elementary superfunctions

[bo198214]

What you're speaking about is simply a fractional iteration.

What you mean is probably regular iteration.
Everything is a fractional iteration that satisfies:
\( f^{[1]} = f \) and \( f^{[s+t]}= f^{[s]}\circ f^{[t]} \)

While regular iteration at a fixed point p is a certain fractional iteration that satisfies that \( f^{[t]} \) is differentiable at the fixed point \( p \). Regular iteration is unique by this demand.

You know that fractional iteration can be expressed in terms of Newton's and Lagrange's series. There are also methods to solve iterational equations that can be extended to fractional iterations.

Ansus, again, the title of this thread is *elementary* *superfunctions*. That implies
1. I am seeking *elementary* functions, while all methods of regular iteration dont return elementary functions per se.
2. I am seeking *superfunctions*, they dont have necessarily to be regular (=the result of regular iteration). So if you have an example of an elementary superfunction that is not by regular iteration at some fixed point, I would highly appreciate it!