page_title: Infinite compositions of analytic functions
section_title: Non-contractive complex functions
hierarchical_section_title: Infinite compositions of analytic functions / Infinite compositions of other functions / Non-contractive complex functions
caption_attribution_description: Topographical (moduli) image of an infinite composition of complex functions.
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: Results involving entire functions include the following, as examples. Set
  Then the following results hold:
  Theorem E1. If aₙ ≡ 1,
  then Fₙ → F, entire.
  Theorem E2.  Set εₙ = |aₙ−1| suppose there exists non-negative δₙ, M₁, M₂, R such that the following holds:
  Then Gₙ(z) → G(z), analytic for |z| < R. Convergence is uniform on compact subsets of {z : |z| < R}.
  Additional elementary results include:
  Theorem GF3. Suppose  where there exist  such that  implies  Furthermore, suppose  and   Then for 
  Theorem GF4. Suppose  where there exist  such that  and  implie  and   Furthermore, suppose  and  Then for 
  Theorem GF5. Let  analytic for |z| < R₀, with |gₙ(z)| ≤ Cβₙ,
  Choose 0 < r < R₀ and define
  Then Fₙ → F uniformly for |z| ≤ R.  Furthermore,
  Example GF1: 
  Example GF2:
caption: Example GF1:Reproductive universe – A topographical (moduli) image of an infinite composition.
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