On meromorphic extendibility

TL;DR

This paper characterizes when a boundary function on a bounded domain in the complex plane can be extended meromorphically inside, using the argument principle and winding number conditions, and relates poles to holomorphic extendibility.

Contribution

It provides a new criterion for meromorphic extendibility based on the argument principle and winding numbers, linking poles to holomorphic extension properties.

Findings

Meromorphic extendibility characterized by winding number bounds.

Extension has at most N poles, where N is determined by the boundary data.

Reduction of meromorphic extension problem to holomorphic extension question.

Abstract

Let D be a bounded domain in the complex plane whose boundary bD consists of finitely many pairwise disjoint real analytic simple closed curves. Let f be an integrable function on bD. In the paper we show how to compute the candidates for poles of a meromorphic extension of f through D and thus reduce the question of meromorphic extendibility to the question of holomorphic extendibility. Let A(D) be the algebra of all continuous functions on the closure of D which are holomorphic on D. For continuous functions f on bD we obtain a characterization of meromorphic extendibility in terms of the argument principle: f extends meromorphically through D if and only if there is a nonnegative integer N such that the winding number of Pf+Q along bD is bounded below by -N for all P, Q in A(D) such that Pf+Q has no zero on bD. If this is the case then the meromorphic extension of f has at most N poles in D, counting multiplicity.