User:MathFacts/Iterational calculus

Iterational calculus, also known as mathematical dynamics is a branch of mathematical analysis that studies self-composition of functions.

Definitions

n-th iteration of function $f(x)\,$ is defined as:

${{f^{[n]}=\ } \atop {\ }}{{\underbrace {f\circ f\circ f\circ \cdots \circ f} } \atop n{\text{ times}}}$

For consistency also usually define that $f^{[0]}(x)=x\,$ (identity function) and $f^{[-1]}(x)\,$ is inverse function of $f(x)\,$. Iterational operation can also be extended into negative number of iterations:

$f^{[-n]}(x)=(f^{[-1]}(x))^{[n]}\,$

For extention to real and complex iterations in iterational calculus usually accepted fractional iteration formula, derived using Carleman matrices:

$f^{[u]}(x)=\sum _{n=0}^{\infty }{\binom {u}{n}}\sum _{m=0}^{n}{\binom {n}{m}}(-1)^{n-m}f^{[n]}(1)$