On Convergence Sets of Power Series with Holomorphic Coefficients

Abstract

We consider convergence sets of formal power series of the form f (z, t) = ∞ n=0 f n (z)t n , where f n (z) are holomorphic functions on a domain Ω in C. A subset E of Ω is said to be a convergence set in Ω if there is a series f (z, t) such that E is exactly the set of points z for which f (z, t) converges as a power series in a single variable t in some neighborhood of the origin. A σ-convex set is defined to be the union of a countable collection of polynomially convex compact subsets. We prove that a subset of C is a convergence set if and only if it is σ-convex.