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arXiv · 0810.0601

On meromorphic extendibility

Josip Globevnik
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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.

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Josip Globevnik. 2008-10-03. On meromorphic extendibility. https://arxiv.org/abs/0810.0601

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