Domitori/Ksexp

Free texts and images.

Fig.1. Tetration

tet b (x)

for

b = e

,

b = 2

,

b = exp(1 / e)

, and $b=\sqrt{2}$ versus

x

.

Solutions of equation $F(z\!+\!1)=\exp\!\Big(F(z)\big)$ in the complex $z$ plane.
by Dmitrii Kouznetsov.

Sources and preface

Henryc Trappmann says that I should not call my function "tetration", and suggests the name ksexp instead. however, I expect that finction ksexp defined below is the same as tetration tet shown in figure, 1 borrowed from Sitizendium.

The most of content of this article is borrowed from the paper "in press" in Mathematics of Computation, http://www.ams.org/cgi-bin/mstrack/accepted_papers/mcom ; The preprint is available at http://www.ils.uec.ac.jp/~dima/PAPERS/2008analuxp99.pdf

Also, some pics from preprint from http://www.ils.uec.ac.jp/~dima/PAPERS/2008ackermann.pdf
and some from http://en.citizendium.org/wiki/Tetration
and from http://math.eretrandre.org/tetrationforum/index.php are expected to be used.

All intents to correct the possible errors are welcommed at any stage, even during the construction. Henryk Trappmann is especially invited to copypast his deduction here and to add his name as coauthor to the header. y. 2008, Dmitrii Kouznetsov.

Abstract

Holomorphic solution of the equation for the iterational exponential are considered.
The existence and the uniqueness are deduced.
The efficient numeric algorithm for the evaluation is suggested.
The application for generalization of exponential is demonstrated.

Introduction

Tetration is fastly growing mathematical function, which was introduced in XX century and suggested for representation of huge numbers in mathematics of computation. For positive integer values of its argument $x$ , tetration $tet b (x)$ on base $b$ can be defined with:

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

For real values of the argument $x$ and various values of base $b$ , this $tet b (x)$ is plotted in FIg.1.

In this article, the generalizaiton of tetration for complex (and, in particular, real) values of its argument is described. Tetration is assumed to be 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 iteratied exponential $exp c (z)$ for the case of non-integer values of the number $c$ of iterations. This part is copied from the http://en.citizendium.org/wiki/Tetration ; however, here the only case with base $b \ge \exp\!\left(\frac{1}{\mathrm{e}}\right)$ will be considered; the most of examples will refer to the case $b = e$ . For this case, ksexp will be defined in a way different from that at sisizendium. The definition below is less general than definition of tetration in sitizendium, but, for now, it allows more rigorous consideration.

Definiton

For real $b\ge \exp(1/\mathrm{e})$ , function $F = ksexp b$ on the base $b$ is function of complex variable, which is holomorphic at least in the set $C= \mathbb{C} \backslash \{x \in \mathbb{R} : x \le -2 \}$ , and satisfies conditions

$F(z\!+\!1) = \exp_b\! \big( F(z) \big) ~ \forall z \in C$

F (0) = 1

$F\!\big(z^*\big) = F\big(z\big)^* ~ \forall z \in C$

$F^{\prime}(x)>0 ~ \forall x>-2$ .

The definition above is not valid for the case $b < exp(1 / e)$ , because the function happens to be periodic, and the perood $T$ is pure imaginary. Therefore, all fhe singularities that are allowed at the negative part of the real axis, happen to be projected from real values $x$ to the complex values $x + m T$ for all integer $m\ne 0$ .

Place of ksexp and tetration in the big picture of math

Sitizendium suggests the following description of tetration, which is generalization of ksexp:

Creation of word tetration is attributed to Englidh mathematician Reuben Louis Goodstein ^[1] ^[2].

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

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

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

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

; $\exp_b(z\!+\!1)=b*(\exp_b(z))$

tet b (1) = b

; $\mathrm{tet}_b(z\!+\!1)=\exp_b\big(\mathrm{tet}_b(z)\big)$

pen b (1) = b

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

Except the zeroth raw, each operation in sequance above is just recurrence of operations from the previous row. Operation ++ could be called zeration (although in the prigramming languages it is called increment), addition (or summation) could be called unation, multiplication (or product) could be called duation , exponentiation coluld be called trination. The following operations ( tetration, pentation) are not used so often, at least up to year 2008. Although tetration could get many other names: superexponentiation ^[3], ultraexponent ^[4], generalized exponent ^[5], 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. It is up-to-last raw.

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 the simlar manner, although tetration, perhaps, already has growth fast enough for the requests of XXI century.

Uniqueness

For the proof of the uniqueness of function ksexp we need the theorem below

Small theorem about almost identical function

Let
$M=\{ n\in \mathbb{N} ~:~ n<-1 \}$
$C=\mathbb{C} \backslash \{ x \in \mathbb{R} ~:~ x \le -2\}$
$h$ is entire 1-periodic function
$\exists x \in \mathbb{R} ~:~ h(x)\ne 0$
$h (0) = 0$
$h\big(z^*\big)=h(z)^* ~\forall z\in \mathbb{C}$
$J(z) = z+h(z) ~ \forall z\in \mathbb{C}$
$J^{\prime}(x)>0~\forall x\in \mathbb{R}$
Then
$\exists z \in C ~:~ J(z) \in M$

Vulgarization: if function $h$ is somehow "small", then, function $J$ looks as "almost identical". The almost identical function $J$ must have negative integer values in some points outside the real axis.

Proof.

Let $E=\{z \in \mathbb{C} : J(z)\in M \}$

Hypothesis 0: For each integer $m < 1$ , there exists only one $z\in \mathbb{C} : J(z)=m$ .

Below, I make some deduction based on Hypothesis 0 and show that this hypothesis is not consistent with Theorem 0.

From Hypothesis (0) it follows that
$v_m(z)=\frac{z-m}{J(z)-m}$ is entire function of $z$ .
$v_m(z)=\frac{z-m}{z-h(z)-m}= \frac{z-m}{z-m-h(z-m)}$
Define $v(z)=\frac{z}{z+h(z)}$
Then $V m (z) = v (z - m)$
Then $v (x) = V m (z + m)$
Theredore $v$ is also entire function.

Function $v$ grows up to infinity along some contours $ζ(t)$ at $t\rightarrow \infty$ . Let contour $ζ$ provides the fastest growth, id est, realizes the maximal $\frac {\mathrm{d}\left| v\big(\zeta(t)\big)\right|} {\left|\mathrm{d}\zeta(t)\right|}= \frac{ \frac{\mathrm{d}} {\mathrm{d} t} \left| v \big(\zeta(t) \big) \right| } { \left|\zeta^{\prime}(t) \right| }$

While $v(z)=\frac{z}{z+h(z)}$ , the combined function $h (ζ(t))$ should approach $ζ(t)$ : $\lim_{t\rightarrow \infty}\Big( h\big(\zeta(t)\big)-\zeta(t)\Big)=0$ The limit $\lim_{t \rightarrow \infty} \zeta(t)$ is not allowed to exist because $\zeta(\infty)$ would be singularity of function $v$ . Hence, $ζ(t)$ grows to infinity. Hence, $h (z)$ should have asymptotically linear growth at infinity.

Among entire functions, the only polynomials are allowed to have polynomial growth at infinity; all other entire funcitons grow faster. Hence, $v$ is linear function.

Then, $h(z)=\frac{z}{v(z)-z}$ is rational function. This property contradicts the conditions of the Theorem, because $h$ is non-trivial periodic function.

In such a qay, Hypothesis 0 contradicts conditions of the Theorem. Hence, there exist more than one point $z$ such that $J(z)\in M$ . Due to the monotonous growth of function $J$ along the real axis, such a point is outside of the real axis.

(end of proof).

Big theorem about almost identical function

Let
$M=\{ n\in \mathbb{N} ~:~ n<-1 \}$
$C=\mathbb{C} \backslash \{ x \in \mathbb{R} ~:~ x \le -2\}$
$h$ is entire 1-periodic function
$\exists x \in \mathbb{R} ~:~ h(x)\ne 0$
$h (0) = 0$
$h\big(z^*\big)=h(z)^* ~\forall z\in \mathbb{C}$
$J(z) = z+h(z) ~ \forall z\in \mathbb{C}$
$J^{\prime}(x)>0~\forall x\in \mathbb{R}$
Then
$\exists z \in C ~:~\Re(z)>0, J(z) \in M$

Example of applicaiton of the theorem about almost identical function

According to the Small Theorem, function $J(z)=z+\alpha~ \sin(2\pi z )$ takes negative integer values not only at the negative part of the real axix, but also somewhere outside the real axis, even at small values of $α$ .

According to the Big theorem, function $J$ takes integer negative value $- 2$ at some values of argument with real part larger than $- 2$ .

Uniqueness of $~\mathrm{ksexp}_b$

Assume some fixed base $b\!>\!\exp(1/\mathrm{e})$ and let $F (z) = ksexp b (z)$ within the range of holomorphism of tetration, id est, in the complex plane except some set of measure zero. This set includes one line $z {\!\le\!}-2$ .

Assume that function $G$ is holomorphic outside the part of the real axis $\{ x \in \mathbb{R} : x\le -2 \}$ and $f = G$ satisfies conditions

$f(z\!+\!1)=\exp_b\big( f(z) \big)$

$f (0) = 1$

$f\big(z^{*}\big)=f(z)^{*}$

$f^{\prime}(x) > 0 \forall x> -2$

and $G$ is not ksexp (although satisfies the same equations above).

Then, there exist function $J$ such that in vicinity of the segment $[ - 1,1]$ function $G$ can be expressed as follows: $G=F\big(J(z)\big)$ and, in some vicinity of the same segment, $J(z)=F^{-1}\big( G(z) \big)$ .

From definition of ksexp, it follows, that it has singulatities along the negative part of the real axis, set of singular points is $M = \{ n \in \mathbb{N} : n\le -2 \}$ .

There exist only one holomorphic extension of a function in a domain of trivial topology. Therefore, we can extend the function $J$ outside of the range of definition, at least until the first singulatity.

Consider $~ h(z)=J(z)-z~$ function of $z$ . Assuming that $F$ and $G$ are not identical in the segment $[ - 1,1]$ , function $h$ should not be identically zero. This function also allows the holomorphic extension. Consider behavior of function $h$ in the complex plane. This extension should be periodic, satisfying conditions of theorem about the almost identical function above. Therefore, function $J$ should take values from set $M$ outside the real axis. Function $F$ has singularities at these points.

Therefore, function $G$ has singularities outside the real axis and does not satisfy the definition of ksexp. In such a way, for a given $b$ , there exist no more than one function $ksexp b$ .

Existence of $ksexp b$

Case $b = exp(1 / e)$

In some sense, the case $b = exp(1 / e)$ is simplest, because $\exp_b(\mathrm{e})=\exp\!\left(\ln\left(\frac{1}{\mathrm{e}}\right)\mathrm{e}\right)=\mathrm{e}$ . It is sufficient to consider a function that slightly deviates from e in some tange, and revonstruct behavior of the function in the whole complex plane as limiting case, taking the sequence of logarithms.

It is sufficient to build up one ksexp; according to the theorem of the uniqueness, there is no another ksexp.

Consider some function $Ψ$ such that

$\Psi (z)=\mathrm{e}- \frac{ 2 \mathrm{e}}{z} + \mathcal{O}(1/z^3)$

DIrect substitution confirms that such a function does not contradict equation $exp b (Ψ(z)) = Ψ(z + 1)$

Define function $Φ$ as limit

$\Phi(z) = \lim_{n\rightarrow \infty} \log_b^n(\Psi(z+n)) = \lim_{n\rightarrow \infty} \log_b^n\left(\mathrm{e} - \frac{ 2 ~\mathrm{e}}{z+n} \right)$

This function satisfies therecurrent equaiton

exp b (Φ(z)) = Φ(z + 1)

In order to get a function, that is unity at zero argument, define

F (z) = Φ(z + z 0)

where $z 0$ is solution of evuation

Φ(z 0) = 0

The only way the logarithmic operation can fail is the the negative value of the argument of zero. Such a value can be realized only at the real axis. Therefore, the function $F$ is ksexp.

$\mathrm{ksexp}_b(x\!+\!\mathrm{i}y)$ at $b=\exp(1\!/\!\mathrm{e})$ in the

x, y

plane.

The way of construction of such a function gives the straightforward algorithm for the evaluation. This algorithm was used to plot the thin curve in figure at the preamble. The same algorothm can be used also for complex values of the argument. The function $f=F(x\!+\!\mathrm{i}y)$ is shown in figure.

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.

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

$F(z)=\mathrm{e}- 2 \mathrm{e}/z + \mathcal{O}(1/z^2)$

Therefore, equilines look similar to circles, typical for the inverse function.

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

Behavior of this function at real values of argument is shown in Figure in the Preamble 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; namely, the condition $F(0)\!=\!1$ cannot be realized.

Exponential asumptotic

Exponential asumptotic is the same for tetration and kslog, to, this section repeats that at citizendium. The exponential convergence of discrete interation of logarithm $z : = log b (z)$ 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)

.

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 / e)

(one real solution

x = e

)

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 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 exist 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 soluition, 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
$L=L_{\rm e}^* \approx 0.318131505204764135312654-1.33723570143068940890116 \!~\rm i$ .

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

Basic properties of the exponential asymptotics

Parameters of asymptotic of tetration versus logarithm of the base

The solutions $L = L 1$ and $L = L 2$ of equation

L = log b (L)

are plotted in figure versus $β = ln(b)$ with thin black lines. 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 $\beta \ge 1/\rm e$ 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 conjugaitons of each other. Requirement of definition of tetration determine the asymptotic of the solution. Parameter $Q$ tetermines periodicity of quasi-periodicity of tetration. The two solutions for $Q$ are shown in figure 3 with green lines.

At $b < exp(1 / e)$ both solutons 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 confition $F (0) = 1$ . 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 fotted lines. At $b > exp(1 / e)$ , only "negative" period corresponds to tetration. At $b > exp(1 / e)$ , the periodicity can be achieved only asymptotically; and $T$ is quasi-period. The real part of quasi-period is markes 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.

At $b = exp(1 / e)$ , the single solution $L = e$ corresponds to tetration with non-exponential asymptotic, which still can be interpreted as ksexp according to the definition above.

At $1 < b < exp(1 / e)$ , the tetrarion is periodic; the multiple lines of singularities do not allow to interpret this tetration as ksexp.

Case $b > exp(1 / e)$ is considered below. Knowledge of the asymptotic behavior gives key to the precise evaluation of ksexp.

Next one

This work is published here under the Creative Commons

license and can be reproduced if non-commercial. cs | de | en | es | fr | ru | zh

Cite error: <ref> tags exist, but no <references/> tag was found

[1]

[2]

[3]

[4]

[5]

NOTICE

Domitori/Ksexp

Free texts and images.

Contents

Sources and preface

Abstract

Introduction

Definiton

Place of ksexp and tetration in the big picture of math

Uniqueness

Small theorem about almost identical function

Big theorem about almost identical function

Example of applicaiton of the theorem about almost identical function

Uniqueness of $~\mathrm{ksexp}_b$

Existence of $ksexp b$

Case $b = exp(1 / e)$

Exponential asumptotic

Fixed points of logarithm

Basic properties of the exponential asymptotics

Next one

Views

Personal tools

Navigation

Search

Toolbox

NOTICE

Domitori/Ksexp

Free texts and images.

Contents

Sources and preface

Abstract

Introduction

Definiton

Place of ksexp and tetration in the big picture of math

Uniqueness

Small theorem about almost identical function

Big theorem about almost identical function

Example of applicaiton of the theorem about almost identical function

Uniqueness of

Existence of ksexpb

Case b = exp(1 / e)

Exponential asumptotic

Fixed points of logarithm

Basic properties of the exponential asymptotics

Next one

Views

Personal tools

Navigation

Search

Toolbox

Uniqueness of $~\mathrm{ksexp}_b$

Existence of $ksexp b$

Case $b = exp(1 / e)$