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Infinite compositions of analytic functions
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<p>In mathematics, infinite <a href="/facts/Function_composition/r5Pvnmth">compositions</a> of <a href="/facts/Analytic_function/JhL3z8FP">analytic functions</a> (ICAF) offer alternative formulations of <a href="/facts/Generalized_continued_fraction/kbj3glnP">analytic continued fractions</a>, <a href="/facts/Series_(mathematics)/yDnlsgIe">series</a>, <a href="/facts/Product_(mathematics)/aXY11nGS">products</a> and other infinite expansions, and the theory evolving from such compositions may shed light on the <a href="/facts/Convergence_(mathematics)/DTok170z">convergence/divergence</a> 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 <a href="/facts/Fixed_point_(mathematics)/vVVddTKc">fixed point</a> equations involving infinite expansions. <a href="/facts/Complex_dynamics/NXHJfR3s">Complex dynamics</a> offers another venue for <a href="/facts/Iterated_function_system/j56g6yEM">iteration of systems of functions</a> rather than a single function. For infinite compositions of a <i>single function</i> see <a href="/facts/Iterated_function/Vzsx64An">Iterated function</a>. For compositions of a finite number of functions, useful in <a href="/facts/Fractal/sXj38JeN">fractal</a> theory, see <a href="/facts/Iterated_function_system/j56g6yEM">Iterated function system</a>. </p><p>Although the title of this article specifies analytic functions, there are results for more general <a href="/facts/Complex_analysis/hgCoQlH0">functions of a complex variable</a> as well. </p>