Infinite compositions of analytic functions

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.
Notation.
There are several notations describing infinite compositions, including the following:
Forward compositions: 
Backward compositions: 
In each case convergence is interpreted as the existence of the following limits:
For convenience, set and .
One may also write formula_2 and
formula_3
Contraction theorem.
Many results can be considered extensions of the following result:
Infinite compositions of contractive functions.
Let {"fn"} be a sequence of functions analytic on a simply-connected domain "S". Suppose there exists a compact set Ω ⊂ "S" such that for each "n", "fn"("S") ⊂ Ω.
Additional theory resulting from investigations based on these two theorems, particularly Forward Compositions Theorem, include location analysis for the limits obtained here . For a different approach to Backward Compositions Theorem, see .
Regarding Backward Compositions Theorem, the example "f"2"n"("z") = 1/2 and "f"2"n"−1("z") = −1/2 for "S" = {"z" : |"z"| < 1} demonstrates the inadequacy of simply requiring contraction into a compact subset, like Forward Compositions Theorem.
For functions not necessarily analytic the Lipschitz condition suffices:
Infinite compositions of other functions.
Non-contractive complex functions.
Results involving entire functions include the following, as examples. Set
Then the following results hold:
Additional elementary results include:
Example GF1: formula_42
Example GF2: formula_43
Linear fractional transformations.
Results for compositions of linear fractional (Möbius) transformations include the following, as examples:
In (a), the sequence converges everywhere in the extended plane. In (b), the sequence converges either everywhere, and to the same value everywhere except at one point, or it converges at only two points. Case (c) can occur with every possible set of convergence.
Examples and applications.
Continued fractions.
The value of the infinite continued fraction
may be expressed as the limit of the sequence {"Fn"(0)} where
As a simple example, a well-known result (Worpitsky Circle*) follows from an application of Theorem (A):
Consider the continued fraction
with
Stipulate that |ζ| < 1 and |"z"| < "R" < 1. Then for 0 < "r" < 1,
Example. formula_51 formula_52
Example. A "fixed-point continued fraction form" (a single variable).
Direct functional expansion.
Examples illustrating the conversion of a function directly into a composition follow:
Example 1. Suppose formula_55 is an entire function satisfying the following conditions:
Then 
Example 2. 
Example 3. 
Example 4.
Calculation of fixed-points.
Theorem (B) can be applied to determine the fixed-points of functions defined by infinite expansions or certain integrals. The following examples illustrate the process:
Example FP1. For |ζ| ≤ 1 let
To find α = "G"(α), first we define:
Then calculate formula_63 with ζ = 1, which gives: α = 0.087118118... to ten decimal places after ten iterations.
Evolution functions.
Consider a time interval, normalized to "I" = [0, 1]. ICAFs can be constructed to describe continuous motion of a point, "z", over the interval, but in such a way that at each "instant" the motion is virtually zero (see Zeno's Arrow): For the interval divided into n equal subintervals, 1 ≤ "k" ≤ "n" set formula_67 analytic or simply continuous – in a domain "S", such that
and formula_69.
Principal example.
implies
where the integral is well-defined if formula_72 has a closed-form solution "z"("t"). Then
Otherwise, the integrand is poorly defined although the value of the integral is easily computed. In this case one might call the integral a "virtual" integral.
Example. formula_74
Example. Let:
Next, set formula_76 and "Tn"("z") = "Tn,n"("z"). Let
when that limit exists. The sequence {"Tn"("z")} defines contours γ = γ("cn", "z") that follow the flow of the vector field "f"("z"). If there exists an attractive fixed point α, meaning |"f"("z") − α| ≤ ρ|"z" − α| for 0 ≤ ρ < 1, then "Tn"("z") → "T"("z") ≡ α along γ = γ("cn", "z"), provided (for example) formula_78. If "cn" ≡ "c" > 0, then "Tn"("z") → "T"("z"), a point on the contour γ = γ("c", "z"). It is easily seen that
and
when these limits exist.
These concepts are marginally related to "active contour theory" in image processing, and are simple generalizations of the Euler method
Self-replicating expansions.
Series.
The series defined recursively by "fn"("z") = "z" + "gn"("z") have the property that the nth term is predicated on the sum of the first "n" − 1 terms. In order to employ theorem (GF3) it is necessary to show boundedness in the following sense: If each "fn" is defined for |"z"| < "M" then |"Gn"("z")| < "M" must follow before |"fn"("z") − "z"| = |"gn"("z")| ≤ "Cβn" is defined for iterative purposes. This is because formula_81 occurs throughout the expansion. The restriction
serves this purpose. Then "Gn"("z") → "G"("z") uniformly on the restricted domain.
Example (S1). Set 
and "M" = ρ2. Then "R" = ρ2 − (π/6) > 0. Then, if formula_84, "z" in "S" implies |"Gn"("z")| < "M" and theorem (GF3) applies, so that
converges absolutely, hence is convergent.
Example (S2): formula_86
Products.
The product defined recursively by
has the appearance
In order to apply Theorem GF3 it is required that:
Once again, a boundedness condition must support
If one knows "Cβn" in advance, the following will suffice:
Then "Gn"("z") → "G"("z") uniformly on the restricted domain.
Example (P1). Suppose formula_92 with formula_93 observing after a few preliminary computations, that |"z"| ≤ 1/4 implies |"Gn"("z")| < 0.27. Then
and
converges uniformly.
Example (P2).
Continued fractions.
Example (CF1): A self-generating continued fraction.
Example (CF2): Best described as a self-generating reverse Euler continued fraction.