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:


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