CONVERGENCE DOMAIN OF SERIES OF HOLOMORPHIC FUNCTIONS AND PROBLEMS OF ANALYTIC CONTINUATION

Abstract

The paper examines the relationship between the convergence domain of a series of holomorphic functions and the domain of holomorphy of its sum - a question resolved differently in one- and several-variable complex analysis. Classical tools (Cauchy-Hadamard formula, Weierstrass and Abel theorems, Hadamard's gap criterion) are combined with the Hartogs extension theorem and plurisubharmonic function techniques. Using an original computed example, it is shown that the convergence domain of a multiple power series can never coincide with a Hartogs figure - it always automatically fills out to a complete, logarithmically convex Reinhardt domain, consistent with and verified against the Hartogs extension theorem. A fundamental qualitative difference between the one- and several-variable settings regarding natural boundaries is demonstrated.