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Concept
Infinite compositions of analytic functions
Résumé
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.
There are several notations describing infinite compositions, including the following:
Forward compositions:
Backward compositions:
In each case convergence is interpreted as the existence of the following limits:
For convenience, set Fn(z) = F1,n(z) and Gn(z) = G1,n(z).
One may also write and
Many results can be considered extensions of the following result:
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:
Results involving entire functions include the following, as examples.
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