Infinite compositions of analytic functions


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. 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 and

Contraction theorem

Many results can be considered extensions of the following result:

Infinite compositions of contractive functions

Let 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 ⊂ 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 f2n = 1/2 and f2n−1 = −1/2 for S = 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:
Example GF2:

Linear fractional transformations

Results for compositions of linear fractional transformations include the following, as examples:
In, the sequence converges everywhere in the extended plane. In, the sequence converges either everywhere, and to the same value everywhere except at one point, or it converges at only two points. Case 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 where
As a simple example, a well-known result follows from an application of Theorem :
Consider the continued fraction
with
Stipulate that |ζ| < 1 and |z| < R < 1. Then for 0 < r < 1,
Example.
]
Example. A fixed-point continued fraction form.

Direct functional expansion

Examples illustrating the conversion of a function directly into a composition follow:
Example 1. Suppose is an entire function satisfying the following conditions:
Then
Example 2.
Example 3.
Example 4.

Calculation of fixed-points

Theorem 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, first we define:
Then calculate with ζ = 1, which gives: α = 0.087118118... to ten decimal places after ten iterations.

Evolution functions

Consider a time interval, normalized to I = . 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 : For the interval divided into n equal subintervals, 1 ≤ k ≤ n set analytic or simply continuous – in a domain S, such that
and.

Principal example

implies
where the integral is well-defined if has a closed-form solution z. 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.
]
Example. Let:
Next, set and Tn = Tn,n. Let
when that limit exists. The sequence defines contours γ = γ that follow the flow of the vector field f. If there exists an attractive fixed point α, meaning |f − α| ≤ ρ|z − α| for 0 ≤ ρ < 1, then Tn → T ≡ α along γ = γ, provided . If cn ≡ c > 0, then Tn → T, a point on the contour γ = γ. 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 + gn have the property that the nth term is predicated on the sum of the first n − 1 terms. In order to employ theorem it is necessary to show boundedness in the following sense: If each fn is defined for |z| < M then |Gn| < M must follow before |fn − z| = |gn| ≤ Cβn is defined for iterative purposes. This is because occurs throughout the expansion. The restriction
serves this purpose. Then Gn → G uniformly on the restricted domain.
Example. Set
and M = ρ2. Then R = ρ2 − > 0. Then, if, z in S implies |Gn| < M and theorem applies, so that
converges absolutely, hence is convergent.
Example :

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 → G uniformly on the restricted domain.
Example. Suppose with observing after a few preliminary computations, that |z| ≤ 1/4 implies |Gn| < 0.27. Then
and
converges uniformly.
Example.

Continued fractions

Example : A self-generating continued fraction.
Example : Best described as a self-generating reverse Euler continued fraction.