Tetration

Fig.1. Tetration $\mathrm {tet} _{b}(x)$ for $b=\mathrm {e}$, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=2} , $b=\exp(1/\mathrm {e} )$, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=\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.

Definiton

For real Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b>1} , Tetration Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F=\mathrm{tet}_b} on the base $b$ is function of complex variable, which is holomorphic at least 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) > -2 \}} , bounded in the range $\{z\in \mathbb {C} :~|\Re (z)|\leq 1\}$, and satisfies conditions

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

$F(0)=1$

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F\!\big(z^*\big) = F\big(z\big)^*}

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

Etymology

Creation of word tetration is attributed to Englidh mathematician Reuben Louis Goodstein . 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=\mathrm{e}} , $b=2$, $b=\exp(1/\mathrm {e} )$, and for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=\sqrt{2}} . It has logarithmic singularity at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2} , and it is monotonously 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

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 $x=\log _{b}(x)$ 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 $\beta \geq 1/{\rm {e}}$ 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 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) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T=\frac{2\pi}{Q}} are shown in Figure 3 with fotted lines. At Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b>\exp(1/ \mathrm{e} )} , only "negative" period forresponds to tetration. At Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b>\exp(1/ \mathrm{ e} )} , the periodicity can be achieved only asymptotically; and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} is quasi-period. The real part of quasi-period is markes with black dotted line; one of two options tor the imaginary part is marked with pink dotted line.

Generally, at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1<b<\exp(1/\mathrm{e})} , tetration is periodic; the period is pure imaginary.

At Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=\exp(1/\mathrm{e})} , tetration is not periodic, and no exponential asymptotic exist.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b>\exp(1/\mathrm{e})} , tetration is quasi–periodic, the quasi-period in the upper complex half-plane is conjugate to that in the lower complex half-plane. The larger is base Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} , the shorter is quasi-period. As the quasi-periods are complex conjugated, the quasi-periodicity takes place away from the real axis.

Evaluation of tetration

Fig.4. Tetration Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f=F(z)} at 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. .

As the asymptoric of tetration is crutually depend of base Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} in ficinity of value Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=\exp(1/\mathrm{e}) } , the evaluation procidure is different foe the cases Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1<b<\exp(1/\mathrm{e}) } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=\exp(1/\mathrm{e}) } , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b>\exp(1/\mathrm{e}) } , and should be considered intependently.

Case Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1<b<\exp(1/\mathrm{e}) }

At Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1<b<\exp(1/\mathrm{e}) } , the period Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} is imaginary. Period with smallest modulus corresponds to the solution that is unity at the origen of coordinates. For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=\sqrt{2}} , function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f=\mathrm{tet}_b(z)} is shown in figure 4 with levels of constant real part and levels of constant imaginary part. Levels Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Re(f)=2,3,4} and levels Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Im(f)=0,\pm 1,\pm2\pm 3, \pm 4} aew shown with thick lines. Intermediate levels are shown with thin lines. There are branch points at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Re(z)=2~,~\Im(z)=2\pi |T| m~ \forall m \in \mathbb{N}} ; the cut lines are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Re(z)<2~,~\Im(z)=2\pi |T| m~ \forall m \in \mathbb{N}} . For this value of base, the period

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T=\frac{2\pi}{\ln^2(2)}\approx −17.1431481793548471041794 ~\mathrm{i}} .

The solution follows asymptotic at large values of real part of the argument, exponentially approaching the limiting value. In particular, for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=\sqrt{2}} , this 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=\exp(1/\mathrm{e}) }

(CC) Image: Dmitrii Kouznetsov
Fig.5. Tetration at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=\exp(1/\mathrm{e})} .

At Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=\exp(1/\mathrm{e}) } , asymptotically

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(z)=\mathrm{e}- 2 \mathrm{e}/z + \mathcal{O}(1/z^2) }

The function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f=F(x+\mathrm{i}y)} is shown in figure 5.

Levels Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Re(f)=-2,-1,0,1,2,3,4} are shown with thick black lines.

Levels Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Im(f)=-1,-2,-3,-4} are shown with thick red lines.

Levels Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Im(f)=1,2,3,4} are shown with thick blue lines.

Intermediate levels are shown with thin lines.

There is cut at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x<-2} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=0} .

Behavior of this function at real values of argument is shown in Figure 1 with thin solid line.

Existence and uniqueness of tetration

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

Iterated exponential and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{\exp}}

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

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