Superfunction

In mathematics, superfunction is a nonstandard name for an iterated function for complexified continuous iteration index. Roughly, for some function f and for some variable x, the superfunction could be defined by the expression

${\displaystyle S(z;x)=\underbrace {f{\Big (}f{\big (}\dots f(x)\dots {\big )}{\Big )}} _{z{\text{ evaluations of the function}}\,f}.}$

Then, S(z; x) can be interpreted as the superfunction of the function f(x). Such a definition is valid only for a positive integer index z. The variable x is often omitted. Much study and many applications of superfunctions employ various extensions of these superfunctions to complex and continuous indices; and the analysis of existence, uniqueness and their evaluation. The Ackermann functions and tetration can be interpreted in terms of superfunctions.