Infinite compositions of analytic functions explained

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} \circ f _{k} \circ ... \circ f _{n} \circ f _{n} (z).

Backward compositions:

G _{k,n} (z) = f _{n} \circ f _{n} \circ ... \circ f _{k} \circ f _{k} (z).

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

\lim _{n\to} F _{1,n} (z), \lim _{n\toinfty} G _{1,n} (z).

For convenience, set and .

One may also write

F _{n(z)=\underset{k=1}{\overset{n}{R}}f} _{k(z)=f} _{1} \circ f _{2\circ} … \circ f _{n(z)} andG _{n(z)=\underset{k=1}{\overset{n}{L}}g} _{k(z)=g} _{n} \circ g _{n-1} \circ … \circ g _{1(z)}

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.^[1] For a different approach to Backward Compositions Theorem, see the following reference.^[2]

Regarding Backward Compositions Theorem, the example f_2n(z) = 1/2 and f_2n−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{align} f _{n(z)&=a} _{n} z + c _{n,2} z ^{2+ c} _{n, 3} z ^{3+ …} \\ \rho _{n} & = \sup _{r} \left\{ \left| c _{n,r} \right| ^{(1)/(r -1)} \right\} \end{align}

Then the following results hold:

Additional elementary results include:

Example GF1:

F _{40} (x + iy) = \underset{k =1 }{\overset{ 40 }{ R}} \left( (x + iy)/(1+ \tfrac{ 1) { 4 ^{k}} (x\cos(y) + iy\sin(x))} \right), [ -20 , 20 ] ^[3]

Example GF2:

G _{40} (x + iy) = \underset{k =1 }{\overset{ 40 }{L}}\left( (x + iy)/(1+ \tfrac{ 1) { 2 ^{k}} (x\cos(y) + iy\sin(x))} \right), [ -20 , 20 ]

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{a _{1}{b} _{1+\cfrac{a} _{2}{b} _{2+ … }}}

may be expressed as the limit of the sequence where

f _{n(z) = (a _{n})/(b _{n+z})} .

As a simple example, a well-known result (Worpitsky's circle theorem^[4]) follows from an application of Theorem (A):

Consider the continued fraction

\cfrac{a _{1\zeta} }{ 1+ \cfrac{a _{2\zeta} }{ 1+ … }}

with

f _{n(z) = (a _{n} \zeta)/(1+ z)} .

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

|a _{n|<rR(1-R) ⇒} \left|f _{n(z)} \right|<rR<R ⇒ (a _{1\zeta})/(1+ (a _{2\zeta})/(1+ …)) = F(\zeta) , analytic for |z| < 1. Set R = 1/2.

Example.

F(z) = ((i -1 )z)/(1+ i + z +) (( 2- i)z)/(1+2 i + z +) (( 3- i)z)/(1+3 i + z +) … , [ -15 , 15 ]

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

f _{k,n} (z) = (\alpha _{k,n} \beta _{k,n})/(\alpha _{k,n} + \beta _{k,n} - z) , \alpha _{k,n} = \alpha _{k,n} (z), \beta _{k,n} = \beta _{k,n} (z), F _{n(z)=} \left (f _{1,n} \circ … \circ f _{n,n} \right ) (z) \alpha _{k,n} = x \cos(ty) + iy \sin(tx), \beta _{k,n} = \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.^[5] Suppose

\phi is an entire function satisfying the following conditions:\begin{cases} \phi (tz) = t\left( \phi (z) + \phi (z) ^{2} \right) & |t|> 1 \ \phi( 0 ) = 0 \ \phi'( 0 ) =1 \end{cases} Then f _{n(z) = z + (z ^{2})/(t ^{n})} \Longrightarrow F _{n(z)\to} \phi (z) .

Example 2.

f _{n(z) = z + (z ^{2})/(2 ^{n})} \Longrightarrow F _{n(z)\to} (1)/(2) \left( e ^{2z} -1 \right)

Example 3.

f _{n(z)=} (z)/(1- \tfrac{z ^{2}) { 4 ^{n}}\Longrightarrow} F _{n(z)\to} \tan (z)

Example 4.

g _{n(z) = (2 ⋅ 4 ^{n})/(z)} \left ( \sqrt{ 1+ (z ^{2})/(4 ^{n}) }-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 ) = (\tfrac{e ^{\zeta})/(4) }

To find α = G(α), first we define:

\begin{align} t ^{\zeta}{ 4 n}}{ 3+ \zeta} _{n(z)& = \cfrac{\tfrac{e} + z} \\ f _{n(\zeta} )& = t _{1\circ} t _{2\circ} … \circ t _{n(0) \end{align}}

Then calculate

G _{n(\zeta} ) = f _{n\circ} … \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 = [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

g _{k,n} (z) = z + \varphi _{k,n} (z) analytic or simply continuous – in a domain S, such that\lim _{n\to} \varphi _{k,n} (z) =0 for all k and all z in S, and g _{k,n} (z) \in S .

Principal example

Source:

\begin{align} g _{k,n} (z) & = z + (1)/(n) \phi \left (z,\tfrac{k}{n} \right ) \\ G _{k,n} (z) & = \left (g _{k,n} \circ g _{k-1,n} \circ … \circ g _{1,n} \right ) (z) \ G _{n(z)} & = G _{n,n} (z) \end{align}

implies

λ _{• n(z) eq} G _{n(z) - z = (1)/(n)} \sum ^{n} _{k =1} \phi \left( G _{k-1,n} (z)\tfrac k n \right) • eq (1)/(n) \sum ^{n} _{k =1} \psi \left (z,\tfrac{k}{n} \right) \sim \int ^{1} _{0} \psi (z,t)dt,

where the integral is well-defined if

\tfrac{dz}{dt} = \phi (z,t) has a closed-form solution z(t). Thenλ _{n(z} _{0) ≈} \int ^{1} _{0} \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) = (2 t - \cos y)/(+ i 1- \sin x\cos y) (1-2 t\sin x)/(1- \sin x\cos y) , \int ^{1} _{0} \psi (z,t) dt

Example. Let:

g _{n(z) = z + (c _{n})/(n)} \phi (z), with f(z) = z + \phi(z).

Next, set

T _{1,n} (z) = g _{n(z),} T _{k,n} (z) = g _{n(T} _{k-1,n} (z)), and T_n(z) = T_n,n(z). LetT(z) = \lim _{n\to} 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{n} . 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 _{n\to} (c)/(n) \sum ^{n} _{k =1} \phi ^{2} \left (T _{k-1,n} (z) \right )

and

L(\gamma (z)) = \lim _{n\to} (c)/(n) \sum ^{n} _{k =1} \left| \phi \left (T _{k-1,n} (z) \right ) \right|,

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

g _{n(G} _{n-1} (z)) occurs throughout the expansion. The restriction|z|<R = M - C\sum ^{inf ty} _{k =1} \beta _{k} > 0

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

Example (S1). Set

f _{n(z) = z + (1)/(\rho n ^{2})} \sqrt{z}, \rho >\sqrt{ (\pi)/(6) } and M = ρ². Then R = ρ² − (π/6) > 0. Then, if S = \left\{ z: |z|<R,\operatorname{Re}(z)> 0 \right\} , z in S implies |G_n(z)| < M and theorem (GF3) applies, so that\begin{align} G _{n(z)} & = z + g _{1(z)+g} _{2(G} _{1(z))+g} _{3(G} _{2(z))+ …} + g _{n(G} _{n-1} (z)) \ & = z + (1)/(\sqrt{z} + \rho ⋅ 1 ^{2}) (1)/(\rho ⋅ 2 ^{2}) \sqrt{G _{1 (z)} + (1)/(\rho ⋅ 3 ^{2})} \sqrt{G _{2(z)}+ …} + (1)/(\rho ⋅ n ^{2}) \sqrt{G _{n-1} (z)} \end{align}

converges absolutely, hence is convergent.

Example (S2):

f _{n(z) = z + (1)/(n ^{2})} ⋅ \varphi (z), \varphi (z) =2 \cos(x/y) + i 2 \sin (x/y), >G _{n(z)=f} _{n} \circ f _{n-1} \circ … \circ f _{1(z),} [ -10 , 10 ], n =50

Products

The product defined recursively by

f _{n(z)=z(} 1+ g _{n(z)),} |z| \leqslant M,

has the appearance

G _{n(z)} = z \prod ^{n} ^{k =1} \left( 1+ g _{k} \left( G _{k-1} (z) \right) \right).

In order to apply Theorem GF3 it is required that:

\left| zg _{n(z)} \right|\le C\beta _{n,} \sum ^{inf ty} _{k =1} \beta _{k<infty.}

Once again, a boundedness condition must support

\left|G _{n-1} (z) g _{n(G} _{n-1} (z))\right|\le C \beta _{n.}

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

|z| \leqslant R = (M)/(P) where P = \prod ^{inf ty} _{n =1} \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 ^{2 }{n} _{n(z) = \tfrac{z} ^{3},} observing after a few preliminary computations, that |z| ≤ 1/4 implies |G_n(z)| < 0.27. Then\left|G _{n(z)} (G ^{2} _{n(z)})/(\right|<( 0 . 02 ) n ^{3}) (1)/(n ^{3}) = C\beta _{n}

and

G _{n(z)=z} \prod ^{n -1} _{k =1} \left( 1+ (G ^{2} _{k(z)})/(n ^{3}) \right)

converges uniformly.

Example (P2).

g _{k,n} (z) = z\left( 1+ (1)/(n) \varphi \left (z,\tfrac k n \right ) \right), G _{n,n} (z) = \left( g _{n,n} \circ g _{n-1,n} \circ … \circ g _{1,n} \right ) (z) = z\prod ^{n} _{k =1} ( 1+ P _{k,n} (z)), P _{k,n} (z) = (1)/(n) \varphi \left (G _{k-1,n} (z),\tfrac{k}{n} \right ), \prod ^{n -1} _{k =1} \left( 1+ P _{k,n} (z) \right) = 1+ P _{1,n} (z) + P _{2,n} (z) + … + P _{k-1,n} (z) + R _{n(z)} \sim \int ^{1} _{0} \pi (z,t) dt + 1+ R _{n(z),} \varphi (z) = x\cos(y) + iy\sin(x), \int ^{1} _{0} (z\pi (z,t) -1 ) dt, [ -15 , 15 ]:

Continued fractions

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

\begin{align} F _{n(z)} & = (\rho (z))/(\delta _{1+}) (\rho (F _{1(z))})/(\delta _{2} +) (\rho (F _{2(z))})/(\delta _{3+}) … (\rho (F _{n-1} (z)))/(\delta _{n}) , \\ \rho (z) & = (\cos(y))/(+ i \cos (y) + \sin (x)) (\sin(x))/(\cos (y) + \sin (x)) , [ 0 <x< 20 ],[ 0 <y< 20 ], \delta _{k\equiv} 1 \end{align}

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

G _{n(z) = (\rho (G _{n-1} (z)))/(1+ \rho (G _{n-1} (z)) -)} (\rho (G _{n-2} (z)))/(1+ \rho (G _{n-2} (z)) -) … (\rho (G _{1(z))})/(1+ \rho (G _{1(z))-}) (\rho (z))/(1+ \rho (z) - z) , \rho (z) = \rho (x + iy) = x\cos(y) + iy\sin(x), [ -15 , 15 ], n =30

See also

• Generalized continued fraction

Notes and References

1. Keen. Linda. Nikola. Lakic. Accumulation constants of iterated function systems with Bloch target domains.. Annales Academiae Scientiarum Fennicae Mathematica. 32. 1. Finnish Academy of Science and Letters. Helsinki. 2007.
2. Book: Keen. Linda. Nikola. Lakic. Forward iterated function systems. Complex dynamics and related topics: lectures from the Morningside Center of Mathematics . 292–299 . Jiang . Yunping . 2003 . International Press . Yuefei. Wang . 1-57146-121-3 . Sommerville . 699694753.
3. https://www.researchgate.net/publication/351764310_A_Short_Note_On_the_Dynamical_System_of_the_Reproductive_Universe
4. Beardon . A. F. . 10.1016/S0377-0427(00)00318-6 . 1-2 . Journal of Computational and Applied Mathematics . 1835708 . 143–148 . Worpitzky's theorem on continued fractions . 131 . 2001.
5. Book: Steinmetz, N. . [{{GBurl|qZGWgVuGHiYC|pg=PR7}} Rational Iteration ]. de Gruyter . 1993 . 978-3-11-088931-4 . 2011.

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