Infinite compositions of analytic functions

Last updated

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.

Contents

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 Fn(z) = F1,n(z) and Gn(z) = G1,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 {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 here . For a different approach to Backward Compositions Theorem, see .

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 [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 an ≡ 1,

then FnF is entire.

Theorem E2 [5]   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 [4]   Suppose where there exist such that implies Furthermore, suppose and Then for

Theorem GF4 [4]   Suppose where there exist such that and implies and Furthermore, suppose and Then for

Theorem GF5 [5]   Let analytic for |z| < R0, with |gn(z)|Cβn,

Choose 0 < r < R0 and define

Then FnF uniformly for

Example GF1:

Example GF1:Reproductive universe - A topographical (moduli) image of an infinite composition. Reproductive universe.jpg
Example GF1:Reproductive universe – A topographical (moduli) image of an infinite composition.

Example GF2:

Example GF2:Metropolis at 30K - A topographical (moduli) image of an infinite composition. Metropolis at 30K.jpg
Example GF2:Metropolis at 30K – A topographical (moduli) image of an infinite composition.

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 {Fn} of non-singular LFTs, the limit function is either:

  1. a non-singular LFT,
  2. a function taking on two distinct values, or
  3. 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 {Fn} converges to an LFT, then fn converge to the identity function f(z) = z.

Theorem LFT3 [9]   If fnf and all functions are hyperbolic or loxodromic Möbius transformations, then Fn(z) → λ, a constant, for all , where {βn} are the repulsive fixed points of the {fn}.

Theorem LFT4 [10]   If fnf where f is parabolic with fixed point γ. Let the fixed-points of the {fn} be {γn} and {βn}. If

then Fn(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 {Fn(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: Continued fraction1 - Topographical (moduli) image of a continued fraction (one for each point) in the complex plane. [-15,15] Continued fraction1.jpg
Example: Continued fraction1 – Topographical (moduli) image of a continued fraction (one for each point) in the complex plane. [−15,15]

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

Example: Infinite Brooch - Topographical (moduli) image of a continued fraction form in the complex plane. (6<x<9.6),(4.8<y<8) Infinite Brooch.jpg
Example: Infinite Brooch - Topographical (moduli) image of a continued fraction form in the complex plane. (6<x<9.6),(4.8<y<8)

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 ζS 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 ≤ kn 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 1: Virtual tunnels - Topographical (moduli) image of virtual integrals (one for each point) in the complex plane. [-10,10] Virtual tunnels.jpg
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 Contours in the vector field f(z) = -Cos(z).jpg
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. [13] Let:

Next, set and Tn(z) = Tn,n(z). Let

when that limit exists. The sequence {Tn(z)} defines contours γ = γ(cn, 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 Tn(z) → T(z) ≡ α along γ = γ(cn, z), provided (for example) . If cnc > 0, then Tn(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 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)| ≤ n is defined for iterative purposes. This is because occurs throughout the expansion. The restriction

serves this purpose. Then Gn(z) → G(z) uniformly on the restricted domain.

Example (S1). Set

and M = ρ2. Then R = ρ2 − (π/6) > 0. Then, if , z in S implies |Gn(z)| < M and theorem (GF3) applies, so that

converges absolutely, hence is convergent.

Example (S2):

Example (S2)- A topographical (moduli) image of a self generating series. Self-generating series3.jpg
Example (S2)- A topographical (moduli) image of a self generating series.

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 n in advance, the following will suffice:

Then Gn(z) → G(z) uniformly on the restricted domain.

Example (P1). Suppose with observing after a few preliminary computations, that |z| ≤ 1/4 implies |Gn(z)| < 0.27. Then

and

converges uniformly.

Example (P2).

Example (P2): Picasso's Universe - a derived virtual integral from a self-generating infinite product. Click on image for higher resolution. Picasso's Universe.jpg
Example (P2): Picasso's Universe – a derived virtual integral from a self-generating infinite product. Click on image for higher resolution.

Continued fractions

Example (CF1): A self-generating continued fraction. [5]

Example CF1: Diminishing returns - a topographical (moduli) image of a self-generating continued fraction. Diminishing returns.jpg
Example CF1: Diminishing returns – a topographical (moduli) image of a self-generating continued fraction.

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

Example CF2: Dream of Gold - a topographical (moduli) image of a self-generating reverse Euler continued fraction. Dream of Gold.jpg
Example CF2: Dream of Gold – a topographical (moduli) image of a self-generating reverse Euler continued fraction.

Related Research Articles

In number theory, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896 using ideas introduced by Bernhard Riemann.

Laplaces equation Second order partial differential equation

In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as

Navier–Stokes equations Equations describing the motion of viscous fluid substances

In physics, the Navier–Stokes equations are a set of partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician George Gabriel Stokes.

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols , , or . In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form. Informally, the Laplacian Δf(p) of a function f at a point p measures by how much the average value of f over small spheres or balls centered at p deviates from f(p).

The theory of functions of several complex variables is the branch of mathematics dealing with complex-valued functions. The function on the complex coordinate space of n-tuples of complex numbers.

Hurwitz zeta function

In mathematics, the Hurwitz zeta function, named after Adolf Hurwitz, is one of the many zeta functions. It is formally defined for complex arguments s with Re(s) > 1 and q with Re(q) > 0 by

Mertens function

In number theory, the Mertens function is defined for all positive integers n as

Riesz function

In mathematics, the Riesz function is an entire function defined by Marcel Riesz in connection with the Riemann hypothesis, by means of the power series

In mathematics, the jet is an operation that takes a differentiable function f and produces a polynomial, the truncated Taylor polynomial of f, at each point of its domain. Although this is the definition of a jet, the theory of jets regards these polynomials as being abstract polynomials rather than polynomial functions.

In mathematics, the von Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt. It is an example of an important arithmetic function that is neither multiplicative nor additive.

In mathematics, the explicit formulae for L-functions are relations between sums over the complex number zeroes of an L-function and sums over prime powers, introduced by Riemann (1859) for the Riemann zeta function. Such explicit formulae have been applied also to questions on bounding the discriminant of an algebraic number field, and the conductor of a number field.

Multiple integral Generalization of definite integrals to functions of multiple variables

In mathematics, a multiple integral is a definite integral of a function of several real variables, for instance, f(x, y) or f(x, y, z). Integrals of a function of two variables over a region in are called double integrals, and integrals of a function of three variables over a region in are called triple integrals. For multiple integrals of a single-variable function, see the Cauchy formula for repeated integration.

The Mason–Weaver equation describes the sedimentation and diffusion of solutes under a uniform force, usually a gravitational field. Assuming that the gravitational field is aligned in the z direction, the Mason–Weaver equation may be written

Oblate spheroidal coordinates

Oblate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the non-focal axis of the ellipse, i.e., the symmetry axis that separates the foci. Thus, the two foci are transformed into a ring of radius in the x-y plane. Oblate spheroidal coordinates can also be considered as a limiting case of ellipsoidal coordinates in which the two largest semi-axes are equal in length.

In fluid mechanics, potential vorticity (PV) is a quantity which is proportional to the dot product of vorticity and stratification. This quantity, following a parcel of air or water, can only be changed by diabatic or frictional processes. It is a useful concept for understanding the generation of vorticity in cyclogenesis, especially along the polar front, and in analyzing flow in the ocean.

In mathematics, the secondary measure associated with a measure of positive density ρ when there is one, is a measure of positive density μ, turning the secondary polynomials associated with the orthogonal polynomials for ρ into an orthogonal system.

In mathematics, vector spherical harmonics (VSH) are an extension of the scalar spherical harmonics for use with vector fields. The components of the VSH are complex-valued functions expressed in the spherical coordinate basis vectors.

In mathematics, the spectral theory of ordinary differential equations is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his dissertation Hermann Weyl generalized the classical Sturm–Liouville theory on a finite closed interval to second order differential operators with singularities at the endpoints of the interval, possibly semi-infinite or infinite. Unlike the classical case, the spectrum may no longer consist of just a countable set of eigenvalues, but may also contain a continuous part. In this case the eigenfunction expansion involves an integral over the continuous part with respect to a spectral measure, given by the Titchmarsh–Kodaira formula. The theory was put in its final simplified form for singular differential equations of even degree by Kodaira and others, using von Neumann's spectral theorem. It has had important applications in quantum mechanics, operator theory and harmonic analysis on semisimple Lie groups.

In fluid dynamics, the Oseen equations describe the flow of a viscous and incompressible fluid at small Reynolds numbers, as formulated by Carl Wilhelm Oseen in 1910. Oseen flow is an improved description of these flows, as compared to Stokes flow, with the (partial) inclusion of convective acceleration.

Mild-slope equation Combined effects of diffraction and refraction for water waves propagating over variable depth and with lateral boundaries

In fluid dynamics, the mild-slope equation describes the combined effects of diffraction and refraction for water waves propagating over bathymetry and due to lateral boundaries—like breakwaters and coastlines. It is an approximate model, deriving its name from being originally developed for wave propagation over mild slopes of the sea floor. The mild-slope equation is often used in coastal engineering to compute the wave-field changes near harbours and coasts.

References

  1. P. Henrici, Applied and Computational Complex Analysis, Vol. 1 (Wiley, 1974)
  2. L. Lorentzen, Compositions of contractions, J. Comp & Appl Math. 32 (1990)
  3. 1 2 J. Gill, The use of the sequence in computing the fixed points of continued fractions, products, and series, Appl. Numer. Math. 8 (1991)
  4. 1 2 3 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
  5. 1 2 3 4 5 6 7 8 9 10 11 J. Gill, John Gill Mathematics Notes, researchgate.net
  6. 1 2 3 S.Kojima, Convergence of infinite compositions of entire functions, arXiv:1009.2833v1
  7. G. Piranian & W. Thron,Convergence properties of sequences of Linear fractional transformations, Mich. Math. J.,Vol. 4 (1957)
  8. J. DePree & W. Thron,On sequences of Mobius transformations, Math. Z., Vol. 80 (1962)
  9. A. Magnus & M. Mandell, On convergence of sequences of linear fractional transformations,Math. Z. 115 (1970)
  10. J. Gill, Infinite compositions of Mobius transformations, Trans. Amer. Math. Soc., Vol176 (1973)
  11. L. Lorentzen, H. Waadeland, Continued Fractions with Applications, North Holland (1992)
  12. N. Steinmetz, Rational Iteration, Walter de Gruyter, Berlin (1993)
  13. J. Gill, Informal Notes: Zeno contours, parametric forms, & integrals, Comm. Anal. Th. Cont. Frac., Vol XX (2014)