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: $F_(z) = f_k \circ f_ \circ \dots \circ f_ \circ f_n (z).$

Backward compositions: $G_(z) = f_n \circ f_ \circ \dots \circ f_ \circ f_k (z).$

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

:$\lim_ F_(z), \qquad \lim_ G_(z).$

For convenience, set and .

One may also write $F_n(z)=\underset\,f_k(z)=f_1 \circ f_2\circ \cdots \circ f_n(z)$ and
$G_n(z)=\underset\,g_k(z)=g_n \circ g_\circ \cdots \circ g_1(z)$

Comment: It is not clear when the first explorations of infinite compositions of analytic functions not restricted to sequences of functions of a specific kind occurred. Possibly in the 1980s.

Contraction theorem

Many results can be considered extensions of the following result:

Infinite compositions of contractive functions

Let 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.

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

Regarding Backward Compositions Theorem, the example ''f''_2''n''(''z'') = 1/2 and ''f''_2''n''−1(''z'') = −1/2 for ''S'' = demonstrates the inadequacy of simply requiring contraction into a compact subset, like Forward Compositions Theorem.

For functions not necessarily analytic the Lipschitz condition suffices:

Infinite compositions of other functions

Non-contractive complex functions

Results involving entire functions include the following, as examples. Set

:$\begin f_n(z)&=a_n z + c_z^2+c_ z^3+\cdots \\ \rho_n &= \sup_r \left\ \end$

Then the following results hold:

Additional elementary results include:

Linear fractional transformations

Results for compositions of linear fractional (Möbius) transformations include the following, as examples:

Examples and applications

Continued fractions

The value of the infinite continued fraction

:$\cfrac$

may be expressed as the limit of the sequence where

:$f_n(z)=\frac.$

As a simple example, a well-known result (Worpitsky's circle theorem) follows from an application of Theorem (A):

Consider the continued fraction

:$\cfrac$

with

:$f_n(z)=\frac.$

Stipulate that , ζ, < 1 and , ''z'', < ''R'' < 1. Then for 0 < ''r'' < 1,

: , a_n, , analytic for , ''z'', < 1. Set ''R'' = 1/2.

Example. $F(z)=\frac\text\frac\text\frac \cdots,$ 15,15/math>
]

Example. A ''fixed-point continued fraction form'' (a single variable).
:$f_(z)=\frac, \alpha_=\alpha_(z), \beta_=\beta_(z), F_n(z)= \left (f_ \circ\cdots \circ f_ \right ) (z)$
:$\alpha_=x \cos(ty)+iy \sin(tx), \beta_= \cos(ty)+i \sin(tx), t=k/n$

Direct functional expansion

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

Example 1. Suppose $\phi$ is an entire function satisfying the following conditions:
:$\begin \phi (tz)=t\left( \phi (z)+\phi (z)^2 \right) & , t, > 1 \\ \phi(0) = 0 \\ \phi'(0) =1 \end$
Then
:$f_n(z)=z+\frac\Longrightarrow F_n(z)\to \phi (z)$.

Example 2.
:$f_n(z)=z+\frac\Longrightarrow F_n(z)\to \frac\left( e^-1 \right)$

Example 3.
:$f_n(z)= \frac\Longrightarrow F_n(z)\to \tan (z)$

Example 4.
:$g_n(z)=\frac \left ( \sqrt-1 \right )\Longrightarrow G_n(z) \to \arctan (z)$

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. For , ''ζ'', ≤ 1 let

:$G(\zeta )=\frac$

To find α = ''G''(α), first we define:

:$\begin t_n(z)&=\cfrac \\ f_n(\zeta )&= t_1\circ t_2\circ \cdots \circ t_n(0) \end$

Then calculate $G_n(\zeta )=f_n\circ \cdots \circ f_1(\zeta )$ with ζ = 1, which gives: α = 0.087118118... to ten decimal places after ten iterations.

Evolution functions

Consider a time interval, normalized to ''I'' = , 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 $g_(z)=z+\varphi_(z)$ analytic or simply continuous – in a domain ''S'', such that

:$\lim_\varphi_(z)=0$ for all ''k'' and all ''z'' in ''S'',
and $g_(z) \in S$.

Principal example

Source:

:$\begin g_(z) &=z+\frac\phi \left (z,\tfrac \right ) \\ G_(z) &= \left (g_\circ g_ \circ \cdots \circ g_ \right ) (z) \\ G_n(z) &=G_(z) \end$

implies

:$\lambda_n(z)\doteq G_n(z)-z=\frac\sum_^n \phi \left( G_(z)\tfrac k n \right)\doteq \frac 1 n \sum_^n \psi \left (z,\tfrac \right) \sim \int_0^1 \psi (z,t)\,dt,$

where the integral is well-defined if $\tfrac=\phi (z,t)$ has a closed-form solution ''z''(''t''). Then

:$\lambda_n(z_0)\approx \int_0^1 \phi ( z(t),t)\,dt.$

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. $\phi (z,t)=\frac+i\frac, \int_0^1 \psi (z,t) \, dt$
]

Example. Let:

:$g_n(z)=z+\frac\phi (z), \quad \text \quad f(z) = z + \phi(z).$

Next, set $T_(z)=g_n(z), T_(z)= g_n(T_(z)),$ and ''T_n''(''z'') = ''T_n,n''(''z''). Let

:$T(z)=\lim_ T_n(z)$

when that limit exists. The sequence 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) $c_n = \sqrt$. 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

:$\oint_\gamma \phi (\zeta ) \, d\zeta =\lim_\frac c n \sum_^n \phi ^2 \left (T_(z) \right )$

and

:$L(\gamma (z))=\lim_ \frac\sum_^n \left, \phi \left (T_(z) \right ) \,$

when these limits exist.

These concepts are marginally related to '' Active contour model, 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 $g_n(G_(z))$ occurs throughout the expansion. The restriction

:, z, 0

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

Example (S1). Set
:$f_n(z)=z+\frac\sqrt, \qquad \rho >\sqrt$
and ''M'' = ρ². Then ''R'' = ρ² − (π/6) > 0. Then, if $S=\left\$, ''z'' in ''S'' implies , ''G_n''(''z''), < ''M'' and theorem (GF3) applies, so that

:$\begin G_n(z) &=z+g_1(z)+g_2(G_1(z))+g_3(G_2(z))+\cdots + g_n(G_(z)) \\ &= z+\frac\sqrt+\frac\sqrt+\frac\sqrt+\cdots +\frac \sqrt \end$

converges absolutely, hence is convergent.

Example (S2): f_n(z)=z+\frac 1 \cdot \varphi (z), \varphi (z)=2\cos(x/y)+i2\sin (x/y), >G_n(z)=f_n \circ f_\circ \cdots \circ f_1(z), \qquad 10,10 n=50

Products

The product defined recursively by

:$f_n(z)=z( 1+g_n(z)), \qquad , z, \leqslant M,$

has the appearance

:$G_n(z) = z \prod _^n \left( 1+g_k \left( G_(z) \right) \right).$

In order to apply Theorem GF3 it is required that:

:$\left, zg_n(z) \\le C\beta_n, \qquad \sum_^\infty \beta_k<\infty.$

Once again, a boundedness condition must support

:$\left, G_(z) g_n(G_(z))\\le C \beta_n.$

If one knows ''Cβ_n'' in advance, the following will suffice:

:$, z, \leqslant R = \frac \qquad \text \quad P = \prod_^\infty \left( 1+C\beta_n\right).$

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

Example (P1). Suppose $f_n(z)=z(1+g_n(z))$ with $g_n(z)=\tfrac,$ observing after a few preliminary computations, that , ''z'', ≤ 1/4 implies , ''G_n''(''z''), < 0.27. Then

:$\left, G_n(z) \frac \<(0.02)\frac=C\beta_n$

and

:$G_n(z)=z \prod_^\left( 1+\frac\right)$

converges uniformly.

Example (P2).

:$g_(z)=z\left( 1+\frac 1 n \varphi \left (z,\tfrac k n \right ) \right),$
:$G_(z)= \left( g_\circ g_\circ \cdots \circ g_ \right ) (z) = z\prod_^n ( 1+P_(z)),$
:$P_(z)=\frac 1 n \varphi \left (G_(z),\tfrac \right ),$
:$\prod_^ \left( 1+P_(z) \right) = 1+P_(z)+P_(z)+\cdots + P_(z) + R_n(z) \sim \int_0^1 \pi (z,t) \, dt + 1+R_n(z),$
:\varphi (z)=x\cos(y)+iy\sin(x), \int_0^1 (z\pi (z,t)-1) \,dt, \qquad 15,15

Continued fractions

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

: \begin
F_n(z) &= \frac \frac \frac \cdots \frac, \\
\rho (z) &= \frac+i\frac, \qquad \qquad\delta_k\equiv 1
\end

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

: $G_n(z)=\frac\ \frac\cdots \frac\ \frac,$
:\rho (z)=\rho (x+iy)=x\cos(y)+iy\sin(x), \qquad 15,15 n=30

See also

*Generalized continued fraction

References

{{reflist

Complex analysis
Analytic functions
Fixed-point theorems
Algorithmic art
Emergence