# Infinite compositions of analytic functions

In mathematics, **infinite compositions of analytic functions (ICAF)** offer alternative formulations of 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.

## Notation

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

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

*z*) =

*f*∘

_{k}*f*

_{k+1}∘ ... ∘

*f*

_{n−1}∘

*f*.

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

*z*) =

*f*∘

_{n}*f*

_{n−1}∘ ... ∘

*f*

_{k+1}∘

*f*

_{k}In each case convergence is interpreted as the existence of the following limits:

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

*z*) =

*F*

_{1,n}(

*z*) and

*G*(

_{n}*z*) =

*G*

_{1,n}(

*z*).

## Contraction theorem

Many results can be considered extensions of the following result:

Contraction Theorem for Analytic Functions.^{[1]}Letfbe analytic in a simply-connected regionSand continuous on the closure Template:Overline ofS. Supposef(Template:Overline) is a bounded set contained inS. Then for allzin Template:Overlinewhere α is the attractive fixed point of

finS.

## Infinite compositions of contractive functions

Let {*f _{n}*} 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*,

*f*(

_{n}*S*) ⊂ Ω.

Forward (inner or right) Compositions Theorem.{F(_{n}z)} converges uniformly on compact subsets ofSto a constant functionF(z) = λ.^{[2]}

Backward (outer or left) Compositions Theorem.{G(_{n}z)} converges uniformly on compact subsets ofSto γ ∈ Ω if and only if the sequence of fixed points {γ_{n}} of the {f} converge to γ._{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.

## Infinite compositions of other functions

### General analytic functions

Results^{[4]} involving **entire functions** include the following, as examples. Set

Then the following results hold:

Theorem E1.^{[5]}Ifa≡ 1,_{n}then

F→_{n}F, entire.

Theorem E2.^{[4]}Set ε_{n}= |a−1| suppose there exists non-negative δ_{n}_{n},M_{1},M_{2},Rsuch that the following holds:Then

G(_{n}z) →G(z), analytic for |z| <R. Convergence is uniform on compact subsets of {z: |z| <R}.

Theorem GF3.^{[4]}Let {f} be a sequence of complex functions defined on_{n}S= {z: |z| <M}. Suppose there exists a non-negative sequence {β_{n}} such thatSet . Then

G(_{n}z) →G(z) for |z| <R, uniformly on compact subsets.

Theorem GF4.^{[4]}Letf(_{n}z) =z(1+g(_{n}z)), analytic for |z| <R_{0}, with |g(_{n}z)| ≤Cβ_{n},Choose 0 <

r<R_{0}and defineThen

F→_{n}Funiformly for |z| ≤R. Furthermore,

### Linear fractional transformations

Results^{[4]} 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.

^{[6]}

Theorem LFT2.If {F} converges to an LFT , then_{n}fconverge to the identity function_{n}f(z) =z.^{[7]}

Theorem LFT3.Iff→_{n}fand all functions arehyperbolicorloxodromicMöbius transformations, thenF(_{n}z) → λ, a constant, for all , where {β_{n}} are the repulsive fixed points of the {f}._{n}^{[8]}

Theorem LFT4.Iff→_{n}fwherefisparabolicwith fixed point γ. Let the fixed-points of the {f} be {γ_{n}_{n}} and {β_{n}}. Ifthen

F(_{n}z) → λ, a constant in the extended complex plane, for allz.^{[9]}

## Examples & applications

### Continued fractions

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*^{[10]}) 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,

Set *R* = 1/2.

### Direct functional expansion

An example illustrating the conversion of a function directly into a composition follows:

Suppose that for |*t*| > 1, , an entire function with *φ*(0) = 0, *φ*′(0) = 1. Then .^{[5]}^{[11]}

### 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)**:^{[3]} For |ζ| ≤ 1 let

To find α = *G*(α), first we define:

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

Theorem (FP2).^{[4]}Let φ(ζ,t) be analytic inS= {z: |z| <R} for alltin [0, 1] and continuous int. SetIf |φ(ζ,

t)| ≤r<Rfor ζ ∈Sandt∈ [0, 1], then

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

#### Example 1

Now, set and . If exists, the initial point z has moved to a new position, *T*(*z*), in a fashion described above (for large values of *n*, ). It is not difficult to show that *f*(*z*) = α*z* + β, α ≥ 0 implies . A byproduct of this derivation is the following representation:

And of course, if *f*(*z*) ≡ *c*, then^{[12]}

#### Example 2

with *f*(*z*) := *z* + φ(*z*). Next, set , and *T _{n}*(

*z*) =

*T*(

_{n,n}*z*). Let

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

*z*)} defines contours γ = γ(

*c*,

_{n}*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

*T*(

_{n}*z*) →

*T*(

*z*) ≡ α along γ = γ(

*c*,

_{n}*z*), provided (for example) . If

*c*≡

_{n}*c*> 0, then it seems apparent - though not rigorously proven for many cases

^{[13]}- that

*T*(

_{n}*z*) →

*T*(

*z*), a point on the contour γ = γ(

*c*,

*z*). It is easily seen that

and

when these limits exist.^{[14]}

These concepts are marginally related to *active contour theory* in image processing.

### Self-replicating series & products

#### Series

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

*z*) =

*z*+

*g*(

_{n}*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

*f*is defined for |

_{n}*z*| <

*M*then |

*G*(

_{n}*z*)| <

*M*must follow before |

*f*(

_{n}*z*)−

*z*| = |

*g*(

_{n}*z*)| ≤

*Cβ*is defined for iterative purposes. This is because occurs throughout the expansion. The restriction

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

*z*) →

*G*(

*z*) uniformly on the restricted domain.

**Example (S1)**: Set

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

*z*)| <

*M*and theorem (GF3) applies, so that

converges absolutely, hence is convergent.

#### Products

The product defined recursively by , |*z*| ≤ *M*, have the appearance

In order to apply theorem (GF3) it is required that where

Once again, a boundedness condition must support

If one knows *Cβ _{n}* in advance, setting |

*z*| ≤

*R*=

*M*/

*P*where

suffices. Then *G _{n}*(

*z*) →

*G*(

*z*) uniformly on the restricted domain.

**Example (P1)**: Suppose that where , observing after a few preliminary computations, that |*z*| ≤ 1/4 implies |*G _{n}*(

*z*)| < 0.27. Then

and

converges uniformly.

## References

- ↑ P. Henrici,
*Applied and Computational Complex Analysis*, Vol. 1 (Wiley, 1974) - ↑ L. Lorentzen, Compositions of contractions, J. Comp & Appl Math. 32 (1990)
- ↑
^{3.0}^{3.1}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) - ↑
^{4.0}^{4.1}^{4.2}^{4.3}^{4.4}^{4.5}J. Gill, Convergence of infinite compositions of complex functions, Comm. Anal. Th. Cont. Frac., Vol XIX (2012) - ↑
^{5.0}^{5.1}^{5.2}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. Zeitschr., Vol. 80 (1962)
- ↑ A. Magnus & M. Mandell, On convergence of sequences of linear fractional transformations,Math. Zeitschr. 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, Zeno's arrow: A mathematical speculation , Comm. Anal. Th. Cont. Frac., Vol XIX (2012)
- ↑ J. Gill, John Gill Mathematics Collection, Scribd.com
- ↑ J. Gill, Progress Report: Zeno Contours in the Complex Plane, Comm. Anal. Th. Cont. Frac., Vol XIX (2012)