infinite compositions of analytic functions

There are several notations describing infinite compositions, including the following:

Forward compositions: $F\_\{k,n\}(z)\; =\; f\_k\; \backslash circ\; f\_\{k\; +\; 1\}\; \backslash circ\; \backslash dots\; \backslash circ\; f\_\{n\; -\; 1\}\; \backslash circ\; f\_n\; (z).$

Backward compositions: $G\_\{k,n\}(z)\; =\; f\_n\; \backslash circ\; f\_\{n\; -\; 1\}\; \backslash circ\; \backslash dots\; \backslash circ\; f\_\{k\; +\; 1\}\; \backslash circ\; f\_k\; (z).$

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

:$\backslash lim\_\{n\backslash to\; \backslash infty\}\; F\_\{1,n\}(z),\; \backslash qquad\; \backslash lim\_\{n\backslash to\backslash infty\}\; G\_\{1,n\}(z).$

For convenience, set {{math|F_n(z) {{=}} F_1,n(z)}} and {{math|G_n(z) {{=}} G_1,n(z)}}.

One may also write $F\_n(z)=\backslash underset\{k=1\}\{\backslash overset\{n\}\{\backslash mathop\; R\}\}\backslash ,f\_k(z)=f\_1\; \backslash circ\; f\_2\backslash circ\; \backslash cdots\; \backslash circ\; f\_n(z)$ and

$G\_n(z)=\backslash underset\{k=1\}\{\backslash overset\{n\}\{\backslash mathop\; L\}\}\backslash ,g\_k(z)=g\_n\; \backslash circ\; g\_\{n-1\}\backslash circ\; \backslash cdots\; \backslash circ\; g\_1(z)$

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. {{cite journal |last=Gill |first=John |title=Compositions of analytic functions of the form Fn(z)=Fn-1(fn(z)),fn->f |journal=Journal of Computational and Applied Mathematics |date=1988 |volume=23 |pages=179-184 |url=https://www.sciencedirect.com/science/article/pii/0377042788902798}}

Many results can be considered extensions of the following result:

{{math theorem|name=Contraction Theorem for Analytic Functions{{cite book |first=P. |last=Henrici |title=Applied and Computational Complex Analysis |volume=1 |publisher=Wiley |orig-date=1974 |isbn=978-0-471-60841-7 |date=1988 |url={{GBurl|ZrJ6GX5a2WMC|pg=PR13}}}}|math_statement= Let f be analytic in a simply-connected region S and continuous on the closure {{overline|S}} of S. Suppose f({{overline|S}}) is a bounded set contained in S. Then for all z in {{overline|S}} there exists an attractive fixed point α of f in S such that:

$$F\_n(z)=(f\backslash circ\; f\backslash circ\; \backslash cdots\; \backslash circ\; f)(z)\backslash to\; \backslash alpha.$$}}

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.

{{math theorem|name=Forward (inner or right) Compositions Theorem|math_statement= {F_n} converges uniformly on compact subsets of S to a constant function F(z) = λ.{{cite journal |last1=Lorentzen |first1=Lisa |title=Compositions of contractions |journal=Journal of Computational and Applied Mathematics |date=November 1990 |volume=32 |issue=1–2 |pages=169–178 |doi=10.1016/0377-0427(90)90428-3 |doi-access=free }} }}

{{math theorem|name=Backward (outer or left) Compositions Theorem|math_statement= {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.{{cite journal |first=J. |last=Gill |title=The use of the sequence F_n(z)=f_n∘⋯∘f₁(z) in computing the fixed points of continued fractions, products, and series |journal=Appl. Numer. Math. |volume=8 |issue=6 |pages=469–476 |date=1991 |doi=10.1016/0168-9274(91)90109-D |url=}}}}

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.{{cite journal|last1=Keen|first1=Linda|first2=Nikola|last2=Lakic|title=Accumulation constants of iterated function systems with Bloch target domains.|journal=Annales Academiae Scientiarum Fennicae Mathematica|volume= 32|number= 1|publisher= Finnish Academy of Science and Letters|location=Helsinki|year=2007|url=https://www.maths.tcd.ie/EMIS/journals/AASF/Vol32/KeenLakic.html}} For a different approach to Backward Compositions Theorem, see the following reference.{{cite book|last1=Keen|first1=Linda|first2=Nikola|last2=Lakic|chapter=Forward iterated function systems|url=http://comet.lehman.cuny.edu/keenl/forwarditer.pdf|pages=292–299 |editor-last=Jiang |editor-first=Yunping |title=Complex dynamics and related topics: lectures from the Morningside Center of Mathematics |date=2003 |publisher=International Press |editor-first2=Yuefei|editor-last2=Wang |isbn=1-57146-121-3 |location=Sommerville |oclc=699694753}}

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:

{{math theorem|name=Theorem{{cite journal |first=J. |last=Gill |title=A Primer on the Elementary Theory of Infinite Compositions of Complex Functions |journal=Communications in the Analytic Theory of Continued Fractions |volume=XXIII |date=2017 |url=https://www.coloradomesa.edu/math-stat/catcf/papers/primerinfcompcomplexfcns.pdf}} |math_statement=Suppose $S$ is a simply connected compact subset of $\backslash Complex$ and let $t\_n\; :\; S\; \backslash to\; S$ be a family of functions that satisfies

Define:

$$\backslash begin\{align\}$$

G_n(z) &= \left (t_n\circ t_{n-1}\circ \cdots \circ t_1 \right ) (z) \\

F_n(z) &= \left (t_1 \circ t_2\circ \cdots \circ t_n \right ) (z)

\end{align}

Then $F\_n(z)\backslash to\; \backslash beta\; \backslash in\; S$ uniformly on $S.$ If $\backslash alpha\_n$ is the unique fixed point of $t\_n$ then $G\_n(z)\backslash to\; \backslash alpha$ uniformly on $S$ if and only if $|\backslash alpha\_n\; -\backslash alpha|\; =\; \backslash varepsilon\_n\; \backslash to\; 0$.}}

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

:$\backslash begin\{align\}$

f_n(z)&=a_n z + c_{n,2}z^2+c_{n,3} z^3+\cdots \\

\rho_n &= \sup_r \left\{ \left| c_{n,r} \right|^{\frac{1}{r-1}} \right\}

\end{align}

Then the following results hold:

{{math theorem|name=Theorem E1{{cite journal |last1=Kojima |first1=Shota |title=On the convergence of infinite compositions of entire functions |journal=Archiv der Mathematik |date=May 2012 |volume=98 |issue=5 |pages=453–465 |doi=10.1007/s00013-012-0385-z |s2cid=121444171 }} |math_statement= If a_n ≡ 1,

then F_n → F is entire.}}

{{math theorem|name=Theorem E2{{cite journal |first=J. |last=Gill |title=Convergence of Infinite Compositions of Complex Functions |journal=Communications in the Analytic Theory of Continued Fractions |volume=XIX |date=2012 |url=https://www.coloradomesa.edu/math-stat/documents/JohnGillResearchnoteInfiniteCompositions2.pdf}}|math_statement= Set ε_n = {{abs|a_n−1}} suppose there exists non-negative δ_n, M₁, M₂, R such that the following holds:

$$\backslash begin\{align\}$$

\sum_{n=1}^\infty \varepsilon_n &< \infty, \\

\sum_{n=1}^\infty \delta_n &< \infty, \\

\prod_{n=1}^\infty (1+\delta_n) &< M_1, \\

\prod_{n=1}^\infty (1+\varepsilon_n) &< M_2, \\

\rho_n &< \frac{\delta_n}{R M_1 M_2}.

\end{align}

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

Additional elementary results include:

{{math theorem|name=Theorem GF3 |math_statement=Suppose $f\_k(z)=z+\backslash rho\_k\; \backslash varphi\_k\; (z)$ where there exist $R,\; M\; >\; 0$ such that implies Furthermore, suppose and $R\; >\; M\backslash sum\_\{k=1\}^\backslash infty\; \backslash rho\_k.$ Then for

{{math theorem|name=Theorem GF4 |math_statement=Suppose $f\_k(z)=z+\backslash rho\_k\; \backslash varphi\_k\; (z)$ where there exist $R,\; M\; >\; 0$ such that

}}

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

{{math theorem|name=Theorem LFT1|math_statement= On the set of convergence of a sequence {F_n} 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.{{cite journal |last1=Piranian |first1=G. |last2=Thron |first2=W. J. |title=Convergence properties of sequences of linear fractional transformations. |journal=Michigan Mathematical Journal |date=1957 |volume=4 |issue=2 |doi=10.1307/mmj/1028989001 |doi-access=free }}

}}

{{math theorem|name=Theorem LFT2{{cite journal |last1=de Pree |first1=J. D. |last2=Thron |first2=W. J. |title=On sequences of Moebius transformations |journal=Mathematische Zeitschrift |date=December 1962 |volume=80 |issue=1 |pages=184–193 |doi=10.1007/BF01162375 |s2cid=120487262 }} |math_statement= If {F_n} converges to an LFT, then f_n converge to the identity function f(z) = z.}}

{{math theorem|name=Theorem LFT3{{cite journal |last1=Mandell |first1=Michael |last2=Magnus |first2=Arne |title=On convergence of sequences of linear fractional transformations |journal=Mathematische Zeitschrift |date=1970 |volume=115 |issue=1 |pages=11–17 |doi=10.1007/BF01109744 |s2cid=119407993 }} |math_statement= If f_n → f and all functions are hyperbolic or loxodromic Möbius transformations, then F_n(z) → λ, a constant, for all $z\; \backslash ne\; \backslash beta\; =\; \backslash lim\_\{n\backslash to\; \backslash infty\}\; \backslash beta\_n$, where {β_n} are the repulsive fixed points of the {f_n}.}}

{{math theorem|name=Theorem LFT4{{cite journal |last1=Gill |first1=John |title=Infinite compositions of Möbius transformations |journal=Transactions of the American Mathematical Society |date=1973 |volume=176 |pages=479 |doi=10.1090/S0002-9947-1973-0316690-6 |doi-access=free }} |math_statement=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.}}

The value of the infinite continued fraction

:$\backslash cfrac\{a\_1\}\{b\_1+\backslash cfrac\{a\_2\}\{b\_2+\backslash cdots\}\}$

may be expressed as the limit of the sequence {F_n(0)} where

:$f\_n(z)=\backslash frac\{a\_n\}\{b\_n+z\}.$

As a simple example, a well-known result (Worpitsky's circle theorem{{cite journal

| last = Beardon | first = A. F.

| doi = 10.1016/S0377-0427(00)00318-6

| issue = 1–2

| journal = Journal of Computational and Applied Mathematics

| mr = 1835708

| pages = 143–148

| title = Worpitzky's theorem on continued fractions

| volume = 131

| year = 2001| bibcode = 2001JCoAM.131..143B

}}) follows from an application of Theorem (A):

Consider the continued fraction

:$\backslash cfrac\{a\_1\backslash zeta\; \}\{1+\backslash cfrac\{a\_2\backslash zeta\; \}\{1+\backslash cdots\}\}$

with

:$f\_n(z)=\backslash frac\{a\_n\; \backslash zeta\; \}\{1+z\}.$

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

: z| < 1. Set R = 1/2.

Example. $F(z)=\backslash frac\{(i-1)z\}\{1+i+z\backslash text\{\; \}+\}\backslash text\{\; \}\backslash frac\{(2-i)z\}\{1+2i+z\backslash text\{\; \}+\}\backslash text\{\; \}\backslash frac\{(3-i)z\}\{1+3i+z\backslash text\{\; \}+\}\; \backslash cdots,$ $[-15,15]$

File:Continued fraction1.jpg]

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

:$f\_\{k,n\}(z)=\backslash frac\{\backslash alpha\_\{k,n\}\; \backslash beta\_\{k,n\}\}\{\backslash alpha\_\{k,n\}+\backslash beta\_\{k,n\}-z\},\; \backslash alpha\_\{k,n\}=\backslash alpha\_\{k,n\}(z),\; \backslash beta\_\{k,n\}=\backslash beta\_\{k,n\}(z),\; F\_n(z)=\; \backslash left\; (f\_\{1,n\}\; \backslash circ\backslash cdots\; \backslash circ\; f\_\{n,n\}\; \backslash right\; )\; (z)$

:$\backslash alpha\_\{k,n\}=x\; \backslash cos(ty)+iy\; \backslash sin(tx),\; \backslash beta\_\{k,n\}=\; \backslash cos(ty)+i\; \backslash sin(tx),\; t=k/n$

File:Infinite Brooch.jpg

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

Example 1.{{cite book |first=N. |last=Steinmetz |title=Rational Iteration |publisher=de Gruyter |orig-date=1993 |isbn=978-3-11-088931-4 |url={{GBurl|qZGWgVuGHiYC|pg=PR7}} |date=2011}} Suppose $\backslash phi$ is an entire function satisfying the following conditions:

:

Then

:$f\_n(z)=z+\backslash frac\{z^2\}\{t^n\}\backslash Longrightarrow\; F\_n(z)\backslash to\; \backslash phi\; (z)$.

Example 2.

:$f\_n(z)=z+\backslash frac\{z^2\}\{2^n\}\backslash Longrightarrow\; F\_n(z)\backslash to\; \backslash frac\{1\}\{2\}\backslash left(\; e^\{2z\}-1\; \backslash right)$

Example 3.

:$f\_n(z)=\; \backslash frac\{z\}\{1-\backslash tfrac\{z^2\}\{4^n\}\}\backslash Longrightarrow\; F\_n(z)\backslash to\; \backslash tan\; (z)$

Example 4.

:$g\_n(z)=\backslash frac\{2\; \backslash cdot\; 4^\{n\}\}\{z\}\; \backslash left\; (\; \backslash sqrt\{1+\backslash frac\{z^2\}\{4^\{n\}\}\}-1\; \backslash right\; )\backslash Longrightarrow\; G\_n(z)\; \backslash to\; \backslash arctan\; (z)$

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

:$G(\backslash zeta\; )=\backslash frac\{\; \backslash tfrac\{e^\backslash zeta\}\{4\}\}\{3+\backslash zeta\; +\backslash cfrac\{\backslash tfrac\{e^\backslash zeta\}\; 8\; \}\{3+\backslash zeta\; +\; \backslash cfrac\{\backslash tfrac\{e^\backslash zeta\}\{12\}\}\{3+\backslash zeta\; +\backslash cdots\}\}\}$

To find α = G(α), first we define:

:$\backslash begin\{align\}$

t_n(z)&=\cfrac{\tfrac{e^\zeta}{4n}}{3+\zeta +z} \\

f_n(\zeta )&= t_1\circ t_2\circ \cdots \circ t_n(0)

\end{align}

Then calculate $G\_n(\backslash zeta\; )=f\_n\backslash circ\; \backslash cdots\; \backslash circ\; f\_1(\backslash zeta\; )$ with ζ = 1, which gives: α = 0.087118118... to ten decimal places after ten iterations.

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

$$f\_n\; (\backslash zeta)=\backslash frac\{1\}\{n\}\; \backslash sum\_\{k=1\}^n\; \backslash varphi\; \backslash left(\; \backslash zeta\; ,\backslash tfrac\{k\}\{n\}\; \backslash right).$$

If {{abs|φ(ζ, t)}} ≤ r < R for ζ ∈ S and t ∈ [0, 1], then

$$\backslash zeta\; =\backslash int\_0^1\; \backslash varphi\; (\backslash zeta\; ,t)\; \backslash ,\; dt$$

has a unique solution, α in S, with $\backslash lim\_\{n\backslash to\; \backslash infty\}\; G\_n(\backslash zeta\; )\; =\; \backslash alpha.$}}

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 $g\_\{k,n\}(z)=z+\backslash varphi\_\{k,n\}(z)$ analytic or simply continuous – in a domain S, such that

:$\backslash lim\_\{n\backslash to\; \backslash infty\}\backslash varphi\_\{k,n\}(z)=0$ for all k and all z in S,

and $g\_\{k,n\}(z)\; \backslash in\; S$.

Source:

:$\backslash begin\{align\}$

g_{k,n}(z) &=z+\frac{1}{n}\phi \left (z,\tfrac{k}{n} \right ) \\

G_{k,n}(z) &= \left (g_{k,n}\circ g_{k-1,n} \circ \cdots \circ g_{1,n} \right ) (z) \\

G_n(z) &=G_{n,n}(z)

\end{align}

implies

:$\backslash lambda\_n(z)\backslash doteq\; G\_n(z)-z=\backslash frac\{1\}\{n\}\backslash sum\_\{k=1\}^n\; \backslash phi\; \backslash left(\; G\_\{k-1,n\}(z)\backslash tfrac\; k\; n\; \backslash right)\backslash doteq\; \backslash frac\; 1\; n\; \backslash sum\_\{k=1\}^n\; \backslash psi\; \backslash left\; (z,\backslash tfrac\{k\}\{n\}\; \backslash right)\; \backslash sim\; \backslash int\_0^1\; \backslash psi\; (z,t)\backslash ,dt,$

where the integral is well-defined if $\backslash tfrac\{dz\}\{dt\}=\backslash phi\; (z,t)$ has a closed-form solution z(t). Then

:$\backslash lambda\_n(z\_0)\backslash approx\; \backslash int\_0^1\; \backslash phi\; (\; z(t),t)\backslash ,dt.$

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. $\backslash phi\; (z,t)=\backslash frac\{2t-\backslash cos\; y\}\{1-\backslash sin\; x\backslash cos\; y\}+i\backslash frac\{1-2t\backslash sin\; x\}\{1-\backslash sin\; x\backslash cos\; y\},\; \backslash int\_0^1\; \backslash psi\; (z,t)\; \backslash ,\; dt$

File:Virtual_tunnels.jpg]

Image:Contours in the vector field f(z) = -Cos(z).jpg

Example. Let:

:$g\_n(z)=z+\backslash frac\{c\_n\}\{n\}\backslash phi\; (z),\; \backslash quad\; \backslash text\{with\}\; \backslash quad\; f(z)\; =\; z\; +\; \backslash phi(z).$

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). Let

:$T(z)=\backslash lim\_\{n\backslash to\; \backslash infty\}\; T\_n(z)$

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) $c\_n\; =\; \backslash sqrt\{n\}$. 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

:$\backslash oint\_\backslash gamma\; \backslash phi\; (\backslash zeta\; )\; \backslash ,\; d\backslash zeta\; =\backslash lim\_\{n\backslash to\; \backslash infty\}\backslash frac\; c\; n\; \backslash sum\_\{k=1\}^n\; \backslash phi\; ^2\; \backslash left\; (T\_\{k-1,n\}(z)\; \backslash right\; )$

and

:$L(\backslash gamma\; (z))=\backslash lim\_\{n\backslash to\; \backslash infty\}\; \backslash frac\{c\}\{n\}\backslash sum\_\{k=1\}^n\; \backslash left|\; \backslash phi\; \backslash left\; (T\_\{k-1,n\}(z)\; \backslash right\; )\; \backslash right|,$

when these limits exist.

These concepts are marginally related to active contour theory in image processing, and are simple generalizations of the Euler method

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

:

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

Example (S1). Set

:$f\_n(z)=z+\backslash frac\{1\}\{\backslash rho\; n^2\}\backslash sqrt\{z\},\; \backslash qquad\; \backslash rho\; >\backslash sqrt\{\backslash frac\{\backslash pi\}\{6\}\}$

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

:$\backslash begin\{align\}$

G_n(z) &=z+g_1(z)+g_2(G_1(z))+g_3(G_2(z))+\cdots + g_n(G_{n-1}(z)) \\

&= z+\frac{1}{\rho \cdot 1^2}\sqrt{z}+\frac{1}{\rho \cdot 2^2}\sqrt{G_1(z)}+\frac{1}{\rho \cdot 3^2}\sqrt{G_2(z)}+\cdots +\frac{1}{\rho \cdot n^2} \sqrt{G_{n-1}(z)}

\end{align}

converges absolutely, hence is convergent.

Example (S2): $f\_n(z)=z+\backslash frac\; 1\; \{n^2\}\backslash cdot\; \backslash varphi\; (z),\; \backslash varphi\; (z)=2\backslash cos(x/y)+i2\backslash sin\; (x/y),\; >G\_n(z)=f\_n\; \backslash circ\; f\_\{n-1\}\backslash circ\; \backslash cdots\; \backslash circ\; f\_1(z),\; \backslash qquad\; [-10,10],\; n=50$

File:Self-generating series3.jpg

The product defined recursively by

:$f\_n(z)=z(\; 1+g\_n(z)),\; \backslash qquad\; |z|\; \backslash leqslant\; M,$

has the appearance

:$G\_n(z)\; =\; z\; \backslash prod\; \_\{k=1\}^n\; \backslash left(\; 1+g\_k\; \backslash left(\; G\_\{k-1\}(z)\; \backslash right)\; \backslash right).$

In order to apply Theorem GF3 it is required that:

:

Once again, a boundedness condition must support

:$\backslash left|G\_\{n-1\}(z)\; g\_n(G\_\{n-1\}(z))\backslash right|\backslash le\; C\; \backslash beta\_n.$

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

:$|z|\; \backslash leqslant\; R\; =\; \backslash frac\{M\}\{P\}\; \backslash qquad\; \backslash text\{where\}\; \backslash quad\; P\; =\; \backslash prod\_\{n=1\}^\backslash infty\; \backslash left(\; 1+C\backslash beta\_n\backslash right).$

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

Example (P1). Suppose $f\_n(z)=z(1+g\_n(z))$ with $g\_n(z)=\backslash tfrac\{z^2\}\{n^3\},$ observing after a few preliminary computations, that |z| ≤ 1/4 implies |G_n(z)| < 0.27. Then

:

and

:$G\_n(z)=z\; \backslash prod\_\{k=1\}^\{n-1\}\backslash left(\; 1+\backslash frac\{G\_k(z)^2\}\{n^3\}\backslash right)$

converges uniformly.

Example (P2).

:$g\_\{k,n\}(z)=z\backslash left(\; 1+\backslash frac\; 1\; n\; \backslash varphi\; \backslash left\; (z,\backslash tfrac\; k\; n\; \backslash right\; )\; \backslash right),$

:$G\_\{n,n\}(z)=\; \backslash left(\; g\_\{n,n\}\backslash circ\; g\_\{n-1,n\}\backslash circ\; \backslash cdots\; \backslash circ\; g\_\{1,n\}\; \backslash right\; )\; (z)\; =\; z\backslash prod\_\{k=1\}^n\; (\; 1+P\_\{k,n\}(z)),$

:$P\_\{k,n\}(z)=\backslash frac\; 1\; n\; \backslash varphi\; \backslash left\; (G\_\{k-1,n\}(z),\backslash tfrac\{k\}\{n\}\; \backslash right\; ),$

:$\backslash prod\_\{k=1\}^\{n-1\}\; \backslash left(\; 1+P\_\{k,n\}(z)\; \backslash right)\; =\; 1+P\_\{1,n\}(z)+P\_\{2,n\}(z)+\backslash cdots\; +\; P\_\{k-1,n\}(z)\; +\; R\_n(z)\; \backslash sim\; \backslash int\_0^1\; \backslash pi\; (z,t)\; \backslash ,\; dt\; +\; 1+R\_n(z),$

:$\backslash varphi\; (z)=x\backslash cos(y)+iy\backslash sin(x),\; \backslash int\_0^1\; (z\backslash pi\; (z,t)-1)\; \backslash ,dt,\; \backslash qquad\; [-15,15]:$

Image:Picasso's Universe.jpg

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

: $\backslash begin\{align\}$

F_n(z) &= \frac{\rho (z)}{\delta_1+} \frac{\rho (F_1(z))}{\delta_2 +} \frac{\rho (F_2(z))}{\delta_3+} \cdots \frac{\rho (F_{n-1}(z))}{\delta_n}, \\

\rho (z) &= \frac{\cos(y)}{\cos (y)+\sin (x)}+i\frac{\sin(x)}{\cos (y)+\sin (x)}, \qquad [0

\end{align}

Image:Diminishing returns.jpg

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

: $G\_n(z)=\backslash frac\{\backslash rho\; (G\_\{n-1\}(z))\}\{1+\backslash rho\; (G\_\{n-1\}(z))-\}\backslash \; \backslash frac\{\backslash rho\; (G\_\{n-2\}(z))\}\{1+\backslash rho\; (G\_\{n-2\}(z))-\}\backslash cdots\; \backslash frac\{\backslash rho\; (G\_1(z))\}\{1+\backslash rho\; (G\_1(z))-\}\backslash \; \backslash frac\{\backslash rho\; (z)\}\{1+\backslash rho\; (z)-z\},$

:$\backslash rho\; (z)=\backslash rho\; (x+iy)=x\backslash cos(y)+iy\backslash sin(x),\; \backslash qquad\; [-15,15],\; n=30$

Image:Dream of Gold.jpg

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