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 ∘ f k + 1 ∘ ⋯ ∘ f n − 1 ∘ f n ( z ) . {\displaystyle F_{k,n}(z)=f_{k}\circ f_{k+1}\circ \dots \circ f_{n-1}\circ f_{n}(z).}
Backward compositions: G k , n ( z ) = f n ∘ f n − 1 ∘ ⋯ ∘ f k + 1 ∘ f k ( z ) . {\displaystyle G_{k,n}(z)=f_{n}\circ f_{n-1}\circ \dots \circ f_{k+1}\circ f_{k}(z).}
In each case convergence is interpreted as the existence of the following limits:
lim n → ∞ F 1 , n ( z ) , lim n → ∞ G 1 , n ( z ) . {\displaystyle \lim _{n\to \infty }F_{1,n}(z),\qquad \lim _{n\to \infty }G_{1,n}(z).}
For convenience, set Fn(z) = F1,n(z) and Gn(z) = G1,n(z). One may also write F n ( z ) = R n k = 1 f k ( z ) = f 1 ∘ f 2 ∘ ⋯ ∘ f n ( z ) {\displaystyle F_{n}(z)={\underset {k=1}{\overset {n}{\mathop {R} }}}\,f_{k}(z)=f_{1}\circ f_{2}\circ \cdots \circ f_{n}(z)} and
G n ( z ) = L n k = 1 g k ( z ) = g n ∘ g n − 1 ∘ ⋯ ∘ g 1 ( z ) {\displaystyle G_{n}(z)={\underset {k=1}{\overset {n}{\mathop {L} }}}\,g_{k}(z)=g_{n}\circ g_{n-1}\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 {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.
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 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:
Infinite compositions of other functions
Non-contractive complex functions Results involving entire functions include the following, as examples. Set
f n ( z ) = a n z + c n , 2 z 2 + c n , 3 z 3 + ⋯ ρ n = sup r { | c n , r | 1 r − 1 } {\displaystyle {\begin{aligned}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{aligned}}}
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
a 1 b 1 + a 2 b 2 + ⋯ {\displaystyle {\cfrac {a_{1}}{b_{1}+{\cfrac {a_{2}}{b_{2}+\cdots }}}}}
may be expressed as the limit of the sequence {Fn(0)} where
f n ( z ) = a n b n + z . {\displaystyle f_{n}(z)={\frac {a_{n}}{b_{n}+z}}.}
As a simple example, a well-known result (Worpitsky's circle theorem) follows from an application of Theorem (A): Consider the continued fraction
a 1 ζ 1 + a 2 ζ 1 + ⋯ {\displaystyle {\cfrac {a_{1}\zeta }{1+{\cfrac {a_{2}\zeta }{1+\cdots }}}}}
with
f n ( z ) = a n ζ 1 + z . {\displaystyle f_{n}(z)={\frac {a_{n}\zeta }{1+z}}.}
Stipulate that |ζ| < 1 and |z| < R < 1. Then for 0 < r < 1,
| a n | < r R ( 1 − R ) ⇒ | f n ( z ) | < r R < R ⇒ a 1 ζ 1 + a 2 ζ 1 + ⋯ = F ( ζ ) {\displaystyle |a_{n}|<rR(1-R)\Rightarrow \left|f_{n}(z)\right|<rR<R\Rightarrow {\frac {a_{1}\zeta }{1+{\frac {a_{2}\zeta }{1+\cdots }}}}=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 + ⋯ , {\displaystyle F(z)={\frac {(i-1)z}{1+i+z{\text{ }}+}}{\text{ }}{\frac {(2-i)z}{1+2i+z{\text{ }}+}}{\text{ }}{\frac {(3-i)z}{1+3i+z{\text{ }}+}}\cdots ,} [ − 15 , 15 ] {\displaystyle [-15,15]}
Example. A fixed-point continued fraction form (a single variable).
f k , n ( z ) = α k , n β k , n α k , n + β k , n − z , α k , n = α k , n ( z ) , β k , n = β k , n ( z ) , F n ( z ) = ( f 1 , n ∘ ⋯ ∘ f n , n ) ( z ) {\displaystyle f_{k,n}(z)={\frac {\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 \cdots \circ f_{n,n}\right)(z)}
α k , n = x cos ( t y ) + i y sin ( t x ) , β k , n = cos ( t y ) + i sin ( t x ) , t = k / n {\displaystyle \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. Suppose ϕ {\displaystyle \phi } is an entire function satisfying the following conditions:
{ ϕ ( t z ) = t ( ϕ ( z ) + ϕ ( z ) 2 ) | t | > 1 ϕ ( 0 ) = 0 ϕ ′ ( 0 ) = 1 {\displaystyle {\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 ⟹ F n ( z ) → ϕ ( z ) {\displaystyle f_{n}(z)=z+{\frac {z^{2}}{t^{n}}}\Longrightarrow F_{n}(z)\to \phi (z)} . Example 2.
f n ( z ) = z + z 2 2 n ⟹ F n ( z ) → 1 2 ( e 2 z − 1 ) {\displaystyle f_{n}(z)=z+{\frac {z^{2}}{2^{n}}}\Longrightarrow F_{n}(z)\to {\frac {1}{2}}\left(e^{2z}-1\right)}
Example 3.
f n ( z ) = z 1 − z 2 4 n ⟹ F n ( z ) → tan ( z ) {\displaystyle f_{n}(z)={\frac {z}{1-{\tfrac {z^{2}}{4^{n}}}}}\Longrightarrow F_{n}(z)\to \tan(z)}
Example 4.
g n ( z ) = 2 ⋅ 4 n z ( 1 + z 2 4 n − 1 ) ⟹ G n ( z ) → arctan ( z ) {\displaystyle g_{n}(z)={\frac {2\cdot 4^{n}}{z}}\left({\sqrt {1+{\frac {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 ( ζ ) = e ζ 4 3 + ζ + e ζ 8 3 + ζ + e ζ 12 3 + ζ + ⋯ {\displaystyle G(\zeta )={\frac {\tfrac {e^{\zeta }}{4}}{3+\zeta +{\cfrac {\tfrac {e^{\zeta }}{8}}{3+\zeta +{\cfrac {\tfrac {e^{\zeta }}{12}}{3+\zeta +\cdots }}}}}}}
To find α = G(α), first we define:
t n ( z ) = e ζ 4 n 3 + ζ + z f n ( ζ ) = t 1 ∘ t 2 ∘ ⋯ ∘ t n ( 0 ) {\displaystyle {\begin{aligned}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{aligned}}}
Then calculate G n ( ζ ) = f n ∘ ⋯ ∘ f 1 ( ζ ) {\displaystyle 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 = [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 + φ k , n ( z ) {\displaystyle g_{k,n}(z)=z+\varphi _{k,n}(z)} analytic or simply continuous – in a domain S, such that
lim n → ∞ φ k , n ( z ) = 0 {\displaystyle \lim _{n\to \infty }\varphi _{k,n}(z)=0} for all k and all z in S, and g k , n ( z ) ∈ S {\displaystyle g_{k,n}(z)\in S} .
Principal example Source:
g k , n ( z ) = z + 1 n ϕ ( z , k n ) G k , n ( z ) = ( g k , n ∘ g k − 1 , n ∘ ⋯ ∘ g 1 , n ) ( z ) G n ( z ) = G n , n ( z ) {\displaystyle {\begin{aligned}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{aligned}}}
implies
λ n ( z ) ≐ G n ( z ) − z = 1 n ∑ k = 1 n ϕ ( G k − 1 , n ( z ) k n ) ≐ 1 n ∑ k = 1 n ψ ( z , k n ) ∼ ∫ 0 1 ψ ( z , t ) d t , {\displaystyle \lambda _{n}(z)\doteq G_{n}(z)-z={\frac {1}{n}}\sum _{k=1}^{n}\phi \left(G_{k-1,n}(z){\tfrac {k}{n}}\right)\doteq {\frac {1}{n}}\sum _{k=1}^{n}\psi \left(z,{\tfrac {k}{n}}\right)\sim \int _{0}^{1}\psi (z,t)\,dt,}
where the integral is well-defined if d z d t = ϕ ( z , t ) {\displaystyle {\tfrac {dz}{dt}}=\phi (z,t)} has a closed-form solution z(t). Then
λ n ( z 0 ) ≈ ∫ 0 1 ϕ ( z ( t ) , t ) d t . {\displaystyle \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. ϕ ( z , t ) = 2 t − cos y 1 − sin x cos y + i 1 − 2 t sin x 1 − sin x cos y , ∫ 0 1 ψ ( z , t ) d t {\displaystyle \phi (z,t)={\frac {2t-\cos y}{1-\sin x\cos y}}+i{\frac {1-2t\sin x}{1-\sin x\cos y}},\int _{0}^{1}\psi (z,t)\,dt}
Example. Let:
g n ( z ) = z + c n n ϕ ( z ) , with f ( z ) = z + ϕ ( z ) . {\displaystyle g_{n}(z)=z+{\frac {c_{n}}{n}}\phi (z),\quad {\text{with}}\quad 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 ) ) , {\displaystyle T_{1,n}(z)=g_{n}(z),T_{k,n}(z)=g_{n}(T_{k-1,n}(z)),} and Tn(z) = Tn,n(z). Let
T ( z ) = lim n → ∞ T n ( z ) {\displaystyle T(z)=\lim _{n\to \infty }T_{n}(z)}
when that limit exists. The sequence {Tn(z)} defines