Infinite Compositions of Analytic Functions

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## Notation

## Contraction theorem

## Infinite compositions of contractive functions

## Infinite compositions of other functions

### Non-contractive complex functions

### Linear fractional transformations

## Examples and applications

### Continued fractions

### Direct functional expansion

### Calculation of fixed-points

### Evolution functions

#### Principal example^{[5]}

### Self-replicating expansions

#### Series

#### Products

#### Continued fractions

## References

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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.

There are several notations describing infinite compositions, including the following:

**Forward compositions:** *F _{k,n}*(

**Backward compositions:** *G _{k,n}*(

In each case convergence is interpreted as the existence of the following limits:

For convenience, set *F _{n}*(

One may also write and

Many results can be considered extensions of the following result:

**Contraction Theorem for Analytic Functions.**^{[1]}Let*f*be analytic in a simply-connected region*S*and continuous on the closure*S*of*S*. Suppose*f*(*S*) is a bounded set contained in*S*. Then for all*z*in*S*there exists an attractive fixed point ? of*f*in*S*such that:

Let {*f _{n}*} be a sequence of functions analytic on a simply-connected domain

**Forward (inner or right) Compositions Theorem.**{*F*} converges uniformly on compact subsets of_{n}*S*to a constant function*F*(*z*) = ?.^{[2]}

**Backward (outer or left) Compositions Theorem.**{*G*} converges uniformly on compact subsets of_{n}*S*to ? ? ? if and only if the sequence of fixed points {*?*} of the {_{n}*f*} converges to_{n}*?*.^{[3]}

Additional theory resulting from investigations based on these two theorems, particularly Forward Compositions Theorem, include location analysis for the limits obtained here [1]. For a different approach to Backward Compositions Theorem, see [2].

Regarding Backward Compositions Theorem, the example *f*_{2n}(*z*) = 1/2 and *f*_{2n-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:

**Theorem.**^{[4]}Suppose is a simply connected compact subset of and let be a family of functions that satisfies- Define:
- Then uniformly on If is the unique fixed point of then uniformly on if and only if .

Results^{[5]} involving **entire functions** include the following, as examples. Set

Then the following results hold:

**Theorem E1.**^{[6]}If*a*? 1,_{n}- then
*F*->_{n}*F*, entire.

**Theorem E2.**^{[5]}Set ?_{n}= |*a*-1| suppose there exists non-negative ?_{n}_{n},*M*_{1},*M*_{2},*R*such that the following holds:- Then
*G*(_{n}*z*) ->*G*(*z*), analytic for |*z*| <*R*. Convergence is uniform on compact subsets of {*z*: |*z*| <*R*}.

**Additional elementary results include:**

**Theorem GF3.**^{[4]}Suppose where there exist such that implies Furthermore, suppose and Then for

**Theorem GF4.**^{[4]}Suppose where there exist such that and implie and Furthermore, suppose and Then for

**Theorem GF5.**^{[5]}Let analytic for |*z*| <*R*_{0}, with |*g*(_{n}*z*)| C?_{n},- Choose 0 <
*r*<*R*_{0}and define - Then
*F*->_{n}*F*uniformly for |*z*| R. Furthermore,

**Example GF1**:

**Example GF2**:

Results^{[5]} for compositions of **linear fractional (Möbius) transformations** include the following, as examples:

**Theorem LFT1.**On the set of convergence of a sequence {*F*} of non-singular LFTs, the limit function is either:_{n}- (a) a non-singular LFT,
- (b) a function taking on two distinct values, or
- (c) a constant.

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.^{[7]}

**Theorem LFT2.**^{[8]}If {*F*} converges to an LFT, then_{n}*f*converge to the identity function_{n}*f*(*z*) =*z*.

**Theorem LFT3.**^{[9]}If*f*->_{n}*f*and all functions are*hyperbolic*or*loxodromic*Möbius transformations, then*F*(_{n}*z*) ->*?*, a constant, for all , where {*?*} are the repulsive fixed points of the {_{n}*f*}._{n}

**Theorem LFT4.**^{[10]}If*f*->_{n}*f*where*f*is*parabolic*with fixed point ?. Let the fixed-points of the {*f*} be {?_{n}_{n}} and {*?*}. If_{n}- then
*F*(_{n}*z*) ->*?*, a constant in the extended complex plane, for all*z*.

The value of the infinite continued fraction

may be expressed as the limit of the sequence {*F _{n}*(0)} where

As a simple example, a well-known result (Worpitsky Circle*^{[11]}) follows from an application of Theorem (A):

Consider the continued fraction

with

Stipulate that |?| < 1 and |*z*| < *R* < 1. Then for 0 < *r* < 1,

- , analytic for |
*z*| < 1. Set*R*= 1/2.

**Example.**

**Example.**^{[5]} A *fixed-point continued fraction form* (a single variable).

Examples illustrating the conversion of a function directly into a composition follow:

**Example 1.**^{[6]}^{[12]} Suppose is an entire function satisfying the following conditions:

Then

- .

**Example 2.**^{[6]}

**Example 3.**^{[5]}

**Example 4.**^{[5]}

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.**^{[3]} For |?|

To find ? = *G*(?), first we define:

Then calculate with ? = 1, which gives = 0.087118118... to ten decimal places after ten iterations.

**Theorem FP2.**^{[5]}Let ?(?,*t*) be analytic in*S*= {*z*: |*z*| <*R*} for all*t*in [0, 1] and continuous in*t*. Set- If |?(?,
*t*)| r <*R*for ? ?*S*and*t*? [0, 1], then - has a unique solution, ? in
*S*, with

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 analytic or simply continuous - in a domain *S*, such that

- for all
*k*and all*z*in*S*,

and .

implies

where the integral is well-defined if 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.**

**Example.**^{[13]} Let:

Next, set and *T _{n}*(

when that limit exists. The sequence {*T _{n}*(

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

The series defined recursively by *f _{n}*(

serves this purpose. Then *G _{n}*(

**Example (S1).** Set

and *M* = ?^{2}. Then *R* = ?^{2} - (?/6) > 0. Then, if , *z* in *S* implies |*G _{n}*(

converges absolutely, hence is convergent.

**Example (S2)**:

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 *C? _{n}* in advance, the following will suffice:

Then *G _{n}*(

**Example (P1).** Suppose with observing after a few preliminary computations, that |*z*| G_{n}(*z*)| < 0.27. Then

and

converges uniformly.

**Example (P2).**

**Example (CF1)**: A self-generating continued fraction.^{[5]}[3]

**Example (CF2)**: Best described as a self-generating reverse Euler continued fraction.^{[5]}

**^**P. Henrici,*Applied and Computational Complex Analysis*, Vol. 1 (Wiley, 1974)**^**L. Lorentzen, Compositions of contractions, J. Comp & Appl Math. 32 (1990)- ^
^{a}^{b}J. Gill, The use of the sequence*F*(_{n}*z*) =*f*? ... ?_{n}*f*_{1}(*z*) in computing the fixed points of continued fractions, products, and series, Appl. Numer. Math. 8 (1991) - ^
^{a}^{b}^{c}J. Gill, A Primer on the Elementary Theory of Infinite Compositions of Complex Functions, Comm. Anal. Th. Cont. Frac., Vol XXIII (2017) and researchgate.net - ^
^{a}^{b}^{c}^{d}^{e}^{f}^{g}^{h}^{i}^{j}^{k}J. Gill, John Gill Mathematics Notes, researchgate.net - ^
^{a}^{b}^{c}S.Kojima, Convergence of infinite compositions of entire functions, arXiv:1009.2833v1 **^**G. Piranian & W. Thron,Convergence properties of sequences of Linear fractional transformations, Mich. Math. J.,Vol. 4 (1957)**^**J. DePree & W. Thron,On sequences of Mobius transformations, Math. Z., Vol. 80 (1962)**^**A. Magnus & M. Mandell, On convergence of sequences of linear fractional transformations,Math. Z. 115 (1970)**^**J. Gill, Infinite compositions of Mobius transformations, Trans. Amer. Math. Soc., Vol176 (1973)**^**L. Lorentzen, H. Waadeland,*Continued Fractions with Applications*, North Holland (1992)**^**N. Steinmetz,*Rational Iteration*, Walter de Gruyter, Berlin (1993)**^**J. Gill, Informal Notes: Zeno contours, parametric forms, & integrals, Comm. Anal. Th. Cont. Frac., Vol XX (2014)

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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