Domains of Holomorphy

Abstract

We give a simple and more elementary proof that the notions of Domain of Holomorphy and Weak Domain of Holomorphy are equivalent. This proof is based on a combination of Baire's Category Theorem and Montel's Theorem. We also obtain generalizations by demanding that the non-extentable functions belong to a particular class of holomorphic functions in the domain. We give an example of a domain in the plane which is a domain of holomorphy with respect to the class of all holomorphic functions but not with respect to the class of holomorphic functions continuously extendable on the closure of the domain.