Tetration

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Fig.1. Tetration $\mathrm{tet}_b(x)$ for $b=\mathrm{e}$ , $b=2$ , $b=\exp(1/\mathrm{e})$ , and $b=\sqrt{2}$ versus $x$ .

This article is currently under construction. While, use article from wikipedia http://en.wikipedia.org/wiki/Tetration

Tetration is fastly growing mathematical function, which was introduced in XX century and suggested for representation of huge numbers in mathematics of computation. However, up to year 2008, this function is not considered as elementary function, it is not implemented in programming languages and it is not used for the internal representation of data, at least in the commercial software.

1 Definiton
2 Etymology
3 Real values of the arguments
4 Asymptotic behavior of tetration
- 4.1 Fixed points of logarithm
5 Evaluation of tetration
6 Existence and uniqueness of tetration
7 Inverse of tetration
8 Iterated exponential and
9 See also
10 References
11 Free online sources

Definiton

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

Etymology

Creation of word tetration is attributed to Englidh mathematician Reuben Louis Goodstein ^[1] ^[2]. This name 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", "sexation", "septation" in the simlar manner, although tetration, perhaps, already has growth fast enough for the requests of XXI century.

Real values of the arguments

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 logarithmic singularity at $-2$ , and it is monotonously increasing function.

At $b\le \exp(1/\mathrm{e})$ 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

At $b > \exp(1/\mathrm{e})$ tetration $\mathrm{tet}_b(x)$ grows faster than any exponential function. For this reason the tetration is suggested for the representation of huge numbers in 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 ${\rm tet}_b$ 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 Ackermann function $A$ is defined for the non-negative integer values of its arguments with equations

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

Asymptotic behavior of tetration

The analytic extension of tetration $~F(z)$ grows fast along the real axis of the complex $z$ -plane, at least for some values of base $b$ . However, it cannot grow infinitely in the direction of imaginary axis. The exponential convergence of discrete interation of logarithm corresponds to the exponential asymptotic behavior

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

where

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

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

(14) $x=\log_{b}(x)$ .

FIg.2. Graphic solution of equation for (two real solutions, and ), (one real solution ) (no real solutions).

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 $x=\rm e$ ) $b=2$ (no real solutions).

Solutions of equation (14) are called fixed points of logarithm.

Fixed points of logarithm

Three examples of graphical solution of equation (14) 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 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 esist two solutions, but they are complex.

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

At $b=2$ , the solutions are
$x=L_2 \approx 0.824678546142074222314065+1.56743212384964786105857 \!~\rm i$ and
$x=L_2^*\approx 0.824678546142074222314065-1.56743212384964786105857 \!~\rm i$ .

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

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

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

The solutions $x=L_1$ and $x=L_2$ of equation (14) are plotted in figure 3 versus $\beta=\ln(b)$ with thin black lines. Let $L_1<L_2$ , and only at $\ln(b)=1/e$ , the equality $L_1=L_2$ takes place.

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 <\math>Q</math>are shown in figure 3 with green lines.

At $b<\exp(1/ \mathrm{e} )$ both solutons for <\math>Q</math> are real. The negative $Q$ corresponds to tetration, decaying to the asymptotic value $L$ in the direction of real axis; positive </math>Q</math> corresponds to the solution growing along the real axis. At the real axix, 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 $b>\exp(1/ \mathrm{e} )$ , only "negative" period forresponds 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 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.

Evaluation of tetration

Fig.4. Tetration at the complex -plane. .

Fig.4. Tetration $f=F(z)$ at the complex $z$ -plane. .

As the asymptoric of tetration is crutually depend of base $b$ in ficinity of value $b=\exp(1/\mathrm{e})$ , the evaluation procidure is different foe the cases $1<b<\exp(1/\mathrm{e})$ , $b=\exp(1/\mathrm{e})$ , and $b>\exp(1/\mathrm{e})$ , and should be considered intependently.

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

At $1<b<\exp(1/\mathrm{e})$ , the period $T$ is imaginary. Period with smallest modulus corresponds to the solution that is unity at the origen of coordinates. For $b=\sqrt{2}$ , 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 $\Im(f)=0,\pm 1,\pm2\pm 3, \pm 4$ 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 base, the period

$T=\frac{2\pi}{\ln^2(2)}\approx −17.1431481793548471041794 ~\mathrm{i}$ .

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 limiting value is 2.

The trace of the solution along the real axis corresponds to the red dotted curve in Figure 1. Other solution of the recursive equaiton () do not satisfy criteria formulated in the definition of tetration.

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

(CC) Image: Dmitrii Kouznetsov
Fig.5. Tetration at $b=\exp(1/\mathrm{e})$ .

At $b=\exp(1/\mathrm{e})$ , asymptotically

Failed to parse (unknown function\mathrme): F(z)=\mathrm e + 2 \mathrme/z + \mathcal{O}(1/z^2)

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 cut at $x<-2$ , $y=0$ .

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

Existence and uniqueness of tetration

If you plan to contribute here, look at draft at User:Dmitrii Kouznetsov/Analytic Tetration.

Inverse of tetration

Iterated exponential and $\sqrt{\exp}$

Especially interesting, and in particular, for the qualitative breakthrough, 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. ^[3]. However, that time, there was no computer facility for the evlauation 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 $z$ plane for various values of $c$ .

References

↑ "TETRATION, a term for repeated exponentiation, was introduced by Reuben Louis Goodstein". Earliest Known Uses of Some of the Words of Mathematics, http://members.aol.com/jeff570/t.html
↑ R.L.Goodstein (1947). "Transfinite ordinals in recursive number theory". Journal of Symbolic Logic 12.
↑ Kneser

Free online sources

http://reglos.de/lars/ffx.html , Lars Kindermann. References about Iterative Roots and Fractional Iteration.
http://math.eretrandre.org/tetrationforum/index.php , Discussion about tetration
http://tetration.itgo.com/ Tetration site by Andrew Robbins
http://www.tetration.org/ Tetration site by Daniel Geisler

Retrieved from "http://en.citizendium.org/wiki/Tetration"

@@ Line 198: / Line 198: @@
 ===Case <math> b=\exp(1/\mathrm{e}) </math>===
+[[Image:TetrationExp1e.jpg|right|thumb|400px|{{#ifexist:Template:TetrationExp1e.jpg/credit|{{TetrationExp1e.jpg/credit}}<br/>|}}Fig.5. Tetration at <math>b=\exp(1/\mathrm{e})</math>.]]
+At <math> b=\exp(1/\mathrm{e}) </math>, asymptotically
+: <math> F(z)=\mathrm e + 2 \mathrme/z + \mathcal{O}(1/z^2) </math>
+The function <math>f=F(x+\mathrm{i}y)</math> is shown in figure 5.
+Levels <math>\Re(f)=-2,-1,0,1,2,3,4</math> are shown with thick black lines.
+Levels <math>\Im(f)=-1,-2,-3,-4</math>      are shown with thick red lines.
+Levels <math>\Im(f)=1,2,3,4</math>             are shown with thick blue lines.
+Intermediate levels are shown with thin lines.
+There is cut at  <math>x<-2</math>, <math>y=0</math>.
 ===Case <math> b>\exp(1/\mathrm{e}) </math>===

Tetration

From Citizendium, the Citizens' Compendium

Revision as of 04:41, 30 October 2008

Contents

Definiton

Etymology

Real values of the arguments

Fast growth

Integer values of the argument

Relation with the Ackermann function

Asymptotic behavior of tetration

Fixed points of logarithm

Evaluation of tetration

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

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

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

Existence and uniqueness of tetration

Inverse of tetration

Iterated exponential and $\sqrt{\exp}$

See also

References

Free online sources

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