Meromorphic function: Difference between revisions

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{{Short description|Class of mathematical function}}

In the mathematical field of [[complex analysis]], a '''meromorphic function''' on an [[open set|open subset]] ''D'' of the [[complex plane]] is a [[function (mathematics)|function]] that is [[holomorphic function|holomorphic]] on all of ''D'' ''except'' for a set of [[isolated point]]s, which are [[pole (complex analysis)|pole''poles'']]s of the function.<ref name=Hazewinkel_2001>{{cite encyclopedia |editor=Hazewinkel, Michiel |year=2001 |orig-year=1994 |article=Meromorphic function |chapter-url=https://www.encyclopediaofmath.org/index.php?title=p/m063460 |encyclopedia=Encyclopedia of Mathematics |title-link=Encyclopedia of Mathematics |publisher=Springer Science+Business Media B.V.; Kluwer Academic Publishers |ISBN=978-1-55608-010-4}} <!-- {{springer|title=Meromorphic function|id=p/m063460}} --></ref> The term comes from the [[Greek language|Greek]] ''meros'' ([[wikt:μέρος|μέρος]]), meaning "part".{{efn|Greek ''meros'' ([[wikt:μέρος|μέρος]]) means "part", in contrast with the more commonly used ''holos'' ([[wikt:ὅλος|ὅλος]]), meaning "whole".}}

Every meromorphic function on ''D'' can be expressed as the ratio between two [[holomorphic function]]s (with the denominator not constant 0) defined on ''D'': any pole must coincide with a zero of the denominator.