Abstract:

We consider convergence sets of formal power series $ f(z,t)=\sum\limits_{n=0}^{\infty} f_n(z)t^n $, where $ f_n(z) $ are holomorphic functions on a domain $ \Omega $ in $ \mathbb{C} $. A subset $ E $ of $ \Omega $ is said to be a convergence set in $ \Omega $ 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 $ \sigma $-convex set is defined to be the union of a countable collection of polynomially convex compact subsets. We prove that a subset of $ \mathbb{C} $ is a convergence set if and only if it is $ \sigma $-convex.

Key words: Formal power series, Analytic functions, Convergence sets