Meromorphic function: Difference between revisions

Revision as of 23:18, 20 April 2008

Suppose $U$ is an open subset of $\mathbb {C}$ . A meromorphic function on $U$ is defined in the following equivalent ways:

A holomorphic function on $U\setminus D$ , where $D$ is a discrete subset of $U$ , such that all points of $D$ are poles for the function
A holomorphic map from $U$ to the Riemann sphere, that is not the constant map to $\infty$

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 ===On an open subset of complex numbers===
-Suppose <math>U</math> is an open subset of <math>\mathbb{C}</math>. A '''meromorphic function''' on <math>U</math> is defined to be a [[holomorphic function]] on <math>U \setminus D</math>, where <math>D</math> is a discrete subset of <math>U</math>, such that all points of <math>D</math> are [[pole]]s for the function.
+Suppose <math>U</math> is an open subset of <math>\mathbb{C}</math>. A '''meromorphic function''' on <math>U</math> is defined in the following equivalent ways:
+* A [[holomorphic function]] on <math>U \setminus D</math>, where <math>D</math> is a discrete subset of <math>U</math>, such that all points of <math>D</math> are [[pole]]s for the function
+* A [[holomorphic map]] from <math>U</math> to the [[Riemann sphere]], that is not the constant map to <math>\infty</math>