Superfunction

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

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

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

$$ {S(z) \atop \,} {= {{\underbrace{f\Big   (t)\Big}} \atop {z {\rm ~evaluations~ of~ function~}f } }$$

{S(z)~=~ \atop {~} {\underbrace{\exp_a\!\Big(\exp_a\!\big(...\exp_a(t) ... )\big)\Big)} \atop ^{z ~\rm exponentials}}

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

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 $$\mathcal{D}$$.