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 function in any real or even complex power. Historically, 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 ${\sqrt {\exp }}$ 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 is the entire analogy of the super-exponential (although it is not real at the real axis).

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 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

Holomorphism declared in the definition is essential for the uniqueness. If no additional requirements on the continuity of the function in the complex plane, the strip $\{z\in \mathbb {C} :-1<\Re (z)\leq 0\}$ can be filled with any function (for example, the Dirichlet function), and extended with the recurrent equation. A little bit more regular approach is fitting of the superfunction at the part of the real axis with some simple function (for example, the linear function), and following extension of this step-vice function to the whole complex plane.

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 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} is superfunction (from 1 to 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} ) of 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 \mathrm{mul}_c } .

Quadratic polynomials

Let 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 H(z)=2 z^2-1} . Then, 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)=\cos( \pi \cdot 2^z) } is a 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 (\mathbb{C},~ 0\! \rightarrow\! 1)} superfunction 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 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 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} is periodic; its 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)} \mathrm{i}\approx 9.0647202836543876194 \!~i }

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

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 \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 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 H} has no need to be entire function. Here is the example with meromorghic 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 H} . Let

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

Then, 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(z)=\tan(\pi 2^z)}

is 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, 0\! \mapsto\! 0)} superfunction of 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 H} , where 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} is the set of complex numbers except singularities of 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} . For the proof, the trigonometric formula

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 \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 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 \alpha=\pi 2^z } , that gives

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

Exponentiation

Let 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} , 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 H(z)= \exp_b(z)} , 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= \{ z \in \mathbb{C} : \Re(z)>-2 \}} . Then, 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 \mathrm{tet}_b } is a 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,~ 0\! \rightarrow\! 1)} superfunction 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 \exp_b} .

Abel function

Inverse of superfunction can be interpreted as the Abel function.

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

The definitionm above does not reuqire 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 A(S(z))=z ~\forall z \in D } ; although, from properties of holomorphic functions, there should exist some subset 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 \mathcal{D}\subseteq D} such 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 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

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(S(z))=S(z\!+\!1)}

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