In mathematics, infinite compositions of analytic functions offer alternative formulations of analytic continued fractions, series and other infinite expansions, the theory evolving from such compositions may shed light on the convergence/divergence of these expansions. Some functions can 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. 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. There are several notations describing infinite compositions, including the following: Forward compositions: Fk,n = fk ∘ fk+1 ∘... ∘ fn−1 ∘ fn. Backward compositions: Gk,n = fn ∘ fn−1 ∘... ∘ fk+1 ∘ fk In each case convergence is interpreted as the existence of the following limits: lim n → ∞ F 1, n, lim n → ∞ G 1, n.
For convenience, set Fn = F1,n and Gn = G1,n. One may write F n = R n k = 1 f k = f 1 ∘ f 2 ∘ ⋯ ∘ f n and G n = L n k = 1 g k = g n ∘ g n − 1 ∘ ⋯ ∘ g 1 Many results can be considered extensions of the following result: Contraction Theorem for Analytic Functions. Let f be analytic in a simply-connected region S and continuous on the closure S of S. Suppose f is a bounded set contained in S. For all z in S there exists an attractive fixed point α of f in S such that: F n = → α, 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, fn ⊂ Ω. Forward Compositions Theorem. Converges uniformly on compact subsets of S to a constant function F = λ. Backward Compositions Theorem. Converges uniformly on compact subsets of S to γ ∈ Ω if and only if the sequence of fixed points of the converges to γ. Additional theory resulting from investigations based on these two theorems Forward Compositions Theorem, include location analysis for the limits obtained here.
For a different approach to Backward Compositions Theorem, see. Regarding Backward Compositions Theorem, the example f2n = 1/2 and f2n−1 = −1/2 for S = demonstrates the inadequacy of requiring contraction into a compact subset, like Forward Compositions Theorem. For functions not analytic the Lipschitz condition suffices: Theorem. Suppose S is a connected compact subset of C and let t n: S → S be a family of functions that satisfies ∀ n, ∀ z 1, z 2 ∈ S, ∃ ρ: | t n − t n | ≤ ρ | z 1 − z 2 |, ρ < 1. Define: G n = F n = ( t 1
Egyptology is the fourth studio album by World Party released in 1997, re-released in 2006. It contained the #31 British single "Beautiful Dream" and the award-winning She's the One, among other songs, but the album was not a commercial success, Karl Wallinger was upset when his label, used "She's the One" as a vehicle for pop artist Robbie Williams. Wallinger wrote: I was so lucky that Robbie recorded "She's the One" because it allowed me to keep going, he gave me enough bacon to live on for four years. He kept my kids in school and me for that I thank him. Due in part to the disagreement over "She's the One", Egyptology would be Wallinger's last album with Chrysalis. "It Is Time" "Beautiful Dream" "Call Me Up" "Vanity Fair" "She's the One" "Vocal Interlude" "Curse of the Mummy's Tomb" "Hercules" "Love Is Best" "Rolling Off a Log" "Strange Groove" "The Whole of the Night" "Piece of Mind" "This World" "Always" Karl Wallinger: All instruments except where noted Chris Sharrock: drums, northern vibes Johnson Somerset: loops on tracks 11 and 15 Anthony Thistlethwaite: "additional massed saxes" on track 3 John Turnbull: guitar on track 12