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: F_{k,n}(z) = f_{k} ∘ f_{k+1} ∘ ... ∘ f_{n−1} ∘ f_{n}(z).
Backward compositions: G_{k,n}(z) = f_{n} ∘ f_{n−1} ∘ ... ∘ f_{k+1} ∘ f_{k}(z)
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).
One may also write and
Contraction theorem
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 α of f in S such that:
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}} converges uniformly on compact subsets of S to a constant function F(z) = λ.[2]
- Backward (outer or left) Compositions Theorem. {G_{n}} converges uniformly on compact subsets of S to γ ∈ Ω if and only if the sequence of fixed points {γ_{n}} of the {f_{n}} converges to γ.[3]
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_{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 .
Infinite compositions of other functions
Non-contractive complex functions
Results[5] involving entire functions include the following, as examples. Set
Then the following results hold:
- Theorem E1.[6] If a_{n} ≡ 1,
- then F_{n} → F, entire.
- Theorem E2.[5] Set ε_{n} = |a_{n}−1| suppose there exists non-negative δ_{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)| ≤ Cβ_{n},
- Choose 0 < r < R_{0} and define
- Then F_{n} → F uniformly for |z| ≤ R. Furthermore,
Example GF1:
Example GF2:
Linear fractional transformations
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_{n}} of non-singular LFTs, the limit function is either:
- (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_{n}} converges to an LFT, then f_{n} converge to the identity function f(z) = z.
- Theorem LFT3.[9] If f_{n} → f and all functions are hyperbolic or loxodromic Möbius transformations, then F_{n}(z) → λ, a constant, for all , where {β_{n}} are the repulsive fixed points of the {f_{n}}.
- Theorem LFT4.[10] If f_{n} → f where f is parabolic with fixed point γ. Let the fixed-points of the {f_{n}} be {γ_{n}} and {β_{n}}. If
- then F_{n}(z) → λ, a constant in the extended complex plane, for all z.
Examples and 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*[11]) 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,
- , analytic for |z| < 1. Set R = 1/2.
Example.
Example.[5] 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.[6][12] Suppose is an entire function satisfying the following conditions:
Then
- .
Example 2.[6]
Example 3.[5]
Example 4.[5]
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.[5] Let φ(ζ, t) be analytic in S = {z : |z| < R} for all t in [0, 1] and continuous in t. Set
- If |φ(ζ, t)| ≤ r < R for ζ ∈ S and t ∈ [0, 1], then
- has a unique solution, α in S, with
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
- for all k and all z in S,
and .
Principal example[5]
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}(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 T_{n}(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 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_{n} is defined for |z| < M then |G_{n}(z)| < M must follow before |f_{n}(z) − z| = |g_{n}(z)| ≤ Cβ_{n} is defined for iterative purposes. This is because occurs throughout the expansion. The restriction
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.
Example (S2):
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 Cβ_{n} in advance, the following will suffice:
Then G_{n}(z) → G(z) uniformly on the restricted domain.
Example (P1). Suppose with observing after a few preliminary computations, that |z| ≤ 1/4 implies |G_{n}(z)| < 0.27. Then
and
converges uniformly.
Example (P2).
Continued fractions
Example (CF1): A self-generating continued fraction.[5]
Example (CF2): Best described as a self-generating reverse Euler continued fraction.[5]
References
- P. Henrici, Applied and Computational Complex Analysis, Vol. 1 (Wiley, 1974)
- L. Lorentzen, Compositions of contractions, J. Comp & Appl Math. 32 (1990)
- 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)
- 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
- J. Gill, John Gill Mathematics Notes, researchgate.net
- 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)