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.

In addition, it is possible to use ICAF to evaluate solutions of fixed point equations involving infinite expansions.

In each case convergence is interpreted as the existence of the following limits: For convenience, set Fn(z) = F1,n(z) and Gn(z) = G1,n(z).

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:

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.

Forward (inner or right) Compositions Theorem — {Fn} converges uniformly on compact subsets of S to a constant function F(z) = λ.

[2] Backward (outer or left) Compositions Theorem — {Gn} converges uniformly on compact subsets of S to γ ∈ Ω if and only if the sequence of fixed points {γn} of the {fn} 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 in the following reference.

[4] For a different approach to Backward Compositions Theorem, see the following reference.

[5] Regarding Backward Compositions Theorem, the example f2n(z) = 1/2 and f2n−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[6] — Suppose

Results involving entire functions include the following, as examples.

Theorem E2[8] — Set εn = |an−1| suppose there exists non-negative δn, M1, M2, R such that the following holds:

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

Additional elementary results include: Theorem GF3[6] — Suppose

Results[8] for compositions of linear fractional (Möbius) transformations include the following, as examples: Theorem LFT1 — On the set of convergence of a sequence {Fn} of non-singular LFTs, the limit function is either: In (a), the sequence converges everywhere in the extended plane.

Theorem LFT4[12] — If fn → f where f is parabolic with fixed point γ.

then Fn(z) → λ, 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 {Fn(0)} where As a simple example, a well-known result (Worpitsky's circle theorem[13]) follows from an application of Theorem (A): Consider the continued fraction with Stipulate that |ζ| < 1 and |z| < R < 1.

[8] A fixed-point continued fraction form (a single variable).

is an entire function satisfying the following conditions: Then Example 2.

[6] Theorem (B) can be applied to determine the fixed-points of functions defined by infinite expansions or certain integrals.

Theorem FP2[8] — Let φ(ζ, t) be analytic in S = {z : |z| < R} for all t in [0, 1] and continuous in t. Set

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

Then Otherwise, the integrand is poorly defined although the value of the integral is easily computed.

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)

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 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.

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.

[8] Example (CF2): Best described as a self-generating reverse Euler continued fraction.

Example: Continued fraction1 – Topographical (moduli) image of a continued fraction (one for each point) in the complex plane. [−15,15]
Example: Infinite Brooch - Topographical (moduli) image of a continued fraction form in the complex plane. (6<x<9.6),(4.8<y<8)
Example 1: Virtual tunnels – Topographical (moduli) image of virtual integrals (one for each point) in the complex plane. [−10,10]
Two contours flowing towards an attractive fixed point (red on the left). The white contour ( c = 2) terminates before reaching the fixed point. The second contour ( c ( n ) = square root of n ) terminates at the fixed point. For both contours, n = 10,000
Example (S2)- A topographical (moduli) image of a self generating series.
Example (P2): Picasso's Universe – a derived virtual integral from a self-generating infinite product. Click on image for higher resolution.
Example CF1: Diminishing returns – a topographical (moduli) image of a self-generating continued fraction.
Example CF2: Dream of Gold – a topographical (moduli) image of a self-generating reverse Euler continued fraction.