Superfunction

Superfunction is smooth exstension of iteration of other function for the case of non-integer number of iterations.

Routgly

Roughly, if $S(z)=f(f(...f(a)))$

${{a+b} \atop \,}{= \atop \,}{a\,+ \atop \,}{{\underbrace {1+1+\cdots +1} } \atop b}$

Failed to parse (syntax error): {\displaystyle {S(z) \atop \,} {= {{\underbrace{f\Big (t)\Big}} \atop {z {\rm ~evaluations~ of~ function~}f } }}

Failed to parse (syntax error): {\displaystyle {S(z)~=~ \atop {~} {\underbrace{\exp_a\!\Big(\exp_a\!\big(...\exp_a(t) ... )\big)\Big)} \atop ^{z ~\rm exponentials}} <math> ==Definition== For complex numbers <math>~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

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

$S(a)=b$.

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 $\mathrm {mul} _{c}$.

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 exost 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 ${\mathcal {D}}$.