Publicacions Matemàtiques

Meromorphic extendibility and the argument principle

Josip Globevnik

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Abstract

Let $\Delta $ be the open unit disc in $\mathbb{C}$. Given a continuous function $\varphi \colon b\Delta \rightarrow \mathbb{C}\setminus \{ 0\}$ denote by $\mathcal{W} (\varphi )$ the winding number of $\varphi$ around the origin. We prove that a continuous function $f\colon b\Delta\rightarrow \mathbb{C}$ extends meromorphically through $\Delta $ if and only if there is a number $N\in \mathbb{N}\cup\{ 0\}$ such that $\mathcal{W} (Pf+Q)\geq -N$ for every pair $P$, $Q$ of polynomials such that $Pf+Q\not= 0$ on $b\Delta$. If this is the case then the meromorphic extension has at most $N$ poles in $\Delta$.

Article information

Source
Publ. Mat., Volume 52, Number 1 (2008), 171-188.

Dates
First available in Project Euclid: 17 December 2007

Permanent link to this document
https://projecteuclid.org/euclid.pm/1197908701

Mathematical Reviews number (MathSciNet)
MR2384845

Zentralblatt MATH identifier
1149.30031

Subjects
Primary: 30E25: Boundary value problems [See also 45Exx]

Keywords
Argument principle meromorphic extensions.

Citation

Globevnik, Josip. Meromorphic extendibility and the argument principle. Publ. Mat. 52 (2008), no. 1, 171--188. https://projecteuclid.org/euclid.pm/1197908701


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