Tetration

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 $x$, tetration $\mathrm {tet} _{b}(x)$ on base $b$ can be defined with:

${\ \mathrm {tet} _{b}(x)=\ \atop {\ }}{{\underbrace {b^{b^{\cdot ^{\cdot ^{b}}}}} } \atop {x}}~$

Fig.1. Tetration $\mathrm {tet} _{b}(x)$ for various $b\!>\!1$ versus $x$.

For real values of the argument $x$ and various values of the base $b$, this $\mathrm {tet} _{b}(x)$ is plotted in Fig.0.

Fig.1. Tetration $f=\mathrm {tet} _{b}(x)$ in the $x,b$ plane with levels $g\!=\!\mathrm {const}$.

The map of $f=\mathrm {tet} _{b}(x)$ in the $x,b$ plane is shown in Fig.1 with levels $f=\mathrm {const}$.

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 $b\!>\!1$, 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 $\exp ^{c}(z)$ for the case of non-integer values of the number $c$ of iterations.

Definition

For real $b>1$, Tetration $F=\mathrm {tet} _{b}$ on the base $b$ is a function of a complex variable, which is holomorphic at least in the range $\{z\in \mathbb {C} :~\Re (z)>-2\}$, bounded in the range $\{z\in \mathbb {C} :~|\Re (z)|\leq 1\}$, and satisfies conditions

$F(z\!+\!1)=\exp _{b}\!{\big (}F(z){\big )}$

$F(0)=1$

$F\!{\big (}z^{*}{\big )}=F{\big (}z{\big )}^{*}$

at least within the range $~\Re (z)>-2~$ .

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 $b=\mathrm {e}$ and $b=2$ .

Etymology and place of tetration in the big picture of math

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

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:

$+\!+$ has only one argument and means unitary increment

$\mathrm {add} _{b}(1)=b+1=+\!+(b)$ ; $\mathrm {add} _{b}(z\!+\!1)=b+(z\!+\!1)=+\!+{\big (}b\!+\!(z){\big )}$

$\mathrm {mul} _{b}(1)=b*1=b$ ; $\mathrm {mul} _{b}(z\!+\!1)=b*(z\!+\!1)=b+{\big (}b*(z){\big )}$

$\exp _{b}(1)=b^{1}=b$ ; $\exp _{b}(z\!+\!1)=b*(\exp _{b}(z))$

$\mathrm {tet} _{b}(1)=b$ ; $\mathrm {tet} _{b}(z\!+\!1)=\exp _{b}{\big (}\mathrm {tet} _{b}(z){\big )}$

$\mathrm {pen} _{b}(1)=b$ ; $\mathrm {pen} _{b}(z\!+\!1)=\mathrm {tet} _{b}{\big (}\mathrm {pen} _{b}(z){\big )}$

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 , ultraexponent , generalized exponent , 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 $b=\mathrm {e}$, $b=2$, $b=\exp(1\!/\!\mathrm {e} )$, and for $b={\sqrt {2}}$. It has a logarithmic singularity at $-2$, and it is a monotonic increasing function.

At $b\leq \exp(1/\mathrm {e} )$ tetration $\mathrm {tet} _{b}(x)$ approaches its limiting value as $x\rightarrow +\infty$, and $\displaystyle \lim _{x\rightarrow +\infty }~\mathrm {tet} _{b}(x)>1$.

Fast growth and application

For $~b\!>\!\exp(1\!/\!\mathrm {e} )~$ tetration $\mathrm {tet} _{b}(x)$ grows faster than any exponential function. For this reason tetration has been proposed for the representation of huge numbers in the mathematics of computation. A number that cannot be stored as floating point could be represented as $\mathrm {tet} _{b}(x)$ for some standard value of $b$ (for example, $b=2$ or $b=\mathrm {e}$) and relatively moderate value of $x$. 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 $z$, tetration ${\rm {tet}}_{b}(z)$ can be interpreted as iterated exponential:

$\mathrm {tet} _{b}(1)=b$

$\mathrm {tet} _{b}(2)=b^{b}$

$\mathrm {tet} _{b}(3)=b^{b^{b}}$

and so on; then, the argument of tetration can be interpreted as number of exponentiations of unity. From definition it follows, that

$\mathrm {tet} _{b}(0)=1$

and

$\mathrm {tet} _{b}(-\!1)=0$

Relation with the Ackermann function

At base $b=2$, tetration is related to the Ackermann function

$\mathrm {tet} _{2}(n+3)=A(4,n)+3$

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

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

The generalization of the 4th Ackermann function $A(4,z)$ for the complex values of $z$ is described in the preprint . Construction of such holomorphic extension is equivalent to construction of tetration $\mathrm {tet} _{b}(z)$ for the base $b=2$.

Asymptotic behavior and properties of tetration

The analytic extension of tetration $~F(z)$ grows rapidly along the real axis of the complex $z$-plane, at least for some values of base $b$. 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

$F(z)=L+\varepsilon (z)+o{\big (}\varepsilon (z)^{2}{\big )}$

where

$\varepsilon (z)=\exp(Qz\!+\!r)$,

$Q$ and $r$ are fixed complex numbers, and $L$ is eigenvalue of logarithm, solution of equation

$x=\log _{b}(x)$.

© Image: Dmitrii Kouznetsov
FIg.2. Graphic solution of equation $x\!=\!\log _{b}(x)$ for $b\!=\!{\sqrt {2}}$ (two real solutions, $x\!=\!2$ and $x\!=\!4$), $b\!=\!\exp(1/{\rm {e)}}$ (one real solution $x\!=\!{\rm {e}}$), and $b\!=\!2$ (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 $b\!=\!{\sqrt {2}}$, $b\!=\!\exp(1\!/\!{\rm {e)}}$, and $b\!=\!2$.

The black line shows function $y=x$ in the $x,y$ plane. The colored curves show function $y=\log _{b}(x)$ for cases $b\!=\!{\sqrt {2}}$ (red), $b\!=\!\exp(1/{\rm {e)}}$ (green), and $b\!=\!2$ (blue).

At $b\!=\!{\sqrt {2}}$, there exist 2 solutions, $x\!=\!2$ and $x\!=\!4$.

At $b\!=\!\exp(1\!/\!{\rm {e)}}$ there exists one solution $x\!=\!{\rm {e}}$.

and $b\!=\!2$, there are no real solutions.

In general,

• at $b\!<\!\exp(1\!/\!{\rm {e)}}$ , there are two real solutions;
• at $b\!=\!\exp(1\!/\!{\rm {e)}}$ , there is one solution, and
• at $b\!>\!\exp(1\!/\!{\rm {e)}}$ , there exist two solutions, but they are complex.

In particular, at $b\!=\!{\sqrt {2}}$, the solutions are
$L\!=\!\!L_{{\sqrt {2}},1}=2$ and $L\!=\!L_{{\sqrt {2}},2}=4$
.

At $b\!=\!2$, the solutions are
$L\!=\!L_{2}\!\approx \!0.824678546142074222314065+1.56743212384964786105857\!~{\rm {i}}$ and
$L\!=\!L_{2}^{*}\!\approx \!0.824678546142074222314065-1.56743212384964786105857\!~{\rm {i}}$.

At $b\!=\!{\rm {e}}$, the solutions are
$L=L_{\rm {e}}\approx 0.318131505204764135312654+1.33723570143068940890116\!~{\rm {i}}$ and
$L=L_{\rm {e}}^{*}\approx 0.318131505204764135312654-1.33723570143068940890116\!~{\rm {i}}$.

A few hundred straightforward iterations of equation $L=\log _{b}(L)$ 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 $L=L_{1}$ and $L=L_{2}$ of equation (14) are plotted in figure 3 versus $\beta =\ln(b)$ with thin black lines. At $\beta <1/\mathrm {e}$, both $L_{1}$ and $L_{2}$ are real and positive. Let $L_{1}<L_{2}$, and only at $\ln(b)=1/e$, the equality $L_{1}=L_{2}$ takes place. Basic properties of tetration are determined by the base $b$. The main parameters versus $\ln(b)$ are plotted in figure 3. The thin black solid curve at $\beta \geq 1/{\rm {e}}$ represents the real part of the solutions $L$ and $L^{*}$ 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 $Q$ determines periodicity of quasi-periodicity of tetration. The two solutions for $Q$ are shown in figure 3 with green lines.

At $b<\exp(1/\mathrm {e} )$ both solutions for $Q$ are real. The negative $Q$ corresponds to tetration, decaying to the asymptotic value $L$ in the direction of real axis; positive $Q$ 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 $F(0)=1$. Therefore, only one negative $Q$ corresponds to the asymptotic behavior of tetration.

At $b\!>\!\exp(1/\mathrm {e} )$, both options for $Q$ 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) $T={\frac {2\pi }{Q}}$ are shown in Figure 3 with dotted lines. At $b>\exp(1/\mathrm {e} )$, only "negative" period corresponds to tetration. At $b>\exp(1/\mathrm {e} )$, the periodicity can be achieved only asymptotically; and $T$ 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 $1\!<\!b\!<\!\exp(1\!/\!\mathrm {e} )$, tetration is periodic; the period is pure imaginary.

At $b\!=\!\exp(1\!/\!\mathrm {e} )$, tetration is not periodic, and no exponential asymptotic exist.

$b\!>\!\exp(1\!/\!\mathrm {e} )$, 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 $b$, 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 $b$ in the vicinity of the value $b=\exp(1/\mathrm {e} )$, the evaluation procedure is different for the cases $1\!<\!b\!<\!\exp(1\!/\!\mathrm {e} )$, $b\!=\!\exp(1/\mathrm {e} )$, and $b\!>\!\exp(1\!/\!\mathrm {e} )$, and these should be considered separately.

Case $1<b<\exp(1/\mathrm {e} )$

Template:CC-Image
Fig.4. Tetration $f=\mathrm {tet} _{b}(x\!+\!\mathrm {i} y)$ for $b\!=\!{\sqrt {2}}$

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

Failed to parse (syntax error): {\displaystyle T=\frac{2\pi}{\ln^2(2)}\approx −17.1431481793548471041794 ~\mathrm{i}} .

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

The solution follows asymptotic at large values of real part of the argument, exponentially approaching the limiting value. In particular, for $b\!=\!{\sqrt {2}}$, 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 $b\!=\!{\sqrt {2}}$, these limiting values are $L\!=\!4$ and $L\!=\!2$.

The trace of the solution along the real axis corresponds to the red dotted curve in Figure 1. Other solutions of the recursive equation $\exp _{b}\!{\big (}F(z){\big )}\!=\!F(z\!+\!1)$, 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, $F(0)\!=\!1$.

Case $b=\exp(1/\mathrm {e} )$

Fig.5. Tetration at $b=\exp(1\!/\!\mathrm {e} )$.

At $b=\exp(1\!/\!\mathrm {e} )$, the limiting value $L=\mathrm {e}$, and, asymptotically,

$F(z)=\mathrm {e} -2\mathrm {e} /z+{\mathcal {O}}\!{\big (}\ln(z)/z^{2}{\big )}$

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

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

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

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

Intermediate levels are shown with thin lines.

There is a cut at $x<-2$, $y=0$, but the hump of the function at the cut reduces as $y$ reduces, id est, with increasing $|z|$. In such a way, everywhere, at $|z|\rightarrow \infty$, the function approaches its limiting value $L=\mathrm {e} \approx 2.71$. almost everywhere, although there is a set of singularities at negative integer values of $z<-1$.

Behavior of this function at real values of argument is shown in Figure 1 with thin solid line. Other solutions of the recursive equation $\exp _{b}\!{\big (}F(z){\big )}\!=\!F(z\!+\!1)$ 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, $F(0)\!=\!1$.

Case $b>\exp(1/\mathrm {e} )$

(CC) Image: Dmitrii Kouznetsov
Fig.6. Tetration $\mathrm {tet} _{b}(x\!+\!\mathrm {i} y)$ at base $b\!=\!2$ and $b\!=\!\mathrm {e}$.

At $b>\exp(1/\mathrm {e} )$, tetration is asymptotically periodic. It decays exponentially to the fixed points $L$ and $L^{*}$ in the upper and lower halfplane. This allows to express it through its values along the imaginary axis, using the Cauchi integral. .

For the base $b=2$ and $b=\mathrm {e}$ tetration $f=\mathrm {tet} _{b}(z)$ is plotted in figure 6. Levels $\Re {f}=0,\pm 1,...\pm 16$ and $\Im {f}=0,\pm 1,...\pm 16$ 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 $\approx \infty$ 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 $L$ and $L^{*}$. Function is quasi-periodic; the same fractal structure reproduces again and again at the translation of argument with quasiperiod $T$ in the upper halfplane and $T^{*}$ in the lower halfplane.

There is cut at $x<-2$, $y=0$. The jump of the function at this cut approaches $L\!-\!L^{*}$ almost everywhere, although there is set of singularities at negative integer values of $x<-2$.

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 $b=\mathrm {e}$.

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 $b={\sqrt {2}},\exp(1/\mathrm {e} ),2,\mathrm {e}$. For $b=\mathrm {e}$, the derivatives of tetration are plotted in figure at the right.

$\mathrm {tet} ^{\prime }$ is plotted with red;

$\mathrm {tet} ^{\prime \prime }$ is plotted with green;

$\mathrm {tet} ^{\prime \prime \prime }$ is plotted with blue;

$\mathrm {tet} ^{\prime \prime \prime \prime }$ is plotted with pink.

Tetration is strictly growing function; its first derivative is positive. For $b=\mathrm {e}$ the minimum of the derivative takes place in vicinity of $-0.5$ and is slightly smaller than unity. At $1\!<\!b\!<\exp(1\!/\mathrm {e} )$, the growth is limited by the minimum of the limiting values $L$. Tetration approaches this limiting value exponentially. In particular, at $b={\sqrt {2}}$, the limiting value is 2.

At $b\!=\!\exp(1\!/\!\mathrm {e} )$, the growth is limited by the fixed point $L\!=\!\mathrm {e}$. Tetration approaches this limiting value as rational function.

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

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

Tetration at the base $b={\sqrt {2}}$

Template:CC-Image
Tetration at the base $b={\sqrt {2}}$

There is hypothesis that at base $b={\sqrt {2}}$, the graphic of tetration $y=\mathrm {tet} _{b}(x)$ is symmetric with respect to the line $y=-x$. This line, together with the graphic, is shown in Figure at let. The asymptotics $x=-2$ and $y=2$ are also shown. The graphic looks symmetric with respect to $x\leftrightarrow y$, but no proof of this hypothesis was suggested. Now this hypothesis is negated; the apparent symmetry is only approximation. However, for $b={\sqrt {2}}$, at the segment $-1\leq x\leq 0$, the relation $\mathrm {tet} _{b}(x)\approx -\mathrm {tet} _{b}^{-1}(-x)$ holds with 4 decimal digits .

Tetration at base e

Holomorphic tetration on the natural base $b\!=\!\mathrm {e}$ is the most developed, at least up to the year 2008. In the rest of this article, it is assumed that $b\!=\!\mathrm {e} \approx 2.71828$ although the majority of results allow a straightforward extension to the case of real $b\!>\!\exp(1/\mathrm {e} )\approx 1.44467$.

Tetration has real fixed point $X_{0}$, id est, solution $z$ of the equation $z=\mathrm {tet} (z)$. Its approximation

$X_{0}\approx 1.85035452902718$

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

Pentation

Complex map of natural pentation, $u\!+\!\mathrm {i} v=\mathrm {pen} (x\!+\!\mathrm {i} y)$ by

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

$\mathrm {pen} (z\!+\!1)=\mathrm {tet} {\big (}\mathrm {pen} (z){\big )}$

$\mathrm {pen} (0)=1$

Expansion of tetration in vicinity of its fixed point $X_{0}$ from the interval $(-2,-1)$ (for real base $b\!>\!1$) 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 $b=\mathrm {e} \approx 2.71$, the fixed point of tetration $X_{0}\!\approx \!-1.850354529$. and pentation to this base approaches this value at large negative values of the real part of the argument . Complex map of natural pentation is shown in figure at left; $u\!+\!\mathrm {i} v=\mathrm {pen} (x\!+\!\mathrm {i} y)$.

Inverse of tetration

Fig.7. slog in the complex plane.

The inverse of tetration $t=\mathrm {slog} (z)=\mathrm {tet} ^{-1}(z)$ can be performed using the Newton method, solving equation $\mathrm {tet} (t)=z$, leading to

$\mathrm {tet} {\big (}\mathrm {kslog} (z){\big )}=z$.

The inverse function has branchpoints $L$ and $L^{*}$. For the kslog, at base $b=\mathrm {e}$, shown in the figure 7, the cuts are placed horizontaly, along the lines

$\Re (z)<\Re (L)$, $\Im (z)=\pm \Im (L)$.

Due to the symmetry $\mathrm {kslog} {\big (}z^{*}{\big )}=\mathrm {kslog} {\big (}z{\big )}^{*}$, 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 $x$ in the shaded region, the relation

$\mathrm {slog} {\big (}\mathrm {tet} (z){\big )}=z$

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 $(-2,\infty )$. The fixed points $L$ is mapped to imaginary infinity, following the shaded strip.

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

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 $\Re (\mathrm {slog} (z))=-2$. 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 $\exp ^{c}$ at non-integer $c$, 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 $\mathrm {slog} (z^{*})=\mathrm {slog} (z)^{*}$. There 3 options are shown in the figure.

The shaded region shows the domain

$G=\{z\in \mathbb {C} ~:~\Re (z)\!>\!\Re (L)~,~|z|<|L|\}$

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 $-2$; 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:

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

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 $\Re (z)>1~,~1<|\Im (z)|<2$. 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:

$\mathrm {tet} (z)=\sum _{n=0}^{\infty }c_{n}(z-z_{0})^{n}$

where the $n$th coefficient

$c_{n}={\frac {1}{n!}}\mathrm {tet} ^{(n)}(z_{0})$

is expressed through the $n$ 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 $z_{0}=0$, the calculation gives the following values

$c$₀ = 1

$c$₁ ≈ 1.091767351258322138

$c$₂ ≈ 0.271483212901696469

$c$₃ ≈ 0.212453248176258214

$c$₄ ≈ 0.069540376139988952

$c$₅ ≈ 0.044291952090474256

$c$₆ ≈ 0.014736742096390039

$c$₇ ≈ 0.008668781817225539

$c$₈ ≈ 0.002796479398385586

$c$₉ ≈ 0.001610631290584341

$c$₁₀≈ 0.000489927231484419

$c$₁₁≈ 0.000288181071154065

$c$₁₂≈ 0.000080094612538551

$c$₁₃≈ 0.000050291141793809

$c$₁₄≈ 0.000012183790344901

$c$₁₅≈ 0.000008665533667382

$c$₁₆≈ 0.000001687782319318

$c$₁₇≈ 0.000001493253248573

$c$₁₈≈ 0.000000198760764204

$c$₁₉≈ 0.000000260867356004

$c$₂₀≈ 0.000000014709954143

$c$₂₁≈ 0.000000046834497327

$c$₂₂≈-0.000000001549241666

$c$₂₃≈ 0.000000008741510781

$c$₂₄≈-0.000000001125787310

$c$₂₅≈ 0.000000001707959267

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

$f=\sum _{n=0}^{25}c_{n}z^{n}$

is shown in the complex $z$ plane.

Levels $\Re (f)=-2,-1,0,1,2,3,4$ are shown with thick black curves.

Levels $\Re (f)=-1.8,-1.6,-1.4,-1.2,-0.8,-0.6,-0.4,-0.2$ are shown with thin red curves.

Levels $\Re (f)=0.2,0.4,0.6,0.8,1.2,1.4,1.6,1.8$ are shown with thin thin blue curves.

Levels $\Im (f)=-2,-1$ are shown with thick red curves.

Levels $\Im (f)=1,2$ are shown with thick blue curves.

Levels $\Im (f)=\pm \pi ,\pm 3\pi$ are shown with thick pink curves.

Levels $\Im (f)=\pm 0.2,\pm 0.4,\pm 0.6,\pm 0.8,\pm 1.2,\pm 1.4,\pm 1.6,\pm 1.8$ are shown with thin green curves.

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

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 $|z|$ 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

$\mathrm {tet} (z)=\sum _{n=0}^{\infty }a_{n}\left({\frac {z-3\mathrm {i} }{2}}\right)^{n}$

can be evaluated in the similar way:

$a_{0}\approx 0.3709065890322851+1.3368216707889140i$

$a_{1}\approx 0.0366009653759846+0.1392221538995050i$

$a_{2}\approx -0.1688843184064154+0.0971853361962927i$

$a_{3}\approx -0.1268131504868087-0.1183162876702863i$

$a_{4}\approx 0.0423580931032393-0.1052093008832072i$

$a_{5}\approx 0.0584830639356318-0.0081022452449608i$

$a_{6}\approx 0.0234003166529485+0.0180777701182038i$

$a_{7}\approx 0.0034426098470138+0.0181510375563591i$

$a_{8}\approx -0.0080369581444167+0.0091742846703500i$

$a_{9}\approx -0.0070469552816877-0.0009395850672747i$

$a_{10}\approx -0.0018461796309531-0.0032234258318168i$

$a_{11}\approx 0.0005406488544310-0.0018967206101561i$

$a_{12}\approx 0.0010224364808881-0.0005596865717924i$

$a_{13}\approx 0.0006471439639805+0.0002598066193583i$

$a_{14}\approx 0.0001044445559337+0.0003719947259883i$

$a_{15}\approx -0.0001117853540434+0.0001678668755219i$

$a_{16}\approx -0.0001063015871081+0.0000207220003313i$

$a_{17}\approx -0.0000507809881911-0.0000357591300574i$

$a_{18}\approx -0.0000031474299869-0.0000352318593759i$

$a_{19}\approx 0.0000134766134413-0.0000133303413745i$

$a_{20}\approx 0.0000098023908240+0.0000004760718415i$

$a_{21}\approx 0.0000035549347545+0.0000038981621220i$

$a_{22}\approx -0.0000002155265265+0.0000029627341324i$

$a_{23}\approx -0.0000013167390363+0.0000009738135453i$

$a_{24}\approx -0.0000008340196081-0.0000001866385871i$

$a_{25}\approx -0.0000002286961098-0.0000003749771677i$

$a_{26}\approx 0.0000000537258461-0.0000002306013659i$

$a_{27}\approx 0.0000001140665665-0.0000000656951029i$

$a_{28}\approx 0.0000000666359546+0.0000000232663057i$

$a_{29}\approx 0.0000000139678685+0.0000000331511830i$

$a_{30}\approx -0.0000000068489056+0.0000000171304198i$

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 $\Re (z)\leq 0.5$. With 50 terms, at $|z-3|<2$, 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 $b\!=\!\mathrm {e}$ is considered; although the generalization to the case $b\!>\!\mathrm {e}$ is straightforward. In this section, the behavior of tetration $\mathrm {tet} (z)$ is considered for moderate values of $\Re (z)$ and large values of $\Im (z)$

At $b\!=\!\mathrm {e}$, in the upper halfplane, tetrations approach the fixed point $L\approx 0.3+1.3\mathrm {i}$ of the logarithm. This approach is exponential. Using the exponential

$\varepsilon =\varepsilon (z)=\exp(Lz+R)$

as a small parameter, for some complex constant $R$, the tetration can be estimated as follows:

$\mathrm {tet} (z)=L+\varepsilon +a_{2}\varepsilon ^{2}+a_{3}\varepsilon ^{3}+{\mathcal {O}}(\varepsilon ^{4})$

Substitution of this expression into the equation $\mathrm {tet} (z\!+\!1)=\exp {\big (}\mathrm {tet} (z){\big )}$ gives
$a_{2}={\frac {1/2}{L-1}}\approx -0.151314-0.2967488\mathrm {i} ~,~$ $a_{3}={\frac {a_{2}+1/6}{L^{2}-1}}\approx -0.36976+0.98730\mathrm {i}$
Value $R\approx 1.077961437526-0.946540963949\mathrm {i}$ can be found fitting the tetration $\mathrm {tet} (z)$ with function $L+\epsilon +a_{2}\epsilon ^{2}+a_{2}\epsilon ^{3}$ at large values of $\Im (z)$. Then, the expansion can be written in the form

$\mathrm {tet} (z)=\sum _{n=0}^{N}a_{n}\varepsilon ^{n}+{\mathcal {(}}O)\!{\big (}\varepsilon ^{N\!+\!1}{\big )}$

assuming that $a_{0}\!=\!L$ and $a_{1}\!=\!1$. At given $N$, 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

$f=\mathrm {tet} (z)-L$,

$f=\mathrm {tet} (z)-{\Big (}L+\varepsilon (z){\Big )}$,

$f=\mathrm {tet} (z)-{\Big (}L+\varepsilon +a_{2}\varepsilon ^{2}{\Big )}$,

$f=\mathrm {tet} (z)-{\Big (}L+\varepsilon +a_{2}\varepsilon ^{2}+a_{3}\varepsilon ^{3}{\Big )}$

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

Levels $\arg(f)=-2,-1$ are shown with thick red lines.

Level $\arg(f)=0$ is shown with thick black lines.

Levels $\arg(f)=1,2$ are shown with thick blue lines.

Levels $\arg(f)=\pm \pi$ are shown with scratched lines. (these lines reveal the step of sampling used by the plotter).

Levels $|f|=\exp(-0.8),\exp(-0.6),\exp(-0.4),\exp(-0.2)$ are shown with thin red lines.

Levels $|f|=\exp(0.2),\exp(0.4),\exp(0.6),\exp(-0.8)$ are shown with thin blue lines.

Levels $|f|=\exp(3),\exp(2),\exp(1),\exp(0),\exp(-\!1),\exp(-\!2),\exp(-\!3),\exp(-\!4),\exp(-\!5),\exp(-\!5),\exp(-\!7),\exp(-\!8)$ are shown with thin thick black lines.

Level $|f|=\exp(-10)$ is shown with thick red line.

Levels $|f|=\exp(-12),\exp(-14),\exp(-16),\exp(-18)$ are shown with thick black lines.

Level $|f|=\exp(-20)$ is shown with thick red line.

Levels $|f|=\exp(-22),\exp(-24),\exp(-26),\exp(-28)$ are shown with thick black lines.

Level $|f|=\exp(-30)$ is shown with thick green line.

Level $|f|=\exp(-31)$ is shown with thick black line.

The plotter tried to draw also

Level $|f|=\exp(-32)$ with thick black line and

Level $|f|=\exp(-33)$ 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

$|f|=\exp(-31)$ 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 $\Im (z)>4$, $\Re (z)<4\Im (z)-25$, the asymptotic approximation

$\mathrm {tet} (z)\approx L+\varepsilon +a_{2}\varepsilon ^{2}+a_{3}\varepsilon ^{3}$

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
$f(z)=\log(z+2)+\sum _{n=0}^{100}c_{n}z^{n}$ in the complex $z$-plane.

Due to recurrent relation $\mathrm {tet} (z)=\exp(\mathrm {tet} (x+1))$, it is sufficient to approximate tetration in any vertical strip of unit with in the complex plane. Some of such approximations are suggested in . In principle, the numerical approximation of tetration with implementation of the Cauchi integral with finite sums 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 $\mathrm {tet} (z)-\log(z+2)$. 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

$f(z)=\log(z+2)+\sum _{n=0}^{100}c_{n}\left({\frac {z}{2}}\right)^{n}$

is shown at the figure in the complex $z$-plane. In vicinity of the origin of coordinates, say, $|z|<1$, the most of terms are negligibly small, and the shortened sum still approximates the tetration. The first coefficients in this expansion are

$c_{0}=0.30685281944005469058$

$c_{1}=1.18353470251664338875$

$c_{2}=1.58593285160678321155$

$c_{3}=1.36629265207672068172$

$c_{4}=1.36264601823980036066$

$c_{5}=1.21734246689515424045$

$c_{6}=1.10981816083559525765$

$c_{7}=0.96674692974769849130$

$c_{8}=0.84089872598668435888$

$c_{9}=0.71353210966804747617$

$c_{10}=0.60168548504001373445$

More coefficients can be extracted from the generator of the figure. While $|z|\leq 2$, 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:

$\mathrm {slog} (z)=\sum _{n=1}^{\infty }c_{n}(z-1)^{n}$.

Approximations for the first 16 coefficients:

$c_{1}\approx 0.91594605649953$

$c_{2}\approx -0.20861842957759$

$c_{3}\approx -0.05450400630209$

$c_{4}\approx 0.07134941925273$

$c_{5}\approx -0.02004387374438$

$c_{6}\approx -0.01101258023037$

$c_{7}\approx 0.01207268318645$

$c_{8}\approx -0.00272922880760$

$c_{9}\approx -0.00269905319156$

$c_{10}\approx 0.00243941500632$

$c_{11}\approx -0.00036220360858$

$c_{12}\approx -0.00070125921262$

$c_{13}\approx 0.00052782155380$

$c_{14}\approx -0.00002987943551$

$c_{15}\approx -0.00018614540434$

$c_{16}\approx 0.00011722843042$

The partial sum with 16 terms (from zero to 16) is plotted in the figure in the complex $z$ 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 $\mathrm {slog} (z)$ with the Taylor series is valid for

$|z-1|<|1-L|$

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

Expansion at fixed point $L$ of logarithm

(CC) Image: Dmitrii Kouznetsov
Fitting of slog with the expansion around $L$.

Superlogarithm can be approximated with expansion

$\mathrm {slog} (z)={\frac {1}{L}}\left(\log(z\!-\!L)+\sum _{m,n}r_{m,n}(z\!-\!L)^{m+2\pi \mathrm {i} n/L}\right)$,

where $L$ 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 $\mathrm {slog} (\exp(z))=\mathrm {slog} (z)+1$. 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

$r_{0,0}\approx \!-\!1.0779614375+0.9465409639i$

$r_{1,0}\approx ~0.1513148972+0.2967488367i$

$r_{2,0}\approx \!-\!0.0607692016+0.0359770148i$

$r_{3,0}\approx \!-\!0.0079378875-0.0176412865i$

$r_{4,0}\approx ~0.0051087546-0.0000718839i$

These coefficients allow to approximate slog in vicinity of the fixed point of logarithm with function

$f={\frac {1}{L}}\left(\log(z\!-\!L)+\sum _{m=0}^{4}r_{m,0}(z-L)^{m}\right)$.

In the figure this $f$ is shown in the complex $z$-plane.

Levels $\Re (f)=0,\pm 1,\pm 2$ are shown with thick black lines.

Levels $\Re (f)=-1.9,..-0.1$ are shown with thin red lines.

Levels $\Im (f)=-1.9,..-1.9$ are shown with thin dark green lines.

Levels $\Im (f)=0$ is shown with thick green line.

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

Level $\Im (f)=-1$ is shown with thick red line

Levels $\Im (f)=1,2,3$ 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 $|z-L|<1$, the deviation of these dashed lines from the levels for function $f$ 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:

$\mathrm {slog} (z)={\frac {\log(z-L)}{L}}+{\frac {\log(z-L^{*})}{L^{*}}}+\sum _{n=0}^{\infty }u_{n}(z-1)^{n}$

The coefficients of this expansion are real. The first coefficients:

$u_{0}\approx \!~1.419225215505$

$u_{1}\approx \!-0.026066290298$

$u_{2}\approx ~0.001733047818$

$u_{3}\approx \!-0.000019521307$

$u_{4}\approx \!-0.000063070064$

$u_{5}\approx ~0.000025678960$

$u_{6}\approx \!-0.000005590100$

$u_{7}\approx \!-0.000000072797$

$u_{8}\approx ~0.000000651489$

$u_{9}\approx \!-0.000000276981$

$u_{10}\!\approx ~0.000000031118$

$u_{11}\!\approx ~0.000000029409$

$u_{12}\!\approx \!-0.000000018969$

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

$f=\mathrm {fslog} (z)={\frac {\log(z-L)}{L}}+{\frac {\log(z-L^{*})}{L^{*}}}+\sum _{n=0}^{50}u_{n}(z-1)^{n}$

is shown in figure in the complex $z$ plane. Lines of constant $\Re (f)$ and those of constant $\Im (f)$ 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

$\mathrm {slog} (z)=\mathrm {slog} (\exp(z))-1$

$\mathrm {slog} (z)=\mathrm {slog} (\log(z))+1$

The residuals

$f=\mathrm {fslog} (z)-\mathrm {fslog} (\exp(z))+1$ and

$f=\mathrm {fslog} (z)-\mathrm {fslog} (\log(z))-1$

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

Iterated exponential and ${\sqrt {\exp }}$

(CC) Image: Dmitrii Kouznetsov
Fig.9. $\exp ^{c}(z)$ in the complex $z$ plane for various $c$.

Pre-historic approach to the iterated exponential

Especially interesting is the case of iteration of natural exponential, id est, $b=\mathrm {e}$. Existence of the fractional iteration, and, in particular, existence of operation ${\sqrt {\exp }}=\exp ^{1/2}$ was demonstrated in 1950 by H.Kneser. . However, that time, there was no computer facility for the evaluation of such an exotic function $F$ that $F(F(z))=\exp(z)$; perhaps, just absence of an apropriate plotter did not allow Kneser to plot the distribution of fractal exponential function $\exp ^{c}(z)$ 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^2(z) =\exp_b\!\Big (\exp_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 \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 $\exp _{b}^{1}=\exp _{b}$. 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 $L^{*}$ 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 Failed to parse (SVG (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^u x^v=x^{u+v}} . 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 $\exp ^{v}(x)$ 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 $F{\big (}z^{*}{\big )}=F(z)^{*}$ 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 , 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 , 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 Failed to parse (SVG (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 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 $J$ 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 ${\mathcal {M}}=\{n\in \mathbb {N} ~:~n<-1\}$.

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 . There is only one tetration, that satisfies requirements of the definition, although the rigorous mathematical proof of the uniqueness is still under development.