Domains of Holomorphy

TL;DR

The paper presents a simplified proof of the equivalence between Domains of Holomorphy and Weak Domains of Holomorphy, utilizing classical theorems, and explores generalizations based on specific classes of holomorphic functions.

Contribution

It provides an elementary proof of the equivalence and introduces generalizations considering particular classes of holomorphic functions.

Findings

Equivalence of Domain of Holomorphy and Weak Domain of Holomorphy proved using Baire's and Montel's Theorems

Generalizations for classes of holomorphic functions that do not extend beyond the domain

Example of a domain of holomorphy for all holomorphic functions but not for extendable ones

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.