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# Tetration

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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:

 has only one argument and means unitary increment
 ; 
 ; 
 ; 
 ; 
 ; 

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 , , , and for . It has a logarithmic singularity at , and it is a monotonic increasing function.

At  tetration  approaches its limiting value as , and .

### Fast growth and application

For  tetration  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  for some standard value of  (for example,  or ) and relatively moderate value of . 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 , tetration  can be interpreted as iterated exponential:





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



and



### Relation with the Ackermann function

At base , tetration is related to the Ackermann function [7]



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



The generalization of the 4th Ackermann function  for the complex values of  is described in the preprint [2] . Construction of such holomorphic extension is equivalent to construction of tetration  for the base .

## Asymptotic behavior and properties of tetration

The analytic extension of tetration  grows rapidly along the real axis of the complex -plane, at least for some values of base . 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



where

,

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

.
© Image: Dmitrii Kouznetsov
FIg.2. Graphic solution of equation  for  (two real solutions,  and ),  (one real solution ), and  (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 , , and .

The black line shows function  in the  plane. The colored curves show function  for cases  (red),  (green), and  (blue).

At , there exist 2 solutions,  and .

At  there exists one solution .

and , there are no real solutions.

In general,

• at  , there are two real solutions;
• at  , there is one solution, and
• at  , there exist two solutions, but they are complex.

In particular, at , the solutions are
 and 
.

At , the solutions are
 and
.

At , the solutions are
 and
.

A few hundred straightforward iterations of equation  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  and  of equation (14) are plotted in figure 3 versus  with thin black lines. At , both  and  are real and positive. Let , and only at , the equality  takes place. Basic properties of tetration are determined by the base . The main parameters versus  are plotted in figure 3. The thin black solid curve at  represents the real part of the solutions  and  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  determines periodicity of quasi-periodicity of tetration. The two solutions for  are shown in figure 3 with green lines.

At  both solutions for  are real. The negative  corresponds to tetration, decaying to the asymptotic value  in the direction of real axis; positive  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 . Therefore, only one negative  corresponds to the asymptotic behavior of tetration.

At , both options for  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)  are shown in Figure 3 with dotted lines. At , only "negative" period corresponds to tetration. At , the periodicity can be achieved only asymptotically; and  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 , tetration is periodic; the period is pure imaginary.

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

, 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 , 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  in the vicinity of the value , the evaluation procedure is different for the cases , , and , and these should be considered separately.

### Case 

Template:CC-Image
Fig.4. Tetration  for 

For , the period  is imaginary. The period with smallest modulus corresponds to the solution that is unity at the origin of coordinates. For , the function  is shown in figure 4 with levels of constant real part and levels of constant imaginary part. Levels  and levels  aew shown with thick lines. Intermediate levels are shown with thin lines. There are branch points at ; the cut lines are . For this value of the base, the period

.

There is a cut at , ; although the jump at this cut reduces at the increase of . In such a way, the function approaches its limiting value  almost everywhere, although there is set of singularities at negative integer values of .

The solution follows asymptotic at large values of real part of the argument, exponentially approaching the limiting value. In particular, for , 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 , these limiting values are  and .

The trace of the solution along the real axis corresponds to the red dotted curve in Figure 1. Other solutions of the recursive equation , 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, .

### Case 

Fig.5. Tetration at .

At , the limiting value , and, asymptotically,



The function  is shown in figure 5.

Levels  are shown with thick black lines.

Levels  are shown with thick red lines.

Levels  are shown with thick blue lines.

Intermediate levels are shown with thin lines.

There is a cut at , , but the hump of the function at the cut reduces as  reduces, id est, with increasing . In such a way, everywhere, at , the function approaches its limiting value . almost everywhere, although there is a set of singularities at negative integer values of .

Behavior of this function at real values of argument is shown in Figure 1 with thin solid line. Other solutions of the recursive equation  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, .

### Case 

(CC) Image: Dmitrii Kouznetsov
Fig.6. Tetration  at base  and .

At , tetration is asymptotically periodic. It decays exponentially to the fixed points  and  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  and  tetration  is plotted in figure 6. Levels  and  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  and . Function is quasi-periodic; the same fractal structure reproduces again and again at the translation of argument with quasiperiod  in the upper halfplane and  in the lower halfplane.

There is cut at , . The jump of the function at this cut approaches  almost everywhere, although there is set of singularities at negative integer values of .

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 .

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 . For , the derivatives of tetration are plotted in figure at the right.

 is plotted with red;
 is plotted with green;
 is plotted with blue;
 is plotted with pink.

Tetration is strictly growing function; its first derivative is positive. For  the minimum of the derivative takes place in vicinity of  and is slightly smaller than unity. At , the growth is limited by the minimum of the limiting values . Tetration approaches this limiting value exponentially. In particular, at , the limiting value is 2.

At , the growth is limited by the fixed point . Tetration approaches this limiting value as rational function.

The growing-up holomorphic solution of equation  can be constructed in the similar way also at , but at the real axis, such a solution remains larger than unity, and the condition  cannot be satisfied. Therefore, this solution cannot be interpreted as extension of iterative exponentiation  for non-integer number  of exponentiations; in this sense, such a solution is not a tetration.

Tetration  shows explosively-fast growth along the real axis only at values .

## Tetration at the base 

Template:CC-Image
Tetration at the base 

There is hypothesis that at base , the graphic of tetration  is symmetric with respect to the line . This line, together with the graphic, is shown in Figure at let. The asymptotics  and  are also shown. The graphic looks symmetric with respect to , but no proof of this hypothesis was suggested. Now this hypothesis is negated; the apparent symmetry is only approximation. However, for , at the segment , the relation  holds with 4 decimal digits [8][9].

## Tetration at base e

Holomorphic tetration on the natural base  is the most developed, at least up to the year 2008. In the rest of this article, it is assumed that  although the majority of results allow a straightforward extension to the case of real .

Tetration has real fixed point , id est, solution  of the equation . Its approximation



can be found, iterating equation , where slog is inverse function of tetration.

## Pentation

Complex map of natural pentation,  by [10]

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




Expansion of tetration in vicinity of its fixed point  from the interval  (for real base ) 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 . 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; .

## Inverse of tetration

The inverse of tetration  can be performed using the Newton method, solving equation , leading to

.

The inverse function has branchpoints  and . For the kslog, at base , shown in the figure 7, the cuts are placed horizontaly, along the lines

, .

Due to the symmetry , 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  in the shaded region, the relation



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 . The fixed points  is mapped to imaginary infinity, following the shaded strip.

In figure 8, the images of the grid lines  and images of the grid lines  are shown. These curves reproduce levels  and  shown in figure 6 for .

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 . 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  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 . There 3 options are shown in the figure.

The shaded region shows the domain



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 ; 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:



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 . 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:



where the th coefficient



is expressed through the  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 , the calculation gives the following values

0 = 1
1 ≈ 1.091767351258322138
2 ≈ 0.271483212901696469
3 ≈ 0.212453248176258214
4 ≈ 0.069540376139988952
5 ≈ 0.044291952090474256
6 ≈ 0.014736742096390039
7 ≈ 0.008668781817225539
8 ≈ 0.002796479398385586
9 ≈ 0.001610631290584341
10≈ 0.000489927231484419
11≈ 0.000288181071154065
12≈ 0.000080094612538551
13≈ 0.000050291141793809
14≈ 0.000012183790344901
15≈ 0.000008665533667382
16≈ 0.000001687782319318
17≈ 0.000001493253248573
18≈ 0.000000198760764204
19≈ 0.000000260867356004
20≈ 0.000000014709954143
21≈ 0.000000046834497327
22≈-0.000000001549241666
23≈ 0.000000008741510781
24≈-0.000000001125787310
25≈ 0.000000001707959267

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



is shown in the complex  plane.

Levels  are shown with thick black curves.
Levels  are shown with thin red curves.
Levels  are shown with thin thin blue curves.
Levels  are shown with thick red curves.
Levels  are shown with thick blue curves.
Levels  are shown with thick pink curves.
Levels  are shown with thin green curves.

In the upper left corner of figure N, the same is shown for function 

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  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



can be evaluated in the similar way:

































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 . With 50 terms, at , 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  is considered; although the generalization to the case  is straightforward. In this section, the behavior of tetration  is considered for moderate values of  and large values of 

At , in the upper halfplane, tetrations approach the fixed point  of the logarithm. This approach is exponential. Using the exponential



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



Substitution of this expression into the equation  gives
 
Value  can be found fitting the tetration  with function  at large values of . Then, the expansion can be written in the form



assuming that  and . At given , 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

,
,
,


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

Levels  are shown with thick red lines.
Level  is shown with thick black lines.
Levels  are shown with thick blue lines.
Levels  are shown with scratched lines. (these lines reveal the step of sampling used by the plotter).
Levels  are shown with thin red lines.
Levels  are shown with thin blue lines.
Levels  are shown with thin thick black lines.
Level  is shown with thick red line.
Levels  are shown with thick black lines.
Level  is shown with thick red line.
Levels  are shown with thick black lines.
Level  is shown with thick green line.
Level  is shown with thick black line.

The plotter tried to draw also

Level  with thick black line and
Level  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
 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 , , the asymptotic approximation



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
 in the complex -plane.

Due to recurrent relation , 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 . 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



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













More coefficients can be extracted from the generator of the figure. While , 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:

.

Approximations for the first 16 coefficients:


















The partial sum with 16 terms (from zero to 16) is plotted in the figure in the complex  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  with the Taylor series is valid for



Obviously, it fails at . For this case, the asymptotic expansion can be used.

### Expansion at fixed point  of logarithm

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

Superlogarithm can be approximated with expansion [12]

,

where  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







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

.

In the figure this  is shown in the complex -plane.

Levels  are shown with thick black lines.
Levels  are shown with thin red lines.
Levels  are shown with thin dark green lines.
Levels  is shown with thick green line.

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

Level  is shown with thick red line
Levels  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 , the deviation of these dashed lines from the levels for function  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:



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















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



is shown in figure in the complex  plane. Lines of constant  and those of constant  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




The residuals

 and


are shown in the figure with levels const. In the voided regions in vicinity of  and , the residual is at the level of . (It is difficult to make the residual smaller, using the arithmetics with double complex variables.) This test indicates, that at , the approximation of slog with two the logarithms and the polynomial of 50th power gives at least 9 correct significant figures.

## Iterated exponential and 

(CC) Image: Dmitrii Kouznetsov
Fig.9.  in the complex  plane for various .

### Pre-historic approach to the iterated exponential

Especially interesting is the case of iteration of natural exponential, id est, . Existence of the fractional iteration, and, in particular, existence of operation  was demonstrated in 1950 by H.Kneser. [12]. However, that time, there was no computer facility for the evaluation of such an exotic function  that ; perhaps, just absence of an apropriate plotter did not allow Kneser to plot the distribution of fractal exponential function  in the complex  plane for various values of , shown in Fig.9.

(CC) Image: Dmitrii Kouznetsov
Fig.10.  versus 

### The Implementation through the tetration

Holomorphic tetration allows to extend the iterated exponential 






For non-integer values of . It can be defined as



If in the notation  the superscript is omitted, it is assumed to be unity; for example . If the subscript is omitted, it is assumed to be , id est, 

### Iterated exponential in the complex plane

Function  is shown in figure 9 with levels of constant real part and levels of constant imaginary part. Levels  and  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  are shown with thin red lines. Levels  are shown with thin green lines. Levels  and Levels  are marked with thick green lines, where  is fixed point of logarithm. At non-integer values of ,  and  are branch points of function ; in figure, the cut is placed parallel to the real axis. At  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  is ploted in figure 10 versus  for values .
in programming languages, inverse function of exp is called log.

For logarithm on base e, notation ln is also used. In particular, ,  and so on.

At least for the real and big enough , the relation  holds, which is qute analogous of relations  and . However, at negative  or negative , value  should be big enough, that  and  and  are defined, see figure 10. For example, at , we need . In particular, at , , we have .

### Application of iterated exponential

The iterated exponential, that can be implemented with holomorphic tetration, may have various applications. In particular, The  at  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  which, at fractal values of , 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 

Withdrawal of the requirement  and  allows the solution by Kneser [12], which is entire and also could be used to build up various powers of the exponential; in particular, . 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  allows to construct solutions, similar to the growing tetration, for base . Although such solutions cannot be interpreted as generalization of exponential iterated  times, they can be useful for generalization of exponential function.

### Non-holomorphic modification of tetration

(CC) Image: Dmitrii Kouznetsov
Fig.11. Almost identical function  in the complex  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

 ,

where , and  is a 1-periodic function. The simple example of such function is



In this case, along the real axis, the function  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  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  is plotted in the complex  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 , the deviation from the identical function becomes visible, and at , the  has many points with real values, including those with various negative integer values. The tetration has cut at negative values of  and singularities there. Therefore, the cuts of the modified tetration are determined by the lines , and modified tetration unavoidable has singularities in points  such that . These singularities are determined by the function  and do not depend on the base  of tetration. In figure 11, the lines  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 -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 , 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  of equation  for integer  are singularities (branchpoints) of the modified tetration.

Let .

The following theorem is suggested: For any entire 1-periodic function  such that ,  is not a constant, , there exist . 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 .

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  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  and , some solution  of equations ,  differs from tetration tet for at least , then this solution is not holomorphic in the range .

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. D.Kouznetsov. (2009). "Solutions of  in the complex plane.". Mathematics of Computation, 78: 1647-1670. DOI:10.1090/S0025-5718-09-02188-7. Research Blogging.
2. 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
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11. D.Kouznetsov. Holomorphic ackermanns. http://mizugadro.mydns.jp/PAPERS/2014acker.pdf under consideration
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