathbb{N}</math>.  For this value of the base, the period
:<math>T=\frac{2\pi}{\ln^2(2)}\approx −17.1431481793548471041794 ~\mathrm{i}</math>.
:<math>T=\frac{2\pi}{\ln^2(2)} \approx -17.1431481793548471041794 ~\mathrm{i} </math>.


There is a cut at  <math>x\!<\!-2</math>, <math>y\!=\!0</math>; although the jump at this cut reduces at the increase of <math>-x</math>. In such a way, the function approaches its limiting value <math>L\!=\!4</math> almost everywhere, although there is set of singularities at negative integer values of <math>x<-2</math>.
There is a cut at  <math>x\!<\!-2</math>, <math>y\!=\!0</math>; although the jump at this cut reduces at the increase of <math>-x</math>. In such a way, the function approaches its limiting value <math>L\!=\!4</math> almost everywhere, although there is set of singularities at negative integer values of <math>x<-2</math>.
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{{Image|TetrationBase2andE.jpg|right|400px|Fig.6. Tetration  <math>\mathrm{tet}_b(x\!+\!\mathrm{i}y)</math> at base <math>b\!=\!2</math> and <math>b\!=\!\mathrm e</math>.}}
{{Image|TetrationBase2andE.jpg|right|400px|Fig.6. Tetration  <math>\mathrm{tet}_b(x\!+\!\mathrm{i}y)</math> at base <math>b\!=\!2</math> and <math>b\!=\!\mathrm e</math>.}}
At <math> b>\exp(1/\mathrm{e}) </math>, tetration is asymptotically periodic. It decays exponentially
At <math> b>\exp(1/\mathrm{e}) </math>, tetration is asymptotically periodic. It decays exponentially
to the fixed points <math>L</math> and <math>L^*</math> in the upper and lower halfplane. This allows to express it through its values along the imaginary axis, using the [[Cauchi integral]]. <ref name="k">D.Kouznetsov. Solutions of <math>F(z+1)=\exp(F(z))</math> in the complex <math>z</math>plane. [[Mathematics of Computation]], 2008, in press; preprint: http://www.ils.uec.ac.jp/~dima/PAPERS/analuxp99.pdf</ref>.  
to the fixed points <math>L</math> and <math>L^*</math> in the upper and lower halfplane. This allows to express it through its values along the imaginary axis, using the [[Cauchi integral]]. <ref name="k">D.Kouznetsov. Solutions of <math>F(z+1)=\exp(F(z))</math> in the complex <math>z</math>plane. Mathematics of Computation, 2008, in press; preprint: http://www.ils.uec.ac.jp/~dima/PAPERS/analuxp99.pdf</ref>.  


For the base <math>b=2</math> and <math>b=\mathrm{e}</math> tetration  
For the base <math>b=2</math> and <math>b=\mathrm{e}</math> tetration  

Latest revision as of 10:22, 14 December 2024

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Tetration is a rapidly growing mathematical function, which was introduced in the 20th century and proposed for the representation of huge numbers in the Mathematics of Computation. For positive integer values of its argument , tetration on base can be defined with:

Fig.1. Tetration for various versus .

For real values of the argument and various values of the base , this is plotted in Fig.0.

Fig.1. Tetration in the plane with levels .

The map of in the plane is shown in Fig.1 with levels .

Up to year 2011, this function has not been listed among elementary functions, it is not implemented in programming languages and it is not used for the internal representation of data in computers.

In this article, the generalizaiton of tetration for complex (and, in particular, real) values of its argument is described. At base , tetration is assumed to be a holomorphic function, at least for positive values of the real part of its argument. This tetration is used to construct the holomorphic extension of the iterated exponential for the case of non-integer values of the number of iterations.

Definition

For real , Tetration on the base is a function of a complex variable, which is holomorphic at least in the range , bounded in the range , and satisfies conditions

at least within the range .

According to this definition, tetration is superfunction of the exponential. This justifies the alternative name "superexponential" for this function and "superlogarithm" for the inverse function. The definition above generalizes the definitions, recently suggested for the specific cases of base [1] and [2].

Etymology and place of tetration in the big picture of math

Creation of word tetration is attributed to the English mathematician Reuben Louis Goodstein [3].

The place of tetration in the mathematical analysis can be seen at the strong zoom-out of the big picture of math. Using mathematical notation, the zoom-out of the mathematical analysis can be drawn as follows:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle +\!+} has only one argument and means unitary 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 \mathrm{add}_b(1)= b+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 \mathrm{add}_b(z\!+\!1)= b+(z\!+\!1)=+\!+\big(b\!+\!(z)\big)}
Failed to parse (SVG (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{mul}_b(1)=b*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 \exp_b(1)=b^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 \exp_b(z\!+\!1)=b*(\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 \mathrm{tet}_b(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 \mathrm{tet}_b(z\!+\!1)=\exp_b\big(\mathrm{tet}_b(z)\big)}
Failed to parse (SVG (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{pen}_b(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 \mathrm{pen}_b(z\!+\!1)=\mathrm{tet}_b\big(\mathrm{pen}_b(z)\big)}

Except the zeroth row, each operation in the sequence above is just a recurrence of operations from the previous row. Operation ++ could be called zeration (although in programming languages it is called increment), addition (or summation) could be called unation, multiplication (or product) could be called duation , exponentiation could be called trination. The following operations ( tetration, pentation) have not been used so often, at least up to the year 2008. Although tetration has been given many other names: superexponentiation [4], ultraexponent [5], generalized exponent [6], other names were not applied to the holomorphic extension of tetration, defined in the previous section.

Manipulation with the holomorphic extensions and the inverses of summation, multiplication, exponentiation form the core of the mathematical analysis. The table above shows the place of tetration in the big picture of math, in the penultimate row.

In the scheme above, each next operation appears as a superfunction with respect to the previous one. In such a way, the name tetration indicates, that this operation is fourth (id est, tetra) in the hierarchy of operations after summation, multiplication, and exponentiation. In principle, one can define "pentation", "sextation", "septation" in a similar manner, although tetration, perhaps, already has a growth rate fast enough for the requirements of the 21st century.

Real values of the arguments, general view

Examples of behavior of this function at the real axis are shown in figure 1 for 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 b=\mathrm{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=\exp(1\!/\!\mathrm{e})} , and 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}} . It has a logarithmic 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 -2} , and it is a monotonic increasing function.

At 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 \mathrm{tet}_b(x)} approaches its limiting value 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 x\rightarrow +\infty} , 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 \displaystyle \lim_{x \rightarrow +\infty} ~ \mathrm{tet}_b(x) > 1} .

Fast growth and application

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\! >\! \exp(1\!/\!\mathrm{e})~} 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 \mathrm{tet}_b(x)} grows faster than any exponential function. For this reason tetration has been proposed for the representation of huge numbers in the Mathematics of Computation[4]. A number that cannot be stored as floating point could be represented 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 \mathrm{tet}_b(x)} for some standard value 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} (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=2} or Failed to parse (SVG (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=\mathrm{e}} ) and relatively moderate value 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} . The analytic properties of tetration could be used for the implementation of arithmetic operations with huge numbers without to convert them to the floating point representation.

Integer values of the argument

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 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 {\rm tet}_b(z)} can be interpreted as iterated exponential:

Failed to parse (SVG (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(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 \mathrm{tet}_b(2)=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 \mathrm{tet}_b(3)=b^{b^b}}

and so on; then, the argument of tetration can be interpreted as number of exponentiations of unity. From definition it follows, 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 \mathrm{tet}_b(0)=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 \mathrm{tet}_b(-\!1)=0}

Relation with the Ackermann function

At 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=2} , tetration is related to the Ackermann function [7]

Failed to parse (SVG (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}_2(n+3)=A(4,n)+3}

where the 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 A} is defined, for non-negative integer values of its arguments, by the following equations

Failed to parse (SVG (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(m, n) =\left\{ \begin{array}{llll} n+1 &,& {\rm if~ }~ m \!=\! 0 \\ A\big(m\!-\!1,~1~\big) &,& {\rm if ~} m\!>\!0 \mbox{ and } n\! =\! 0 \\ A\Big(m\!-\!1,A(m,n\!-\!1)\Big) &,& {\rm if ~} m\!>\! 0 \mbox{ and } n\! >\! 0 \end{array} \right. }

The generalization of 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 A(4,z)} for the complex 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} is described in the preprint [2] . Construction of such holomorphic extension is equivalent to construction 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 \mathrm{tet}_b(z)} for the 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=2} .

Asymptotic behavior and properties of tetration

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)} grows rapidly 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 does not grow infinitely in the direction of imaginary axis. The asymptotic behavior determines the basic properties of tetration.

The exponential convergence of discrete iteration of logarithm corresponds to the exponential asymptotic 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 F(z)=L+\varepsilon(z) + o \big(\varepsilon(z)^{2}\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 \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

Failed to parse (SVG (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.2. 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} ), 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} (no real solutions).

Solutions of this equation are called fixed points of logarithm.

Fixed points of logarithm

Three examples of graphical solution of equation for fixed points of logarithm are shown in figure 2 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 exists 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 solution, 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 exist 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 L\!=\!\!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 L\!=\!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 L\!=\!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 L\!=\!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 L=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 L=L_{\rm e}^* \approx 0.318131505204764135312654-1.33723570143068940890116 \!~\rm i} .

A few hundred straightforward iterations 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)} are sufficient to get the error smaller than the last decimal digit in the approximations above.

Basic properties

FIg.3. parameters of asymptotic of tetration versus logarithm of the base

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=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 L=L_2 } of equation (14) are plotted in figure 3 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. 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<1/\mathrm e} , 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 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 L_2} are real and positive. 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. Basic properties of tetration are determined by the 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} . The main parameters 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)} are plotted in figure 3. 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 conjugations of each other. Requirement of definition of tetration determine the asymptotic of the solution. 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} determines periodicity of quasi-periodicity of tetration. The two solutions 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 Q} are shown in figure 3 with green lines.

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/ \mathrm{e} )} both solutions 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 Q} are real. The negative Failed to parse (SVG (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} corresponds to tetration, decaying to the asymptotic 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} in the direction of real axis; positive Failed to parse (SVG (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} corresponds to the solution growing along the real axis. At the real axis, such a solution remains larger than unity; this does not allow 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 F(0)=1} . Therefore, only one negative Failed to parse (SVG (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} corresponds to the asymptotic behavior of tetration.

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/ \mathrm{e} )} , both options 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 Q} are mutually complex conjugate. The real part is shown thif thick green line; one option of the imaginary part is shown with dashed line.

Possibilities for the period (or 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=\frac{2\pi}{Q}} are shown in Figure 3 with dotted lines. 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/ \mathrm{e} )} , only "negative" period corresponds to tetration. 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/ \mathrm{ e} )} , the periodicity can be achieved only asymptotically; 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 T} is quasi-period. The real part of quasi-period is marked with black dotted line; one of two options tor the imaginary part is marked with pink dotted line.

Generally, 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 1\!<\!b\!<\!\exp(1\!/\!\mathrm{e})} , tetration is periodic; the period is pure imaginary.

At , tetration is not periodic, and no exponential asymptotic exist.

Failed to parse (SVG (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\!/\!\mathrm{e})} , tetration is quasi–periodic, the quasi-period in the upper complex half-plane is conjugate to that in the lower complex half-plane. The larger is 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} , the shorter is quasi-period. As the quasi-periods are complex conjugated, the quasi-periodicity takes place away from the real axis.

Evaluation of tetration

As the asymptotics of tetration are critically dependent on the 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} in the vicinity of the 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 b=\exp(1/\mathrm{e}) } , the evaluation procedure is different for the 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 1\!<\!b\!<\!\exp(1\!/\!\mathrm{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\!=\!\exp(1/\mathrm{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\!>\!\exp(1\!/\!\mathrm{e}) } , and these should be considered separately.

Case Failed to parse (SVG (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/\mathrm{e}) }

Template:CC-Image
Fig.4. 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=\mathrm{tet}_b(x\!+\!\mathrm{i}y)} 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}}

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 1<b<\exp(1/\mathrm{e}) } , 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} is imaginary. The period with smallest modulus corresponds to the solution that is unity at the origin of coordinates. 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 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=\mathrm{tet}_b(z)} is shown in figure 4 with levels of constant real part and levels of constant imaginary part. 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 \Re(f)=2,3,4} 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 \Im(f)=0,\pm 1,\pm2\pm 3, \pm 4} aew shown with thick lines. Intermediate levels are shown with thin lines. There are branch points 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 \Re(z)=2~,~\Im(z)=2\pi |T| m~ \forall m \in \mathbb{N}} ; the cut lines 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 \Re(z)<2~,~\Im(z)=2\pi |T| m~ \forall m \in \mathbb{N}} . For this value of the base, 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}{\ln^2(2)} \approx -17.1431481793548471041794 ~\mathrm{i} } .

There is a 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 x\!<\!-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 y\!=\!0} ; although the jump at this cut reduces at the increase 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 such a way, the function approaches its limiting 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\!=\!4} almost everywhere, although there is set of singularities at negative 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<-2} .

The solution follows asymptotic at large values of real part of the argument, exponentially approaching the limiting value. In particular, 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}} , this maximum limiting value is in the left hand side of the figure, and to its minimum limiting value in the right hand side. 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}} , these limiting values 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 L\!=\!4} 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\!=\!2} .

The trace of the solution along the real axis corresponds to the red dotted curve in Figure 1. Other solutions of the recursive 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 \exp_b\!\big(F(z)\big)\!=\!F(z\!+\!1)} , that may grow up along the real axis, can be constructed in a similar way, but they do not satisfy the criteria formulated in the definition of tetration; in particular, Failed to parse (SVG (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} .

Case Failed to parse (SVG (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/\mathrm{e}) }

Fig.5. Tetration 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\!/\!\mathrm{e})} .

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\!/\!\mathrm{e}) } , the limiting 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=\mathrm{e}} , and, asymptotically,

Failed to parse (SVG (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)=\mathrm{e}- 2 \mathrm{e}/z + \mathcal{O}\!\big(\ln(z)/z^2\big) }

The 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=F(x\!+\!\mathrm{i}y)} is shown in figure 5.

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 \Re(f)=-2,-1,0,1,2,3,4} are shown with thick black lines.

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 \Im(f)=-1,-2,-3,-4} are shown with thick red lines.

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 \Im(f)=1,2,3,4} are shown with thick blue lines.

Intermediate levels are shown with thin lines.

There is a 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 x<-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 y=0} , but the hump of the function at the cut reduces 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 y} reduces, id est, with increasing Failed to parse (SVG (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|} . In such a way, everywhere, 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|\rightarrow \infty} , the function approaches its limiting 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=\mathrm{e}\approx 2.71} . almost everywhere, although there is a set of singularities at negative 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<-1} .

Behavior of this function at real values of argument is shown in Figure 1 with thin solid line. Other solutions of the recursive 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 \exp_b\!\big(F(z)\big)\!=\!F(z\!+\!1)} can be constructed in the similar way; they may grow up along the real axis, but they do not satisfy criteria formulated in the definition of tetration; in particular, Failed to parse (SVG (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} .

Case Failed to parse (SVG (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/\mathrm{e}) }

(CC) Image: Dmitrii Kouznetsov
Fig.6. 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 \mathrm{tet}_b(x\!+\!\mathrm{i}y)} at 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\!=\!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\!=\!\mathrm e} .

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/\mathrm{e}) } , tetration is asymptotically periodic. It decays exponentially to the fixed points Failed to parse (SVG (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 halfplane. This allows to express it through its values along the imaginary axis, using the Cauchi integral. [1].

For the 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=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=\mathrm{e}} 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=\mathrm{tet}_b(z)} is plotted in figure 6. 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 \Re{f}=0, \pm 1, ... \pm 16 } 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 \Im{f}=0, \pm 1, ... \pm 16 } are drawn with thick lines. The function has a logarithmic singularity at point -2 and cut at real values of the argument, smaller than -2. In the right hand side, symbols mean huge values that cannot be stored in the conventional floating point representation (logarithm, mantissa). In the upper left and lower left part of each graphic, the function approaches its asymptotic 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 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^*} . Function is quasi-periodic; the same fractal structure reproduces again and again at the translation of argument with quasiperiod Failed to parse (SVG (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} in the upper halfplane 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 T^*} in the lower halfplane.

There is 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 x<-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 y=0} . The jump of the function at this cut approaches Failed to parse (SVG (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\!-\!L^*} almost everywhere, although there is set of singularities at negative 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<-2} .

Along the real axis, tetration for these values of the base is plotted also in figure 1 with thick solid and dashed lines.

Behavior along the real axis

(CC) Image: Dmitrii Kouznetsov
Derivatives of tetration 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=\mathrm{e}} .

The growth of tetration along the real axis is crucially determined by its base. The graphic of this function is shown at the top of the article 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}, \exp(1/\mathrm{e}), 2, \mathrm{e}} . 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=\mathrm{e}} , the derivatives of tetration are plotted in figure at the right.

is plotted with 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 \mathrm{tet}^{\prime\prime}} is plotted with green;
Failed to parse (SVG (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}^{\prime\prime\prime}} is plotted with blue;
Failed to parse (SVG (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}^{\prime\prime\prime\prime}} is plotted with pink.

Tetration is strictly growing function; its first derivative is positive. 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=\mathrm{e}} the minimum of the derivative takes place in vicinity 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 -0.5} and is slightly smaller than unity. 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 1\! <\!b \!< \exp(1 \! / \mathrm{e} )} , the growth is limited by the minimum of 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 L} . Tetration approaches this limiting value exponentially. 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 limiting value is 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 b \! =\! \exp(1 \! / \! \mathrm{e} )} , the growth is limited by the fixed point Failed to parse (SVG (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\!=\!\mathrm{e}} . Tetration approaches this limiting value as rational function.


The growing-up holomorphic 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 \exp_b(F(z))\!=\!F(z\!+\!1)} can be constructed in the similar way 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 1\! \le \! b \! \le \! \exp(1\!/\!{\mathrm e})} , but at the real axis, such a solution remains larger than unity, and the 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 F(0)\!=\!1} cannot be satisfied. Therefore, this solution cannot be interpreted as extension of iterative exponentiation Failed to parse (SVG (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( \exp_b\big(... 1 ..)\big) } for non-integer number Failed to parse (SVG (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} of exponentiations; in this sense, such a solution is not a tetration.

Tetration shows explosively-fast growth along the real axis only 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 b \! >\! \exp(1 \! / \! \mathrm{e} )} .

Tetration at the 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} }

Template:CC-Image
Tetration at the 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} }

There is hypothesis that at 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} } , the graphic 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 y=\mathrm{tet}_b(x)} is symmetric with respect to the line Failed to parse (SVG (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} . This line, together with the graphic, is shown in Figure at let. The asymptotics Failed to parse (SVG (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 y=2} are also shown. The graphic looks symmetric with respect 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 x \leftrightarrow y} , but no proof of this hypothesis was suggested. Now this hypothesis is negated; the apparent symmetry is only approximation. However, 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} } , at the segment Failed to parse (SVG (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\le x\le 0} , the relation Failed to parse (SVG (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(x)\approx -\mathrm{tet}_b^{-1}(-x)} holds with 4 decimal digits [8][9].

Tetration at base e

Holomorphic tetration on the natural 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\!=\!\mathrm{e}} is the most developed, at least up to the year 2008. In the rest of this article, it is assumed 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 b\!=\!\mathrm{e}\approx 2.71828} although the majority of results allow a straightforward extension to the case of real Failed to parse (SVG (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/\mathrm{e})\approx 1.44467} .

Tetration has real fixed point Failed to parse (SVG (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_0} , id est, 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 z} of the 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 z=\mathrm{tet}(z)} . Its approximation

Failed to parse (SVG (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_0\approx 1.85035452902718}

can be found, iterating 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 z=\mathrm{slog}(z)} , where slog is inverse function of tetration.

Pentation

Complex map of natural pentation, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u\!+\!\mathrm i v=\mathrm{pen}(x\!+\!\mathrm i y)} by [10]

Superfunction of tetration can be called pentation; it is solution of equations

Failed to parse (SVG (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{pen}(z\!+\!1)=\mathrm{tet} \big( \mathrm{pen}(z)\big) }
Failed to parse (SVG (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{pen}(0)=1}

Expansion of tetration in vicinity of its fixed point Failed to parse (SVG (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_0} from the interval Failed to parse (SVG (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,-1)} (for real 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\!>\!1} ) allows to define unique (and the only "true") pentation. Other superfunctions of tetration can be obtained by the periodic modification of the argument of pentation.

For base , the fixed point 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 X_0\!\approx\! -1.850354529} . and pentation to this base approaches this value at large negative values of the real part of the argument [11][10]. Complex map of natural pentation is shown in figure at left; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u\!+\!\mathrm i v=\mathrm{pen}(x\!+\!\mathrm i y)} .

Inverse of tetration

Fig.7. slog in the complex plane.

The inverse 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 t=\mathrm{slog}(z)=\mathrm{tet}^{-1}(z)} can be performed using the Newton method, solving 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 \mathrm{tet}(t)=z} , leading 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 \mathrm{tet}\big(\mathrm{kslog}(z)\big)=z} .

The inverse function has branchpoints Failed to parse (SVG (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^*} . For the kslog, at 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=\mathrm{e}} , shown in the figure 7, the cuts are placed horizontaly, 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(z)<\Re(L)} , Failed to parse (SVG (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(z)=\pm \Im(L)} .

Due to the symmetry Failed to parse (SVG (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{kslog}\big(z^*\big)=\mathrm{kslog}\big(z\big)^*} , it is sufficient to plot only half of the complex plane.

Fig.8. Shaded region is image of the upper halfplane at the mapping with slog function.

The mapping with function kslog is shown in figure 8. 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 x} in the shaded region, the relation

Failed to parse (SVG (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{slog}\big(\mathrm{tet}(z)\big)=z}

takes place. The upper part of the complex plane is mapped into the upper halfplane, and the lower halfplane is mapped to the lower halfplane. The real axis is mapped into the halfline Failed to parse (SVG (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,\infty)} . The fixed points Failed to parse (SVG (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 mapped to imaginary infinity, following the shaded strip.

In figure 8, the images of the grid 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(z)=-1,0,\Re(L),1,2,3} and images of the grid 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 \Im(z)=1,\Im(L),2,3,4} are shown. These curves reproduce 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 \Re(F(z))=\rm const} 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 \Im(F(z))=\rm const} shown in figure 6 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=\rm e} .

Together, the pair of functions tet and kslog allow to evaluate any iteration (including negative, fractional and even complex) of the exponential function.

Beyond the cutlines of slog

(CC) Image: Dmitrii Kouznetsov
3 options to put cutlines of the superlogarithmic function

The function slog has branchpoints, and there are many options to pit the cutlines. The cutlines, parallel to the abscias axis, considered above is one of many options. One can put the cutlines along the level Failed to parse (SVG (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(\mathrm{slog}(z))=-2} . This cases simplify the plotting of the slog function, while the only tetration is available for the efficient evaluation. On the other hand, there is no simple expression for the parametrization of such a cutline, and the calculation of the position of the cutlines slows down the algorithm of evaluation of slog. In addition, there is tradition, that cutlines of the special functions are placed parallel to the real or imaginary axis. One could place the cutline from the branchpoints to the abscissa axis. In this case, the additional cutlines should go along the negative part of the real axis, which is not convenient for the applications. (For example, for the evaluation of the generalized exponential Failed to parse (SVG (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^c} at non-integer , it is important, that the real axis belongs to the range of holomorphism of slog. There are 3 other options to place the cutlines parallel to the coordinate axes, keeping 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 \mathrm{slog}(z^*)=\mathrm{slog}(z)^*} . There 3 options are shown in the figure.

The shaded region shows the domain

Failed to parse (SVG (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 \in \mathbb{C}~:~ \Re(z)\!>\!\Re(L)~,~ |z|<|L| \}}

For evaluation of any of these superlogarithms, it is sufficient to have the efficient algorithm for the evaluation in this domain.

The left hand side picture corresponds to the slog, sa it is defined above. Within the strip between the cutlines, the function approaches its limiting 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} ; it becomes infinite in vicinity of the branch points and varies very slow in the rest of the complex plane.

The central part of the picture represents slog with vertical cutlines, let us call this function slogv. In the left hand side of the complex halfplane, the evaluation of slog begins with exponentiation of its argument. The exponential is periodic function; therefore, in this part, slog is periodic:

Failed to parse (SVG (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{slogv}(z+2\pi\mathrm{i} m)= \mathrm{slogv}(z)~ \forall m\in \mathbb{N} ~,~ \forall z\in \mathbb{C}~:~ \Re(z)<0 }

The third option, let us denote it slogr, is to put the cutlines in the direction of the real axis. Along the real axis, the function remains holomorphic, but there are additional branchpoints in the ranges Failed to parse (SVG (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)>1~,~ 1<|\Im(z)|<2} . In these regions, the function slogr show the fractal behavior.

The slog, defined at the beginning, seems to be simpler than slogv and slogr; therefore, namely slog with horizontal cutlines, parallel to the real axis, going in the direction, opposite to the real axis, is used below.

Polynomial approximation

The Taylor series for the tetration can be written in the usual form:

Failed to parse (SVG (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}(z)=\sum_{n=0}^{\infty} c_n (z-z_0)^n}

where 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 n} th coefficient

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_n=\frac{1}{n!}\mathrm{tet}^{(n)}(z_0)}

is expressed through 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 n} derivative of the function. The coefficients of the expansion can be calculated using the straightforward differentiation of the representation through the Cauchy integral.

Taylor expansion at zero

(CC) Image: Dmitrii Kouznetsov
Fig.N. Approximation of tetration with polynomial of 25th power

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 z_0=0} , the calculation gives the following 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 c} 0 = 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 c} 1 ≈ 1.091767351258322138
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} 2 ≈ 0.271483212901696469
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} 3 ≈ 0.212453248176258214
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} 4 ≈ 0.069540376139988952
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} 5 ≈ 0.044291952090474256
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} 6 ≈ 0.014736742096390039
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} 7 ≈ 0.008668781817225539
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} 8 ≈ 0.002796479398385586
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} 9 ≈ 0.001610631290584341
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} 10≈ 0.000489927231484419
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} 11≈ 0.000288181071154065
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} 12≈ 0.000080094612538551
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} 13≈ 0.000050291141793809
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} 14≈ 0.000012183790344901
15≈ 0.000008665533667382
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} 16≈ 0.000001687782319318
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} 17≈ 0.000001493253248573
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} 18≈ 0.000000198760764204
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} 19≈ 0.000000260867356004
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} 20≈ 0.000000014709954143
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} 21≈ 0.000000046834497327
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} 22≈-0.000000001549241666
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} 23≈ 0.000000008741510781
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} 24≈-0.000000001125787310
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} 25≈ 0.000000001707959267

The truncated Taylor series gives the polynomial approximation. In the upper right hand side of the Figure N, the polynomial

Failed to parse (SVG (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=\sum_{n=0}^{25} c_n z^n}

is shown 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.

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 \Re(f)=-2,-1,0,1,2,3,4} are shown with thick black curves.
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 \Re(f)=-1.8,-1.6,-1.4,-1.2,-0.8,-0.6,-0.4,-0.2} are shown with thin red curves.
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 \Re(f)=0.2,0.4,0.6,0.8,1.2,1.4,1.6,1.8} are shown with thin thin blue curves.
Levels are shown with thick red curves.
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 \Im(f)=1,2} are shown with thick blue curves.
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 \Im(f)=\pm \pi,\pm 3\pi} are shown with thick pink curves.
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 \Im(f)= \pm 0.2, \pm 0.4, \pm 0.6, \pm 0.8, \pm 1.2, \pm 1.4, \pm 1.6, \pm 1.8} are shown with thin green curves.

In the upper left corner of figure N, the same is shown for 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 \log(f)}

At the bottom left, the overlap of the upper two images is shown.

At the bottom right, lines of constant modulus and constant phase of holomorphic tetration in the same range.

The good approximation of tetration takes place in 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 |z|} of order of unity or smaller; the radius of convergence of the series is 2.

Expansion at 3i

Taylor expansion of tetration developed at 3i, truncated at 30th power, plotted in the complex plane.

Coefficients of the expansion

Failed to parse (SVG (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}(z)=\sum_{n=0}^{\infty} a_n \left(\frac{z-3 \mathrm{i}}{2}\right)^n}

can be evaluated in the similar way:

Failed to parse (SVG (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_{ 0}\approx 0.3709065890322851 + 1.3368216707889140 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 a_{ 1}\approx 0.0366009653759846 + 0.1392221538995050 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 a_{ 2}\approx-0.1688843184064154 + 0.0971853361962927 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 a_{ 3}\approx-0.1268131504868087 -0.1183162876702863 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 a_{ 4}\approx 0.0423580931032393 -0.1052093008832072 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 a_{ 5}\approx 0.0584830639356318 -0.0081022452449608 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 a_{ 6}\approx 0.0234003166529485 + 0.0180777701182038 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 a_{ 7}\approx 0.0034426098470138 + 0.0181510375563591 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 a_{ 8}\approx-0.0080369581444167 + 0.0091742846703500 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 a_{10}\approx-0.0018461796309531 -0.0032234258318168 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 a_{11}\approx 0.0005406488544310 -0.0018967206101561 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 a_{12}\approx 0.0010224364808881 -0.0005596865717924 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 a_{13}\approx 0.0006471439639805 + 0.0002598066193583 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 a_{14}\approx 0.0001044445559337 + 0.0003719947259883 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 a_{15}\approx-0.0001117853540434 + 0.0001678668755219 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 a_{16}\approx-0.0001063015871081 + 0.0000207220003313 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 a_{17}\approx-0.0000507809881911 -0.0000357591300574 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 a_{18}\approx-0.0000031474299869 -0.0000352318593759 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 a_{19}\approx 0.0000134766134413 -0.0000133303413745 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 a_{20}\approx 0.0000098023908240 + 0.0000004760718415 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 a_{21}\approx 0.0000035549347545 + 0.0000038981621220 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 a_{22}\approx-0.0000002155265265 + 0.0000029627341324 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 a_{23}\approx-0.0000013167390363 + 0.0000009738135453 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 a_{24}\approx-0.0000008340196081 -0.0000001866385871 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 a_{25}\approx-0.0000002286961098 -0.0000003749771677 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 a_{26}\approx 0.0000000537258461 -0.0000002306013659 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 a_{27}\approx 0.0000001140665665 -0.0000000656951029 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 a_{29}\approx 0.0000000139678685 + 0.0000000331511830 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 a_{30}\approx-0.0000000068489056 + 0.0000000171304198 i}

The plot of approximation of tetration with the resulting polynomial of 30th power is shown in figure. This approximation can be used for plotting of camera-ready pictures of tetration, using it and its conjugation 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 \Re(z)\le 0.5 } . With 50 terms, 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-3|<2} , such approximation returns 14 significant figures.

Asymptotic expansion at large values of the imaginary part of the argument

(CC) Image: Dmitrii Kouznetsov
Fig.As. Deviation of tetration from its asymptotic expansion.

Here only the case 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\!=\!\mathrm{e}} is considered; although the generalization to the case Failed to parse (SVG (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\!>\!\mathrm{e}} is straightforward. In this section, 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 \mathrm{tet}(z)} is considered for moderate 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)} and large 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 \Im(z)}

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\!=\!\mathrm{e}} , in the upper halfplane, tetrations approach the fixed point Failed to parse (SVG (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\approx 0.3+1.3\mathrm{i}} of the logarithm. This approach is exponential. Using the exponential

Failed to parse (SVG (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=\varepsilon(z)=\exp(L z +R) }

as a small parameter, for some complex constant Failed to parse (SVG (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} , the tetration can be estimated as follows:

Failed to parse (SVG (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}(z)=L+\varepsilon+a_2 \varepsilon^2 + a_3 \varepsilon^3 + \mathcal{O}(\varepsilon^4)}

Substitution of this expression into the 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 \mathrm{tet}(z\!+\!1)=\exp\big(\mathrm{tet}(z)\big)} 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 a_2=\frac{1/2}{L-1}\approx -0.151314 - 0.2967488 \mathrm{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 a_3=\frac{a_2+1/6}{L^2-1}\approx -0.36976 + 0.98730 \mathrm{i}}
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 R\approx 1.077961437526 -0.946540963949 \mathrm{i}} can be found fitting 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 \mathrm{tet}(z)} with function at large 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 \Im(z)} . Then, the expansion can be written in the form

Failed to parse (SVG (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}(z)=\sum_{n=0}^N a_n \varepsilon^n + \mathcal(O)\!\big(\varepsilon^{N\!+\!1}\big)}

assuming 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 a_0\!=\!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 a_1\!=\!1} . At given Failed to parse (SVG (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} , such a representation indicates the possible approximations of tetration. The deviations of tetration from these approximations are shown in figure Fig.As.

The four plots in fig.As correspond to the four asymptotic approximations. The deviations

Failed to parse (SVG (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=\mathrm{tet}(z)-L} ,
Failed to parse (SVG (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=\mathrm{tet}(z)-\Big(L+\varepsilon(z)\Big)} ,
Failed to parse (SVG (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=\mathrm{tet}(z)-\Big(L+\varepsilon+a_2 \varepsilon^2 \Big)} ,
Failed to parse (SVG (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=\mathrm{tet}(z)-\Big(L+\varepsilon+a_2 \varepsilon^2 + a_3 \varepsilon^3\Big)}

are shown in the complex plane with lines of constant phase and constant modulus.

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 \arg(f)=-2,-1} are shown with thick red lines.
Level Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \arg(f)=0} is shown with thick black lines.
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 \arg(f)=1,2} are shown with thick blue lines.
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 \arg(f)=\pm \pi} are shown with scratched lines. (these lines reveal the step of sampling used by the plotter).
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(-0.8),\exp(-0.6),\exp(-0.4),\exp(-0.2)} are shown with thin red lines.
Levels are shown with thin blue lines.
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(3), \exp(2), \exp(1),\exp(0), \exp(-\!1), \exp(-\!2), \exp(-\!3), \exp(-\!4), \exp(-\!5), \exp(-\!5), \exp(-\!7), \exp(-\!8)} are shown with thin thick black lines.
Level Failed to parse (SVG (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(-10)} is shown with thick red line.
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(-12),\exp(-14), \exp(-16),\exp(-18)} are shown with thick black lines.
Level Failed to parse (SVG (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(-20)} is shown with thick red line.
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(-22),\exp(-24), \exp(-26),\exp(-28)} are shown with thick black lines.
Level Failed to parse (SVG (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(-30)} is shown with thick green line.
Level Failed to parse (SVG (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(-31)} is shown with thick black line.

The plotter tried to draw also

Level Failed to parse (SVG (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(-32)} with thick black line and
Level Failed to parse (SVG (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(-33)} with thin dark green line, which are seen a the upper left hand side corners of the two last pictures, but the precision of evaluation of tetration is not sufficient to plot the smooth lines; for the same reason, the curve 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 |f|=\exp(-31)} in the last picture, in the upper right band side looks a little bit irreguler; also, the pattenn in the upper left corner of the last two pictures looks chaotic; the plotter cannot distinguish the function from its asymptotic approximation.

The figure indicates 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 \Im(z)>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 \Re(z)<4\Im(z) - 25} , the asymptotic approximation

Failed to parse (SVG (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}(z)\approx L+\varepsilon+a_2 \varepsilon^2 + a_3 \varepsilon^3}

gives at least 14 correct significant figures. At large values of the imaginary part, this approximation is more precise than the evaluation of tetration through the contour integral.

Approximation of tetration with elementary functions

(CC) Image: Dmitrii Kouznetsov
Failed to parse (SVG (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)=\log(z+2)+\sum_{n=0}^{100}c_n z^n} 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.

Due to recurrent relation Failed to parse (SVG (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}(z)=\exp(\mathrm{tet}(x+1))} , it is sufficient to approximate tetration in any vertical strip of unit with in the complex plane. Some of such approximations are suggested in [1]. In principle, the numerical approximation of tetration with implementation of the Cauchi integral with finite sums [1] also should be considered as approximation with elementary function. However for the practical evaluation of tetration, shorter expressions are more suitable. One of such approximation comes from the Taylor expansion 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 \mathrm{tet}(z)-\log(z+2)} . The subtraction of logarithm remove the closest singularity that limits the radius of convergence of the Taylor series, and makes precise the approximation with finite sum. One of such approximation with one logarithm and polynomial of 100th power

Failed to parse (SVG (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)=\log(z+2)+\sum_{n=0}^{100}c_n\left(\frac{z}{2}\right)^n}

is shown at the figure 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. In vicinity of the origin of coordinates, say, Failed to parse (SVG (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|<1 } , the most of terms are negligibly small, and the shortened sum still approximates the tetration. The first coefficients in this expansion 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 c_0 = 0.30685281944005469058 }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_1 = 1.18353470251664338875}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_2 = 1.58593285160678321155 }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_4 = 1.36264601823980036066 }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_5 = 1.21734246689515424045 }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_6 = 1.10981816083559525765 }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_7 = 0.96674692974769849130 }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_8 = 0.84089872598668435888 }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_9 = 0.71353210966804747617 }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{10}=0.60168548504001373445 }

More coefficients can be extracted from the generator of the figure. 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| \le 2 } , the approximation with 101 terms returns at least 14 correct significant figures.

Approximation of slog

Function slog, which is inverse of tetration, allows the approximation with elementary functions.

Taylor expansion

(CC) Image: Dmitrii Kouznetsov
Approximation of slog with polynomial of 16th power from the Taylor expansion at unity.

The Taylor series for the tetration can be inverted, gaining the expansion of the superlogarithm:

Failed to parse (SVG (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{slog}(z)=\sum_{n=1}^\infty c_n (z-1)^n} .

Approximations for the first 16 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 c_{ 1}\approx 0.91594605649953 }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{ 2}\approx -0.20861842957759 }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{ 3}\approx -0.05450400630209 }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{ 4}\approx 0.07134941925273 }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{ 5}\approx -0.02004387374438 }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{ 6}\approx -0.01101258023037 }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{ 7}\approx 0.01207268318645 }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{ 8}\approx -0.00272922880760 }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{ 9}\approx -0.00269905319156 }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{10}\approx 0.00243941500632 }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{11}\approx -0.00036220360858}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{12}\approx -0.00070125921262}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{13}\approx 0.00052782155380 }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{14}\approx -0.00002987943551}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{15}\approx -0.00018614540434}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{16}\approx 0.00011722843042 }

The partial sum with 16 terms (from zero to 16) is plotted in the figure 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. Lines of constant real part and constant imaginary part are drawn.

The radius of convergence of this series is determined by the distance to the closest singularity; the representation 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 \mathrm{slog}(z)} with the Taylor series is valid 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 |z-1|< |1-L|}

Obviously, it fails 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=L} . For this case, the asymptotic expansion can be used.

Expansion at fixed point Failed to parse (SVG (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

(CC) Image: Dmitrii Kouznetsov
Fitting of slog with the expansion around Failed to parse (SVG (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} .

Superlogarithm can be approximated with expansion [12]

Failed to parse (SVG (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{slog}(z)=\frac{1}{L} \left( \log(z\!-\!L)+ \sum_{m,n} r_{m,n} (z\!-\!L)^{m+2\pi \mathrm{i} n/L} \right)} ,

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 L} is fixed point of logarithm. This expansion indicates the ways to construct the asymptotic approximations of slog. The coefficients can be expressed from the asymptotical analysis of equation . Also, they can be expressed from the asymptotical estimate of tetration at large values of the imaginary part of the argument. The evaluation of first coefficients 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 r_{0,0}\approx \! -\!1.0779614375 + 0.9465409639 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 r_{1,0}\approx ~ 0.1513148972 + 0.2967488367 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 r_{2,0}\approx \! -\!0.0607692016 + 0.0359770148 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 r_{3,0}\approx \! -\!0.0079378875 -0.0176412865 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 r_{4,0}\approx ~ 0.0051087546 -0.0000718839 i }

These coefficients allow to approximate slog in vicinity of the fixed point of logarithm with 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=\frac{1}{L} \left( \log(z\!-\!L)+ \sum_{m=0}^4 r_{m,0} (z-L)^{m} \right)} .

In the figure this Failed to parse (SVG (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 shown 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.

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 \Re(f)=0,\pm 1, \pm2} are shown with thick black lines.
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 \Re(f)=-1.9, .. -0.1} are shown with thin red lines.
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 \Im(f)=-1.9, .. -1.9} are shown with thin dark green lines.
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 \Im(f)=0} is shown with thick green line.

(Deviation of this line from the real axis indicates the error of the approximation.)

Level Failed to parse (SVG (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)=-1} is shown with thick red line
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 \Im(f)=1,2,3} are shown with thin dark blue lines.

For comparison, dashed lines show the precise evaluation for some of the levels above for the robust implementation of the slog function as inverse of tetration. 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-L|<1 } , the deviation of these dashed lines from the levels for 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 not seen even at the strong zooming-in of the central part of the figure.

Approximation of slog with elementary functions

(CC) Image: Dmitrii Kouznetsov
Approximation of slog with function fslog.
(CC) Image: Dmitrii Kouznetsov
Numerical test of approximation of slog with function fslog.

The precision of approximation of slog (with fixed precision of the arithmetics used for the tvaluation) can be extended, takung unto account the singularitues of slog at the fixed points. From the asumptotical representation above, one can guess the robust representation for slog:

Failed to parse (SVG (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{slog}(z)=\frac{\log(z-L)}{L} + \frac{\log(z-L^*)}{L^*} + \sum_{n=0}^\infty u_n (z-1)^n}

The coefficients of this expansion are real. The first 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 u_1\approx\! -0.026066290298}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_2\approx~ 0.001733047818}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_3\approx\! -0.000019521307}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_4\approx\! -0.000063070064}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_5\approx~ 0.000025678960}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_6\approx\! -0.000005590100}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_7\approx\! -0.000000072797}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_8\approx~ 0.000000651489}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_9\approx\! -0.000000276981}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_{10}\!\approx~ 0.000000031118}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_{11}\!\approx~ 0.000000029409}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_{12}\!\approx\! -0.000000018969}

This representation allows construction of approximations, truncating the series. One of such approximations

Failed to parse (SVG (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=\mathrm{fslog}(z)=\frac{\log(z-L)}{L} + \frac{\log(z-L^*)}{L^*} + \sum_{n=0}^{50} u_n (z-1)^n}

is shown in figure 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. Lines of constant Failed to parse (SVG (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 those of constant Failed to parse (SVG (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 plotted.

The range of approximation of slog with this function is wider than that with the Taylor expansion at unity. The extended range of approximation allows its validation with the numerical test of identities

Failed to parse (SVG (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{slog}(z)=\mathrm{slog}(\exp(z))-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 \mathrm{slog}(z)=\mathrm{slog}(\log(z))+1}

The residuals

Failed to parse (SVG (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=\mathrm{fslog}(z)-\mathrm{fslog}(\exp(z))+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 f= \mathrm{fslog}(z)-\mathrm{fslog}(\log(z))-1}

are shown in the figure with 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|=} const. In the voided regions in vicinity 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=0} and , the residual is at the level 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 10^{-14}} . (It is difficult to make the residual smaller, using the arithmetics with double complex variables.) This test indicates, 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 |z-1| \le |L|} , the approximation of slog with two the logarithms and the polynomial of 50th power gives at least 9 correct significant figures.

Iterated exponential 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 \sqrt{\exp}}

(CC) Image: Dmitrii Kouznetsov
Fig.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 \exp^c(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 for various Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} .

Pre-historic approach to the iterated exponential

Especially interesting is the case of iteration of natural exponential, id est, Failed to parse (SVG (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=\mathrm{e}} . Existence of the fractional iteration, and, in particular, existence of operation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{\exp}=\exp^{1/2}} was demonstrated in 1950 by H.Kneser. [12]. However, that time, there was no computer facility for the evaluation of such an exotic 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} 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(F(z))=\exp(z)} ; perhaps, just absence of an apropriate plotter did not allow Kneser to plot the distribution of fractal exponential 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 \exp^c(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 for various 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 c} , shown in Fig.9.

(CC) Image: Dmitrii Kouznetsov
Fig.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^c(x)} 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}

The Implementation through the tetration

Holomorphic tetration allows to extend the iterated exponential Failed to parse (SVG (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^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 \exp_b^0(z) =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 \exp_b^1(z) =\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 \exp_b^{c+1}(z) =\exp_b\!\Big(\exp_b^c(z)\Big) }

For non-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 c} . It can be defined 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 \exp_b^c(z) = \mathrm{tet}_b\!\Big(c+ {\mathrm{tet}_b}^{-1}(z)\Big) }

If in the notation Failed to parse (SVG (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^c} the superscript is omitted, it is assumed to be unity; 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 \exp_b^1=\exp_b} . If the subscript is omitted, it is assumed to 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 \mathrm{e}} , id est, Failed to parse (SVG (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^c=\exp_\mathrm{e}^c}

Iterated exponential in the complex plane

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=\exp^c(z)} is shown in figure 9 with levels of constant real part and levels of constant imaginary part. 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 \Re(f)=-3,-2,-1,0,1,2,3,4,5,6,7,8,9} 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 \Im(f)=-3,..14} are drown with thick lines. Red corresponds to a negative value of the real or the imaginaryt part, black corresponds to zero, and blue corresponds to the positeive values. 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 \Re(f)=-0.2, -0.4, -0.6, -0.8} are shown with thin red lines. 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 \Im(f)= 0.2, 0.4, 0.6, 0.8} are shown with thin green lines. 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 \Re(f)=\Re(L)} 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 \Im(f)=\Im(L)} are marked with thick green lines, 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 L\approx 0.31813150520476413 +1.3372357014306895~ \mathrm{i}} is fixed point of logarithm. At non-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 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 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^*} are branch points 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 \exp^c} ; in figure, the cut is placed parallel to the real axis. 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 c<0} there is an additional cut which goes along the negative part of the real axis. In the figure, the cuts are marked with pink lines.

Iterated exponential of a real argument

For real values of the argument, 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=\exp^c(x)} is ploted in figure 10 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} for 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 c=0,\pm 0.1, \pm 0.5, \pm 0.9, \pm 1, \pm 2} .
in programming languages, inverse function of exp is called log.

For logarithm on base e, notation ln is also used. In particular, Failed to parse (SVG (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^{-1}(x)=\ln(x)} , Failed to parse (SVG (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^{-2}(x)=\ln\big(\ln(x)\big)} and so on.

At least for the real and big enough Failed to parse (SVG (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} , the relation Failed to parse (SVG (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^u\big(\exp^v(x)\big)=\exp^{u+v}(x)} holds, which is qute analogous of relations Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ux\!+\!vx=(u\!+\!v)x} and . However, at negative Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} or negative Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v} , 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 x} should be big enough, 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 \exp^u(x)} 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^v(x)} 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^{u+v}(x)} are defined, see figure 10. For example, 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 \min(u,v,u\!+\!v)=-2} , we need Failed to parse (SVG (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\!>\!1} . 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 u\!=\!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 v\!=\!-1} , we have Failed to parse (SVG (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^1\!\big(\exp^{-1}(x)\big)=\exp\!\big(\ln(x)\big)=x} .

Application of iterated exponential

The iterated exponential, that can be implemented with holomorphic tetration, may have various applications. In particular, 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 \exp_b^c(x)} 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\!<\!c\!<\!1} could describe a process that grows faster than any polynomial, but slower than any exponential. In such a way, the iterated exponential, at the proper implementation, should greatly extend the abilities of fast and precise fitting of functions. This is just analogy 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(x)=x^a} which, at fractal 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 a} , may be good for description of a function that grows faster than any linear function but slower than any quadratic function.

Similar functions

Withdrawal of some of requirements from the definition of tetration allows the huge variety of similar functions.

Entire solutions 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(z\!+\!1)=\exp(F(z))}

Withdrawal of the requirement Failed to parse (SVG (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(z^*\big)=F(z)^*} 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} allows the solution by Kneser [12], which is entire and also could be used to build up various powers of the exponential; in particular, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{\exp}} . Such entire function is shown in upper part of figure 1c in [1], in order to reveal the asymptotic behavior of holomotphic tetration.

Withdrawal of 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 F(0)\!=\!1} allows to construct solutions, similar to the growing tetration, 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 1\!<\!b\!\le\!\exp(1/\mathrm{e})} . Although such solutions cannot be interpreted as generalization of exponential iterated Failed to parse (SVG (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} times, they can be useful for generalization of exponential function.

Non-holomorphic modification of tetration

(CC) Image: Dmitrii Kouznetsov
Fig.11. Almost identical 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(z)\!=\! z \!+\! 10^{-9} \sin(2 \pi 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\!=\!x\!+\!\mathrm{i} y} plane.
(CC) Image: Dmitrii Kouznetsov
Fig.12. Motified tetration at the complex plane.
(CC) Image: Dmitrii Kouznetsov
Fig.13. Zoom in of fig.12

Withdrawal of requirement of holomorphicity from the definition of tetration allows functions, which look like the tetration, at least along the real axis. Even the reduction of the range of holomorphism in the requirement allows to consider tetration with modified argument. The modified tetration can be defined 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 \mathrm{tem}(z)=\mathrm{tet}\!\big(J(z)\big)= \mathrm{tet}\!\big(z\!+\! h(z)\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 J(z)=z+h(z)} , and is a 1-periodic function. The simple example of such function is

Failed to parse (SVG (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)=10^{-9}\sin(2 \pi z)}

In this case, along the real axis, the 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 J} is almost identical to its argument; and values of the modified tetration are close to values of tetration. Being plotted at figure 1 or in figure 10, the deviation of such 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 J} from the identity is small, and the deviation of the modified tetration from tetration is also small. If the figures are printed in the real scale, then the deviation of the curves would be of order of atomic size.

However the difference becomes seen at the complex values of the argument. In figure 11, 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 J(z)=z\!+\!h(z)} is plotted 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\!=\!x\!+\!\mathrm{i} y} plane. Levels of constant real part and those of constant imaginary part are drawn. In vicinity of the real axis, these lines almost coincide with the gridline; the grid is drawn with step unity and extended one step to the right and one step to the left from the graphic. In order to show that it behaves as it if would be a continuation of the plot. 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 |\Im(z)|\approx 3} , the deviation from the identical function becomes visible, 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 |\Im(z)|>3} , 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 J(z)} has many points with real values, including those with various negative integer values. The tetration has cut at 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 z<-2} and singularities there. Therefore, the cuts of the modified tetration are determined by 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 J(z) \le -2 } , and modified tetration unavoidable has singularities in points Failed to parse (SVG (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 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 J(z) \in \{ n\in \mathbb{N} ~:~ n<-1 \} } . These singularities are determined by the 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 h} and do not depend on the 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} of tetration. In figure 11, 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 J(z)<-2} are seen not only along the real axis, but also at the top and at the bottom of the figure. In such a way, figure 11 shows the cutlines of the modified tetration. One has no need to evaluate tetration in order to see the margin of the change of holomorphism of the modified tetration.

In figure 12, the modified tetration is plotted 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 additional cuts are seen in the upper and lower parts of the figure 12. Only within the strip along the real axis, the function is holomorphic. While the amplitude of sinusoidal is of order 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 10^{-9}} , the strip of holomorphism is wider than unity, although this width slightly reduces along the real axis.

In order to see the behavior of the modified tetration in vicinity of the additional singularities, in fig.13, the zooming-in of the part of figure 12 is shown. The zoom has improved resolution, so, in its turn, it can be zoomed in to the size of the screen of a computer to see the details. In each cell of the grid, the small and deformed image of the central part of the fig.12 appears.

One has no need to evaluate tetration in order to reveal its singularities outside the real axis. All 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 z} 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 z+h(z)=n} 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<-1} are singularities (branchpoints) of the modified tetration.

Let .

The following theorem is suggested: For any entire 1-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 h} 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 h(0)\!=\!0} , Failed to parse (SVG (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} is not a constant, Failed to parse (SVG (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\big(z^*\big)=h(z)^*~\forall z\in \mathbb{C} } , there exist Failed to parse (SVG (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\in \mathbb{C} : |\Re(z)|\!<\!1~,~ z\!+\!h(z)~ \in \mathcal{M} } . sequence Although this theorem is not yet proved, the intents to construct at least one example of function, that would contradict it, were not successful. This theorem is somehow independent from the theory of tetration, but it indicates, that any modified tetration cannot be holomorphic in 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 \{z\in \mathrm{C}~:~\Re(z)>-2 \} } .

According to the theorem above, the modified tetration does not satisfy the condition of quasi-periodicity, and does not satisfy the criterion of holomorphism in the definition of tetration. The sequence of cutlines for the specific example of modified tetration is seen in figure 13; the modified tetration is not even continuous. An addition to 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 h} some highest sinusoidals brings the discontinuities even closer to the real axis. This indicates, that if in at least one point at the real axis between Failed to parse (SVG (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} 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 1} , some 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} of equations Failed to parse (SVG (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\!+\!1)\!=\!\exp(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(0)\!=\!1} differs from tetration tet for at least Failed to parse (SVG (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^{-9}} , then this solution is not holomorphic in 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 \{z\in \mathbb{C} : |\Re(z)|<1, |\Im(z)|<4 \}} .

Even small deformation of tetration tet breaks its continuity. Similar reasons in favor of uniqueness tetration are suggested also in [1]. There is only one tetration, that satisfies requirements of the definition, although the rigorous mathematical proof of the uniqueness is still under development.


References

  1. ↑ 1.0 1.1 1.2 1.3 1.4 1.5 D.Kouznetsov. (2009). "Solutions 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(z+1)=\exp(F(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.". Mathematics of Computation, 78: 1647-1670. DOI:10.1090/S0025-5718-09-02188-7. Research Blogging. Cite error: Invalid <ref> tag; name "k" defined multiple times with different content Cite error: Invalid <ref> tag; name "k" defined multiple times with different content Cite error: Invalid <ref> tag; name "k" defined multiple times with different content Cite error: Invalid <ref> tag; name "k" defined multiple times with different content Cite error: Invalid <ref> tag; name "k" defined multiple times with different content
  2. ↑ 2.0 2.1 D.Kouznetsov. Ackermann functions of complex argument. Preprint of the Institute for Laser Science, UEC, 2008. http://www.ils.uec.ac.jp/~dima/PAPERS/2008ackermann.pdf Cite error: Invalid <ref> tag; name "k2" defined multiple times with different content
  3. ↑ R.L.Goodstein (1947). "Transfinite ordinals in recursive number theory". Journal of Symbolic Logic 12.
  4. ↑ 4.0 4.1 P.Walker (1991). "Infinitely differentiable generalized logarithmic and exponential functions". Mathematics of Computation 196: 723-733. Cite error: Invalid <ref> tag; name "w" defined multiple times with different content
  5. ↑ M.H.Hooshmand. (2006). "Ultra power and ultra exponential functions". Integral Transforms and Special Functions 17 (8): 549-558.
  6. ↑ N.Bromer. Superexponentiation. Mathematics Magazine, 60 No. 3 (1987), 169-174
  7. ↑ W.Ackermann. ”Zum Hilbertschen Aufbau der reellen Zahlen”. Mathematische Annalen 99(1928), 118-133.
  8. ↑ D.Kouznetsov, H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756. http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html http://mizugadro.mydns.jp/PAPERS/2010q2.pdf
  9. ↑ https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0 http://www.ils.uec.ac.jp/~dima/BOOK/202.pdf http://mizugadro.mydns.jp/BOOK/202.pdf Д.Кузнецов. Суперфункции. Lambert Academic Press, 2014. (In Russian, 328 pages.)
  10. ↑ 10.0 10.1 https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0 http://mizugadro.mydns.jp/BOOK/202.pdf Д.Кузнецов. Суперфункции. Lambert Academic Publishing, 2014. (In Russian)
  11. ↑ D.Kouznetsov. Holomorphic ackermanns. http://mizugadro.mydns.jp/PAPERS/2014acker.pdf under consideration
  12. ↑ 12.0 12.1 12.2 H.Kneser. “Reelle analytische L¨osungen der Gleichung Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi(\varphi(x)) = e^x } und verwandter Funktionalgleichungen”. Journal fur die reine und angewandte Mathematik, 187 (1950), 56-67. Cite error: Invalid <ref> tag; name "kneser" defined multiple times with different content Cite error: Invalid <ref> tag; name "kneser" defined multiple times with different content
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