Compositions of analytic functions of the form Fn(z) = Fn−1(fn(z)), fn(z) → f(z)

摘要

Frequently, in applications, a function is iterated in order to determine its fixed point, which represents the solution of some problem. In the variation of iteration presented in this paper fixed points serve a different purpose. The sequence {Fn(z)} is studied, where F1(z) = f1(z) and Fn(z) = Fn−1(fn(z)), with fn → f. Many infinite arithmetic expansions exhibit this form, and the fixed point, α, of f may be used as a modifying factor (z = α) to influence the convergence behaviour of these expansions. Thus one employs, rather than seeks the fixed point of the function f.