Superfunction

Superfunction comes from iteration of another function. Roughly, for some function $f$ and for some constant $t$, the superfunction could be defined with expression

${{S(z)} \atop \,}{= \atop \,}{{\underbrace {f{\Big (}f{\big (}...f(t)...{\big )}{\Big )}} } \atop {z\mathrm {~evaluations~of~function~} f\!\!\!\!\!\!}}$

then $S$ can be interpreted as superfunction of function $f$. Such definition is valid only for positive integer $z$. The most research and applications around the superfunctions are related with various extensions of superfunction; and analysis of the existence, uniqueness and ways of the evaluation. For simple function $f$, such as addition of a constant or multiplication by a constant, the superfunction can be expressed in terms of elementary function. In particular, the Ackernann functions and tetration can be interpreted in terms of super-functions.

History

logos of the Phys.Dept. of the MSU and that of Math.Dep.

${\sqrt {!}}$ and ${\sqrt {\exp }}$ in the complex plane

Analysis of superfunctions came from the application to the evaluation of fractional iterations of functions. Superfunctions and their inverse functions allow evaluation of not only minus-first power of a function (inverse function), but also any real and even complex iteration of the function. Historically, the first function of such kind considered was ${\sqrt {\exp }}~$; then, function ${\sqrt {!~}}~$ was used as logo of the Physics department of the Moscow State University . (Mathematicians of the same University were not so arrogant and used the symbol of integral and the Moebius surface at their logo, see the figure at left).

That time, researchers did not have computational facilities for evaluation of such functions, but the ${\sqrt {\exp }}$ was more lucky than the $~{\sqrt {!~}}~~$; at least the existence of holomorphic function $\varphi$ such that $\varphi (\varphi (z))=\exp(z)$ has been demonstrated in 1950 by Helmuth Kneser . Actually, for his proof, Kneser had constructed the superfunction of exp and the corresponding Abel function ${\mathcal {X}}$, satisfying the Abel equation

${\mathcal {X}}(\exp(z))={\mathcal {X}}(z)+1$ .

The inverse function, id est $F={\mathcal {\chi }}^{-1}$ is an entire super-exponential, although it is not real at the real axis; it cannot be interpreted as tetrational, because the condition $F(0)=1$ cannot be realized for the entire super-exponential. The real ${\sqrt {\exp }}$ can be constructed with the tetrational (which is also a superexponential), and the real ${\sqrt {\rm {Factorial}}}$ can be constructed with the superfactorial. The plots of ${\sqrt {\rm {Factorial}}}$ and ${\sqrt {\exp }}$ in the complex plane are shown in the right hand side figure.

Extensions

The recurrent formula of the preamble can be written as equations

$S(z\!+\!1)=f(S(z))~\forall z\in \mathbb {N} :z>0$

$S(1)=f(t)$.

Instead of the last equation, one could write

$S(0)=t$

and extend the range of definition of superfunction $S$ to the non-negative integers. Then, one may postulate

$S(-1)=f^{-1}(t)$

and extend the range of validity to the integer values larger than $-2$. The following extension, for example,

$S(-2)=f^{-2}(t)$

is not trivial, because the inverse function may happen to be not defined for some values of $t$. In particular, tetration can be interpreted as super-function of exponential for some real base $b$; in this case,

$f=\exp _{b}$

then, at $t=1$,

$S(-1)=\log _{b}(1)=0$.

but

$S(-2)=\log _{b}(0)~\mathrm {is~not~defined}$.

For extension to non-integer values of the argument, superfunction should be defined in different way.

Definition

For complex numbers $~a~$ and $~b~$, such that $~a~$ belongs to some connected domain $D\subseteq \mathbb {C}$,
superfunction (from $a$ to $b$) of holomorphic function $~f~$ on domain $D$ is function $S$, holomorphic on domain $D$, such that

$S(z\!+\!1)=f(S(z))~\forall z\in D:z\!+\!1\in D$

$S(a)=b$.

Uniqueness

In general, the super-function is not unique. For a given base function $H$, from given $(a\mapsto d)$ superfunciton $F$, another $(a\mapsto d)$ super-function $G$ could be constructed as

$G(z)=F(z+\mu (z))$

where $\mu$ is any 1-periodic function, holomorphic at least in some vicinity of the real axis, such that $\mu (a)=0$.

The modified super-function may have narrowed range of holomorphism. The variety of possible super-functions is especially large in the limiting case, when the width of the range of holomorphizm becomes zero; in this case, one deals with the real-analytic superfunctions .

If the range of holomorphism required is large enough, then, the super-function is expected to be unique, at least in some specific base functions $H$. In particular, the $(C,0\mapsto 1)$ super-function of $\exp _{b}$, for $b>1$, is called tetration and is believed to be unique at least for $C=\{z\in \mathbb {C} ~:~\Re (z)>-2\}$; for the case $b>\exp(1/\mathrm {e} )$, see ; but up to year 2009, the uniqueness is rather conjecture than a theorem with the formal mathematical proof.

Examples

The short table of superfunctions is suggested in . A little bit more extended table is available at TORI . Some superfunctions can be expressed with elementary functions, they are used without to mention that they are superfunctions. For example, for the transfer function "++", which means unity increment, the superfunction is just addition of a constant.

Addition

Chose a complex number $c$ and define function $\mathrm {add} _{c}$ with relation $\mathrm {add} _{c}(z)=c\!+\!z~\forall z\in \mathbb {C}$ . Define function $\mathrm {mul_{c}}$ with relation $\mathrm {mul_{c}} (z)=c\!\cdot \!z~\forall z\in \mathbb {C}$.

Then, function $~\mathrm {mul_{c}} ~$ is superfunction ($~0$ to $~c~$) of function $~\mathrm {add_{c}} ~$ on $~\mathbb {C} ~$.

Multiplication

Exponentiation $\exp _{c}$ is superfunction (from 1 to $c$) of function $\mathrm {mul} _{c}$.

Quadratic polynomials

Let $H(z)=2z^{2}-1$. Then, $f(z)=\cos(\pi \cdot 2^{z})$ is a $(\mathbb {C} ,~0\!\rightarrow \!1)$ superfunction of $H$.

Indeed,

$f(z+1)=\cos(2\pi \cdot 2^{z})=2\cos(\pi \cdot 2^{z})^{2}-1=H(f(z))$

and

$f(0)=\cos(2\pi )=1$

In this case, the superfunction $f$ is periodic; its period

$T={\frac {2\pi }{\ln(2)}}\mathrm {i} \approx 9.0647202836543876194\!~i$

and the superfunction approaches unity also in the negative direction of the real axis,

$\lim _{x\rightarrow -\infty }f(x)=1$

The example above and the two examples below are suggested at .

More general case refers to the logistic transfer function $H(z)=u~z~(1-z)$ where $u>1$ is parameter; the corresponding superfunction $F$ is called logistic sequence ; $F(z\!+\!1)=H(F(z))$. Within wide range of values of parameter $u$, the logistic sequence $H(z)$ is holomorphic function of $z$; at $u\!=\!4$, it can be expressed in a way, equivalent to the representation above .

Rational function

In general, the transfer function $H$ has no need to be entire function. Here is the example with meromorphic function $H$. Let

$H(z)={\frac {2z}{1-z^{2}}}~\forall z\in D~$; $~D=\mathbb {C} \backslash \{-1,1\}$

Then, function

$F(z)=\tan(\pi 2^{z})$

is $(C,0\!\mapsto \!0)$ superfunction of function $H$, where $C$ is the set of complex numbers except singularities of function $F$. For the proof, the trigonometric formula

$\tan(2\alpha )={\frac {2\tan(\alpha )}{1-\tan(\alpha )^{2}}}~~\forall \alpha \in \mathbb {C} \backslash \{\alpha \in \mathbb {C} :\cos(\alpha )=0||\sin(\alpha )=\pm \cos(\alpha )\}$

can be used at $\alpha =\pi 2^{z}$, that gives

$H(F(z))={\frac {2\tan(\pi 2^{z})}{1-\tan(\pi 2^{z})}}=\tan(2\pi 2^{z})=F(z+1)$

Algebraic function

In the similar way one can consider the transfer function

$H(z)=2z{\sqrt {1-z^{2}}}$

and

$F(z)=\sin(\pi 2^{z})$

which is $(C,~0\!\rightarrow \!0)$ superfunction of $H$ for $C=\{z\in \mathbb {C} :\Re (\cos(\pi 2^{z}))>0\}$.

Exponentiation

Let $b>1$, $H(z)=\exp _{b}(z)$, $C=\{z\in \mathbb {C} :\Re (z)>-2\}$. Then, tetration $\mathrm {tet} _{b}$ is a $(C,~0\!\rightarrow \!1)$ superfunction of $\exp _{b}$.

Abel function

Inverse of superfunction can be interpreted as the Abel function.

For some domain $E\subseteq \mathbb {C}$ and some $u\in E$,$v\in \mathbb {C}$,
Abel function (from $u$ to $v$ ) of function $F$ with respect to superfunction $S$ on domain $E\in \mathbb {C}$ is holomorphic function $A$ from $E$ to $D$ such that

$S(A(z))=z~\forall z\in E$

$A(u)=c$

The definitionm above does not reuqire that $A(S(z))=z~\forall z\in D$; although, from properties of holomorphic functions, there should exist some subset ${\mathcal {D}}\subseteq D$ such that $A(S(z))=z~\forall z\in {\mathcal {D}}$. In this subset, the Abel function satisfies the Abel equation.

Abel equation

The Abel equation is some equivalent of the recurrent equation

$F(S(z))=S(z\!+\!1)$

in the definition of the superfunction. However, it may hold for $z$ from the reduced domain ${\mathcal {D}}$.

Applications of superfunctions and the Abel functions

Superfunctions, usially the superexponentials, are proposed as a fast-growing function for an upgrade of the floating point representation of numbers in computers. Such an upgrade would greatly extend the range of huge numbers which are still distinguishable from infinity.

Other applications refer to the calculation of fractional iterates (or fractional power) of a function. Any holomorphic function can be declared as a "transfer function", then its superfunctions and corresponding Abel functions can be considered.

Transition from a function to its inverse function

$\exp ^{c}(x)$ versus $x$ for various $c$

$\exp ^{c}$ in the complex plane for various $c$

A superfunction $S$ allows to calculate the fractional iteration $H^{c}$ of some transfer function $H$. Once the superfunction $S$ and the Abel function $A=S^{-1}$ are established, the fractional iteration can be defined as $H^{c}(z)=S(c+A(z))$. Then, as $c$ changes from 1 to $-1$, the holomorphic transition from function $H$ to $H^{-1}$ is relalised. The figure at left shows an example of transition from $\exp ^{1}\!=\!\exp$ to $\exp ^{\!-1}\!=\!\ln$. Function $\exp ^{c}$ versus real argument is plotted for $c=2,1,0.9,0.5,0.1,-0.1,-0.5,-0.9,-1,-2$. The tetrational and ArcTetrational were used as superfunction $S$ and Abel function $A$ of the exponential. The figure at right shows these functions in the complex plane. At non-negative integer number of iteration, the iterated exponential is entire function; at non-integer values, it has two branch points, thich correspond to the fixed points $L$ and $L^{*}$ of natural logarithm. At $c\!\geq \!0$, function $\exp ^{c}(z)$ remains holomorphic at least in the strip $|\Im (z)|<\Im (L)\approx 1.3$ along the real axis.

Nonlinear Optics

In the investigation of the nonlinear response of optical materials, the sample is supposed to be optically thin, in such a way, that the intensity of the light does not change much as it goes through. Then one can consider, for example, the absorption as function of the intensity. However, at small variation of the intensity in the sample, the precision of measurement of the absorption as function of intensity is not good. The reconstruction of the superfunction from the Transfer Function allows to work with relatively thick samples, improving the precision of measurements. In particular, the Transfer Function of the similar sample, which is half thiner, could be interpreted as the square root (id est, half-iteration) of the Transfer Function of the initial sample.

Similar example is suggested for a nonlinear optical fiber .

Nonlinear Acoustics

It may have sense to characterize the nonlinearities in the attenuation of shock waves in a homogeneous tube. This could find an application in some advanced muffler, using nonlinear acoustic effects to withdraw the energy of the sound waves without to disturb the flux of the gas. Again, the analysis of the nonlinear response, id est, the Transfer Function, may be boosted with the superfunction.

Vaporization and condensation

In analysis of condensation, the growth (or vaporization) of a small drop of liquid can be considered, as it diffuses down through a tube with some uniform concentration of vapor. In the first approximation, at fixed concentration of the vapor, the mass of the drop at the output end can be interpreted as the Transfer Function of the input mass. The square root of this Transfer Function will characterize the tube of half length.

Snow avalanche

The mass of a snowball, that rolls down from the hill, can be considered as a function of the path it already have passed. At fixed length of this path (that can be determined by the altitude of the hill) this mass can be considered also as a Transfer Function of the input mass. The mass of the snowball could be measured at the top of the hill and at thе bottom, giving the Transfer Function; then, the mass of the snowball as a function of the length it passed is superfunction.

Operational element

If one needs to build-up an operational element with some given transfer function $H$, and wants to realize it as a sequential connection of a couple of identical operational elements, then, each of these two elements should have transfer function $h={\sqrt {H}}$. Such a function can be evaluated through the superfunction and the Abel function of the transfer function $H$.

The operational element may have any origin: it can be realized as an electronic microchip, or a mechanical couple of curvilinear grains), or some asymmetric U-tube filled with different liquids, and so on.