A Characterization of Weierstrass Analytic Functions

Abstract

The sum theorem and other results of dimension theory make it possible to characterize the graphs of Weierstrass analytic functions among the subsets of C² (the set of all ordered pairs of complex numbers) without assuming local complex parametrizations and in fact without any reference to plane topology. A complete Weierstrass function F is a set consisting of a power series and all its analytic continuations. Its graph, ϕ(F), is the set of all points (Z ₀,W ₀) such that Fincludes at least one power series ${{w}_{0}} + {{a}_{1}}(z - {{z}_{0}}) +\ldots+ {{a}_{n}}{{(z - {{z}_{0}})}^{n}} +\ldots$ of positive radius of convergence.