page_title: Infinite compositions of analytic functions
section_title: Continued fractions
hierarchical_section_title: Infinite compositions of analytic functions / Examples and applications / Self-replicating expansions / Continued fractions
caption_attribution_description: English: Self-generating continued fraction: topographical image
context_page_description: In mathematics, infinite compositions of analytic functions 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.
  Although the title of this article specifies analytic functions, there are results for more general functions of a complex variable as well.
context_section_description: Example (CF1): A self-generating continued fraction.
  Example (CF2): Best described as a self-generating reverse Euler continued fraction.
caption: Example CF1: Diminishing returns – a topographical (moduli) image of a self-generating continued fraction.
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