Residue of function at point: Difference between revisions

Revision as of 15:03, 20 April 2008

Definition

Suppose $U \subset C$ is an open subset and $f : U \to C$ be a holomorphic function. Suppose $z_{0}$ is a point in $C ∖ U$ such that there exists an open neighborhood $V ∋ z_{0}$ such that $V ∖ z_{0} \subset U$ (in other words, $z_{0}$ is an isolated singularity of $f$ ). The residue of $f$ at $z_{0}$ is defined in the following equivalent ways:

It is the coefficient of $1 / (z - z_{0})$ in the Laurent series expansion of $f$ about <mtah>z_0</math>
It is given by the following formula, where $γ$ is a small counter-clockwise circular loop about $z_{0}$ that lies completely inside $U$ :

$r e s (f; z_{0}) : = \frac{1}{2 π i} \oint_{γ} f (z) d z$

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 ==Definition==
-Suppose <math>U \subset \mathbb{C}</math> is an open subset and <math>f:U \to \mathbb{C}</math> be a [[holomorphic function]]. Suppose <math>z_0</math> is a point in <math>\C \setminus U</math> such that there exists an open neighborhood <math>V \ni z_0</math> such that <math>V \setminus z_0 \subset U</math> (in other words, <math>z_0</math> is an isolated singularity of <math>f</math>). The '''residue''' of <math>f</math> at <math>z_0</math> is defined in the following equivalent ways:
+Suppose <math>U \subset \mathbb{C}</math> is an open subset and <math>f:U \to \mathbb{C}</math> be a [[holomorphic function]]. Suppose <math>z_0</math> is a point in <math>\mathbb{C} \setminus U</math> such that there exists an open neighborhood <math>V \ni z_0</math> such that <math>V \setminus z_0 \subset U</math> (in other words, <math>z_0</math> is an isolated singularity of <math>f</math>). The '''residue''' of <math>f</math> at <math>z_0</math> is defined in the following equivalent ways:
 * It is the coefficient of <math>1/(z - z_0)</math> in the Laurent series expansion of <math>f</math> about <mtah>z_0</math>