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| In mathematics, '''infinite [[composition (mathematics)|compositions]] of [[analytic function]]s (ICAF)''' offer alternative formulations of [[continued fractions]], [[series (mathematics)|series]], [[product (mathematics)|products]] and other infinite expansions, and the theory evolving from such compositions may shed light on the [[convergence (mathematics)|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 (mathematics)|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]].
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| ==Notation==
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| There are several notations describing infinite compositions, including the following:
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| '''Forward compositions''':
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| : <math> F_{k,n} (z)= f_k \circ f_{k+1} \circ \cdots \circ f_{n-1} \circ f_n(z)</math>
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| '''Backward compositions''':
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| : <math>G_{k,n}(z) = f_n\circ f_{n-1} \circ \cdots \circ f_{k+1} \circ f_k(z).</math>
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| Convergence is interpreted as the existence of <math> \lim_{n\to \infty} F_{1,n}(z)</math> and <math> \lim_{n\to\infty} G_{1,n}(z).</math>
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| For convenience, set ''F<sub>n</sub>''(''z'') = ''F''<sub>1,''n''</sub>(''z'') and ''G<sub>n</sub>''(''z'') = ''G''<sub>1,''n''</sub>(''z'').
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| ==Contraction theorem==
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| Many results can be considered extensions of the following result:
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| <blockquote>'''Contraction Theorem for Analytic Functions.'''<ref>P. Henrici, ''Applied and Computational Complex Analysis'', Vol. 1 (Wiley, 1974)</ref> 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
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| :<math>F_n(z)=f\circ f\circ \cdots \circ f(z)\to \alpha,</math>
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| the attractive fixed point of ''f'' in ''S'', for all ''z'' in {{overline|''S''}}.</blockquote>
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| ==Infinite compositions of contractive functions==
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| Let {''f<sub>n</sub>''} 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<sub>n</sub>''(''S'') ⊂ Ω.
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| <blockquote>'''Forward (inner or right) Compositions Theorem.''' {''F<sub>n</sub>''(''z'')} converges uniformly on compact subsets of ''S'' to a constant function ''F''(''z'') = λ.<ref>L. Lorentzen, Compositions of contractions, J. Comp & Appl Math. 32 (1990)</ref></blockquote>
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| <blockquote>'''Backward (outer or left) Compositions Theorem.''' {''G<sub>n</sub>''(''z'')} converges uniformly on compact subsets of ''S'' to γ ∈ Ω if and only if the sequence of fixed points {γ<sub>''n''</sub>} of the {''f<sub>n</sub>''} converge to γ.<ref name = Gilla>J. Gill, The use of the sequence ''F<sub>n</sub>''(''z'') = ''f<sub>n</sub>'' o … o ''f''<sub>1</sub>(''z'') in computing the fixed points of continued fractions, products, and series, Appl. Numer. Math. 8 (1991)</ref></blockquote>
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| Additional theory resulting from investigations based on these two theorems, particularly Forward Compositions Theorem, include location analysis for the limits obtained here [http://comet.lehman.cuny.edu/keenl/blochconstantsfinalversion.pdf]. For a different approach to Backward Compositions Theorem, see [http://comet.lehman.cuny.edu/keenl/forwarditer.pdf].
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| Regarding Backward Compositions Theorem, the example ''f''<sub>2''n''</sub>(''z'') = 1/2 and ''f''<sub>2''n''−1</sub>(''z'') = −1/2 for ''S'' = {''z'' : |''z''| < 1} demonstrates the inadequacy of simply requiring contraction into a compact subset, like Forward Compositions Theorem.
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| ==Infinite compositions of other functions==
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| === General analytic functions ===
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| Results<ref name=Gillb /> involving '''[[entire function]]s''' include the following, as examples. Set
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| :<math>\begin{align}
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| f_n(z)&=a_n z + c_{n,2}z^2+c_{n,3} z^3+\cdots \\
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| \rho_n &= \sup_r \left\{ \left| c_{n,r} \right|^{\frac{1}{r-1}} \right\}
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| \end{align}</math>
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| Then the following results hold:
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| <blockquote>'''Theorem E1.'''<ref name=Kojima>S.Kojima, Convergence of infinite compositions of entire functions, arXiv:1009.2833v1</ref> If ''a<sub>n</sub>'' ≡ 1,
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| :<math>\sum_{n=1}^\infty \rho_n < \infty</math>
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| then ''F<sub>n</sub>'' → ''F'', entire.</blockquote>
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| <blockquote>'''Theorem E2.'''<ref name=Gillb>J. Gill, Convergence of infinite compositions of complex functions, Comm. Anal. Th. Cont. Frac., Vol XIX (2012)</ref> Set ε<sub>''n''</sub> = |''a<sub>n</sub>''−1| suppose there exists non-negative δ<sub>''n''</sub>, ''M''<sub>1</sub>, ''M''<sub>2</sub>, ''R'' such that the following holds:
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| :<math>\begin{align}
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| \sum_{n=1}^{\infty} \varepsilon_n &<\infty, \\
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| \sum_{n=1}^{\infty} \delta_n&<\infty, \\
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| \prod_{n=1}^{\infty} (1+\delta_n) &<M_1, \\
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| \prod_{n=1}^{\infty} (1+\varepsilon_n) &< M_2, \\
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| \rho_n &< \frac{\delta_n}{R M_1 M_2}.
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| \end{align}</math>
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| Then ''G<sub>n</sub>''(''z'') → ''G''(''z''), analytic for |''z''| < ''R''. Convergence is uniform on compact subsets of {''z'' : |''z''| < ''R''}.</blockquote>
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| <blockquote>'''Theorem GF3.'''<ref name=Gillb /> Let {''f<sub>n</sub>''} be a sequence of complex functions defined on ''S'' = {''z'' : |''z''| < ''M''}. Suppose there exists a non-negative sequence {β<sub>''n''</sub>} such that
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| :<math>C\sum_{n=1}^{\infty}\beta_n<M,</math>
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| :<math>\left|f_n(z)-z \right|<C\beta_n, \qquad z \in S.</math>
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| Set <math>R=M-C\sum_{n=1}^{\infty}\beta_n>0</math>. Then ''G<sub>n</sub>''(''z'') → ''G''(''z'') for |''z''| < ''R'', uniformly on compact subsets.</blockquote>
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| <blockquote>'''Theorem GF4.'''<ref name=Gillb /> Let ''f<sub>n</sub>''(''z'') = ''z''(1+''g<sub>n</sub>''(''z'')), analytic for |''z''| < ''R''<sub>0</sub>, with |''g<sub>n</sub>''(''z'')| ≤ ''C''β<sub>''n''</sub>,
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| :<math>\sum_{n=1}^{\infty} \beta_n<\infty.</math>
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| Choose 0 < ''r'' < ''R''<sub>0</sub> and define
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| :<math>R=R(r)=\frac{R_0-r}{\prod_{n=1}^{\infty} \left( 1+C\beta_n \right)}.</math>
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| Then ''F<sub>n</sub>'' → ''F'' uniformly for |''z''| ≤ ''R''. Furthermore,
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| :<math>\left| F'(z) \right|\le \prod_{n=1}^{\infty } {\left( 1+\tfrac{R_0}{r}C\beta_n \right)}</math>.</blockquote>
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| === Linear fractional transformations ===
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| Results<ref name=Gillb /> for compositions of '''[[möbius transformation|linear fractional (Möbius) transformations]]''' include the following, as examples:
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| <blockquote>'''Theorem LFT1.''' On the set of convergence of a sequence {''F<sub>n</sub>''} of non-singular LFTs, the limit function is either
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| *(a) a non-singular LFT,
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| *(b) a function taking on two distinct values, or
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| *(c) a constant.
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| 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.<ref>G. Piranian & W. Thron,Convergence properties of sequences of Linear fractional transformations, Mich. Math. J.,Vol. 4 (1957)</ref></blockquote>
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| <blockquote>'''Theorem LFT2.''' If {''F<sub>n</sub>''} converges to an LFT , then ''f<sub>n</sub>'' converge to the identity function ''f''(''z'') = ''z''.<ref>J. DePree & W. Thron,On sequences of Mobius transformations, Math. Zeitschr., Vol. 80 (1962)</ref></blockquote>
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| <blockquote>'''Theorem LFT3.''' If ''f<sub>n</sub>'' → ''f'' and all functions are ''hyperbolic'' or ''loxodromic'' Möbius transformations, then ''F<sub>n</sub>''(''z'') → λ, a constant, for all <math>z\ne \beta = \lim_{n\to \infty} \beta_n</math>, where {β<sub>''n''</sub>} are the repulsive fixed points of the {''f<sub>n</sub>''}.<ref>A. Magnus & M. Mandell, On convergence of sequences of linear fractional transformations,Math. Zeitschr. 115 (1970)</ref></blockquote>
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| <blockquote>'''Theorem LFT4.''' If ''f<sub>n</sub>'' → ''f'' where ''f'' is ''parabolic'' with fixed point γ. Let the fixed-points of the {''f<sub>n</sub>''} be {γ<sub>''n''</sub>} and {β<sub>''n''</sub>}. If
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| :<math>\begin{align}
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| \sum_{n=1}^{\infty} \left|\gamma_n-\beta_n \right| &<\infty \\
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| \sum_{n=1}^{\infty} n \left|\beta_{n+1}-\beta_n \right|&<\infty
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| \end{align}</math>
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| then ''F<sub>n</sub>''(''z'') → λ, a constant in the extended complex plane, for all ''z''.<ref>J. Gill, Infinite compositions of Mobius transformations, Trans. Amer. Math. Soc., Vol176 (1973)</ref></blockquote>
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| ==Examples & applications==
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| === Continued fractions ===
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| The value of the infinite continued fraction
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| :<math>\frac{a_1}{b_1+\frac{a_2}{b_2+\ldots}}</math>
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| may be expressed as the limit of the sequence {''F<sub>n</sub>''(0)} where
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| :<math>f_n(z)=\frac{a_n}{b_n+z}.</math>
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| As a simple example, a well-known result (Worpitsky Circle*<ref>L. Lorentzen, H. Waadeland, ''Continued Fractions with Applications'', North Holland (1992)</ref>) follows from an application of Theorem (A):
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| Consider the continued fraction
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| :<math>\frac{a_1\zeta }{1+\frac{a_2\zeta }{1+\ldots}} </math>
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| with
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| :<math>f_n(z)=\frac{a_n \zeta }{1+z}.</math>
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| Stipulate that |ζ| < 1 and |''z''| < ''R'' < 1. Then for 0 < ''r'' < 1,
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| :<math>|a_n|<rR(1-R)\Rightarrow \left|f_n(z) \right|<rR<R\Rightarrow \frac{a_1\zeta }{1+\frac{a_2\zeta }{1+\ldots}} = F(\zeta )</math>, analytic for |''z''| < 1.
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| Set ''R'' = 1/2.
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| === Direct functional expansion ===
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| An example illustrating the conversion of a function directly into a composition follows:
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| Suppose that for |''t''| > 1, <math>\varphi (tz)=t\left( \varphi (z)+\varphi (z)^2 \right)</math>, an entire function with '''φ(0) = 0, φ′(0) = 1'''. Then <math>f_n(z)=z+\frac{z^2}{t^n}\Rightarrow F_n(z)\to \varphi (z)</math>.<ref name=Kojima /><ref>N. Steinmetz, ''Rational Iteration'', Walter de Gruyter, Berlin (1993)</ref>
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| '''Example.''' <math>f_n(z)=z+\frac{z^2}{2^n}\Rightarrow F_n(z)\to \frac{1}{2}\left( e^{2z}-1 \right)</math><ref name=Kojima />
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| === Calculation of fixed-points ===
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| 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:
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| '''Example (FP1)''':<ref name=Gilla /> For |ζ| ≤ 1 let
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| :<math>G(\zeta )=\frac{ \tfrac{e^{\zeta}}{4}}{3+\zeta +\frac{\tfrac{e^{\zeta}}{8}}{3+\zeta +\frac{\tfrac{e^{\zeta}}{12}}{3+\zeta +\ldots}}}</math>
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| To find α = ''G''(α), first we define:
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| :<math>\begin{align}
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| t_n(z)&=\frac{\tfrac{e^{\zeta}}{4n}}{3+\zeta +z} \\
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| f_n(\zeta )&= t_1\circ t_2\circ \cdots \circ t_n(0)
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| \end{align}</math>
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| Then calculate <math>G_n(\zeta )=f_n\circ \cdots \circ f_1(\zeta )</math> with ζ = 1, which gives: α = 0.087118118... to ten decimal places after ten iterations.
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| <blockquote>'''Theorem (FP2).'''<ref name=Gillb /> Let φ(ζ, ''t'') be analytic in ''S'' = {''z'' : |''z''| < ''R''} for all ''t'' in [0, 1] and continuous in ''t''. Set
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| :<math>f_n (\zeta)=\frac{1}{n}\sum_{k=1}^{n}{\varphi \left( \zeta ,\tfrac{k}{n} \right)}.</math>
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| If |φ(ζ, ''t'')| ≤ ''r'' < ''R'' for ζ ∈ ''S'' and ''t'' ∈ [0, 1], then
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| :<math>\zeta =\int_0^1 \varphi (\zeta ,t)dt</math>
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| has a unique solution, α in ''S'', with <math>\underset{n\to \infty }{\mathop{\lim }}\,{{G}_{n}}(\zeta )=\alpha </math>.</blockquote>
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| === Evolution functions ===
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| 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 paradoxes|Zeno's Arrow]]): For the interval divided into n equal subintervals, 1 ≤ ''k'' ≤ ''n'' set <math>g_{k,n}(z)=z+\varphi_{k,n}(z)</math> analytic - or simply continuous - in a domain ''S'', such that
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| :<math>\lim_{n\to \infty}\varphi_{k,n}(z)=0</math>
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| for all ''k'' and ''z'' in ''S'' imply <math>g_{k,n}(z)\in S</math>.
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| ====Example 1====
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| <math>g_{k,n}(z)=z+\frac{k}{n^2}f(z).</math>
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| Now, set <math>T_{1,n}(z)=g_{1,n}(z)</math> and <math>T_{k,n}(z)=g_{k,n}\left(T_{k-1,n}(z) \right)</math>. If <math>\lim_{n\to \infty}T_{n,n}(z)=T(z)</math> exists, the initial point z has moved to a new position, ''T''(''z''), in a fashion described above (for large values of ''n'', <math>g_{k,n}(z)\approx z</math>). It is not difficult to show that ''f''(''z'') = α''z'' + β, α ≥ 0 implies <math>T_{n,n}(z)\to e^{\frac{\alpha}{2}}z+b\beta </math>. A byproduct of this derivation is the following representation:
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| :<math>\lim_{n\to \infty} \prod_{k=1}^n \left( 1+\frac{2k}{n^2}x \right)=e^x, \qquad x \in \mathbf{R}.</math>
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| And of course, if ''f''(''z'') ≡ ''c'', then<ref>J. Gill, Zeno's arrow: A mathematical speculation , Comm. Anal. Th. Cont. Frac., Vol XIX (2012)</ref>
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| :<math>T(z)=z+c\int_0^1 tdt.</math>
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| [[Image:Contours in the vector field f(z) = -Cos(z).jpg|thumb|right|350px|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]]
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| ====Example 2====
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| :<math>g_n(z)=z+\frac{c_n}{n}\varphi (z),</math> | |
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| with ''f''(''z'') := ''z'' + φ(''z''). Next, set <math>T_{1,n}(z)=g_n(z)</math>, <math>T_{k,n}(z)= g_n\left(T_{k-1,n}(z) \right)</math>, and ''T<sub>n</sub>''(''z'') = ''T<sub>n,n</sub>''(''z''). Let
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| :<math>T(z)=\lim_{n\to \infty}T_n(z)</math>
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| when that limit exists. The sequence {''T<sub>n</sub>''(''z'')} defines contours γ = γ(''c<sub>n</sub>'', ''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<sub>n</sub>''(''z'') → ''T''(''z'') ≡ α along γ = γ(''c<sub>n</sub>'', ''z''), provided (for example) <math>c_n = \sqrt{n}</math>. If ''c<sub>n</sub>'' ≡ ''c'' > 0, then it seems apparent - though not rigorously proven for many cases<ref>J. Gill, Zeno Contours, Parametric Forms & Integrals, Scribd.com</ref> - that ''T<sub>n</sub>''(''z'') → ''T''(''z''), a point on the contour γ = γ(''c'', ''z''). It is easily seen that
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| :<math>\oint_{\gamma}\varphi (\zeta )d\zeta =\lim_{n\to \infty}\frac{c}{n}\sum_{k=1}^{n}\varphi^2 \left (T_{k-1,n}(z) \right )</math>
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| and
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| :<math>L(\gamma (z))=\lim_{n\to \infty} \frac{c}{n}\sum_{k=1}^n \left| \varphi \left (T_{k-1,n}(z) \right ) \right|, </math>
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| when these limits exist.<ref>J. Gill, Progress Report: Zeno Contours in the Complex Plane, Comm. Anal. Th. Cont. Frac., Vol XIX (2012)</ref>
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| These concepts are marginally related to ''[[Active contour model|active contour theory]]'' in image processing.
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| === Self-replicating series & products ===
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| ====Series====
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| The series defined recursively by ''f<sub>n</sub>''(''z'') = ''z'' + ''g<sub>n</sub>''(''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<sub>n</sub>'' is defined for |''z''| < ''M'' then |''G<sub>n</sub>''(''z'')| < ''M'' must follow before |''f<sub>n</sub>''(''z'')−''z''| = |''g<sub>n</sub>''(''z'')| ≤ ''C''β<sub>''n''</sub> is defined for iterative purposes. This is because <math>g_n(G_{n-1}(z))</math> occurs throughout the expansion. The restriction
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| :<math>|z|<R=M-C\sum_{k=1}^{\infty} \beta_k >0</math>
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| serves this purpose. Then ''G<sub>n</sub>''(''z'') → ''G''(''z'') uniformly on the restricted domain.
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| '''Example (S1)''': Set
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| :<math>f_n(z)=z+\frac{1}{\rho n^2}\sqrt{z}, \qquad \rho >\sqrt{\frac{\pi }{6}}</math>
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| and ''M'' = ρ<sup>2</sup>. Then ''R'' = ρ<sup>2</sup>−(π/6) > 0. Then, if <math>S=\left\{ z: |z|<R,\operatorname{Re}(z)>0 \right\}</math>, ''z'' in ''S'' implies <math>\left|G_n(z) \right|<M</math> and theorem (GF3) applies, so that | |
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| :<math>\begin{align}
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| G_n(z) &=z+g_1(z)+g_2(G_1(z))+g_3(G_2(z))+\cdots + g_n(G_{n-1}(z)) \\
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| &= 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}</math>
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| converges absolutely, hence is convergent.
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| ====Products====
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| The product defined recursively by <math>f_n(z)=z\left( 1+g_n(z) \right)</math>, |''z''| ≤ ''M'', have the appearance
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| :<math>G_n(z) = z \prod _{k=1}^n \left( 1+g_k \left( G_{k-1}(z) \right) \right).</math>
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| In order to apply theorem (GF3) it is required that <math>\left| z\cdot g_n(z) \right|\le C\beta_n</math> where
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| :<math>\sum_{k=1}^{\infty} \beta_k<\infty.</math>
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| Once again, a boundedness condition must support
| |
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| :<math>\left|G_{n-1}(z)\cdot g_n(G_{n-1}(z))\right|\le C \beta_n.</math>
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| If one knows ''C''β<sub>''n''</sub> in advance, setting |''z''| ≤ ''R'' = ''M''/''P'' where
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| :<math>\prod_{n=1}^{\infty} \left( 1+C\beta_n\right) =P</math>
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| suffices. Then ''G<sub>n</sub>''(''z'') → ''G''(''z'') uniformly on the restricted domain.
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| '''Example (P1)''': Suppose that <math>f_n(z)=z(1+g_n(z))</math> where <math>g_n(z)=\frac{z^2}{n^3}</math>, observing after a few preliminary computations, that |''z''| ≤ 1/4 implies |''G<sub>n</sub>''(''z'')| < 0.27. Then
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| :<math>\left|G_n(z)\cdot \frac{G_n(z)^2}{n^3} \right|<(0.02)\frac{1}{n^3}=C\beta_n</math>
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| and
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| :<math>G_n(z)=z\cdot \prod_{k=1}^{n-1}\left( 1+\frac{G_k(z)^2}{n^3}\right)</math>
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| converges uniformly.
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| ==References==
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| {{reflist}}
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| [[Category:Complex analysis]]
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| [[Category:Analytic functions]]
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| [[Category:Fixed-point theorems]]
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