One valued primitives and the F. and M. Riesz theorem

TL;DR

This paper characterizes when holomorphic functions on non-simply connected domains have multi-valued primitives, linking this to their extension to the simply connected envelope, and generalizes the F. and M. Riesz theorem.

Contribution

It establishes a criterion for the existence of multi-valued primitives based on holomorphic extension, generalizing the classical F. and M. Riesz theorem.

Findings

Functions admit multi-valued primitives iff they extend holomorphically to the envelope

Generalization of the F. and M. Riesz theorem to non-simply connected domains

Provides a new perspective on holomorphic extension and primitives

Abstract

For a non-simply connected domain Omega in C and f a holomorphic function on Omega we prove that f admits one-valued primitives of any order in Omega, if and only if, it extends holomorphically in the simply connected envelope of Omega . This leads to a generalization of the F. and M. Riesz theorem.