In mathematics, infinite compositions of analytic functions (ICAF) offer alternative formulations of analytic continued fractions, series, products and other infinite expansions, and the theory evolving from such compositions may shed light on the convergence/divergence of these expansions. Some functions can actually be expanded directly as infinite compositions. In addition, it is possible to use ICAF to evaluate solutions of fixed point equations involving infinite expansions. Complex dynamics offers another venue for iteration of systems of functions rather than a single function. For infinite compositions of a single function see Iterated function. For compositions of a finite number of functions, useful in fractal theory, see Iterated function system.
Although the title of this article specifies analytic functions, there are results for more general functions of a complex variable as well.
There are several notations describing infinite compositions, including the following:
Forward compositions: Fk,n(z) = fk ∘ fk+1 ∘ ... ∘ fn−1 ∘ fn(z).
Backward compositions: Gk,n(z) = fn ∘ fn−1 ∘ ... ∘ fk+1 ∘ fk(z)
In each case convergence is interpreted as the existence of the following limits: