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 appllications around the superfunctions is 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

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 . 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 corresponding Abel function ${\mathcal {X}}$, satisfying the Abel equation

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

the inverse function $F$ is the entire super-exponential, although it is not real at the real axis; it cannot be interpreted as tetration, because the condition $F(0)=1$ cannot be realized for the entire super-exponential.

Extensions

The recurrence above 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 trifial, 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

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

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 $x$ from the reduced domain ${\mathcal {D}}$.