On meromorphic extendibility

Let D be a bounded domain in the complex plane whose boundary 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 D ¯ which are holomorphic on D. We prove that a continuous function f on bD extends meromorphically through D if and only if there is an N ∈ N ∪ { 0 } such that the change of argument of P f + Q along bD is bounded below by − 2 π N for all P , Q ∈ A ( D ) such that P f + Q ≠ 0 on bD. If this is the case then the meromorphic extension of f has at most N poles in D, counting multiplicity.