Frequently,
continued fractions,
series,
products and other infinite expansions can be characterized as infinite
compositions of
analytic functions (ICAF), and the theory evolving from such compositions may shed light on the
convergence/divergence of these expansions. It addition, it is possible to use ICAF to evaluate solutions of
fixed point equations involving infinite expansions.
There are several notations describing infinite expansions including the following: and . Convergence is interpreted as the existence of and .
Most results can be considered extensions of the following contraction theorem for analytic functions:
Let be analytic in a simply-connected region and continuous on the closure of . Suppose is a bounded set contained in . Then , the attractive fixed point of in , for all . [1]
Forward (or inner or right) compositions:
Let be a sequence of functions analytic on a simply-connected domain . Suppose there exists a compact set such that for each n, . Then converges uniformly on to a constant function . [2]
Backward (or outer or left) compositions:
Let be a sequence of functions analytic on a simply-connected domain . Suppose there exists a compact set such that for each n, . Then converges uniformly on to if and only if the sequence of fixed points of the converge to . [3]
Frequently,
continued fractions,
series,
products and other infinite expansions can be characterized as infinite
compositions of
analytic functions (ICAF), and the theory evolving from such compositions may shed light on the
convergence/divergence of these expansions. It addition, it is possible to use ICAF to evaluate solutions of
fixed point equations involving infinite expansions.
There are several notations describing infinite expansions including the following: and . Convergence is interpreted as the existence of and .
Most results can be considered extensions of the following contraction theorem for analytic functions:
Let be analytic in a simply-connected region and continuous on the closure of . Suppose is a bounded set contained in . Then , the attractive fixed point of in , for all . [1]
Forward (or inner or right) compositions:
Let be a sequence of functions analytic on a simply-connected domain . Suppose there exists a compact set such that for each n, . Then converges uniformly on to a constant function . [2]
Backward (or outer or left) compositions:
Let be a sequence of functions analytic on a simply-connected domain . Suppose there exists a compact set such that for each n, . Then converges uniformly on to if and only if the sequence of fixed points of the converge to . [3]