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If it wasn't discrete you could simply choose the complex plane as set and every function is meromorphic |
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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 discrete set of [[isolated point]]s, which are [[pole (complex analysis)|pole]]s of the function. This terminology comes from the [[Ancient Greek]] ''meros'' ([[wikt:μέρος|μέρος]]), meaning "part," as opposed to ''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.
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