Residue of function at point: Difference between revisions

Latest revision as of 19:18, 18 May 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 $z_{0}$
It is given by the following formula, where $γ$ is a small counter-clockwise circular loop about $z_{0}$ that lies completely inside $U$ :

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

If the following limit is a finite complex number, then that complex number equals the residue at $z_{0}$ :

$lim_{z \to z_{0}} (z - z_{0}) f (z)$

(the limit is zero iff the function is holomorphic at $z_{0}$ , and is finite nonzero iff it has a simple pole at $z_{0}$ ).

@@ Line 4: / Line 4: @@
 * It is the coefficient of <math>1/(z - z_0)</math> in the Laurent series expansion of <math>f</math> about <math>z_0</math>
-* It is given as a limit:
-<math>\lim_{z \to z_0} (z - z_0)f(z)</math>
 * It is given by the following formula, where <math>\gamma</math> is a small counter-clockwise circular loop about <math>z_0</math> that lies completely inside <math>U</math>:
 <math>res(f;z_0) := \frac{1}{2 \pi i} \oint_\gamma f(z) \, dz</math>
+If the following limit is a finite complex number, then that complex number equals the residue at <math>z_0</math>:
+<math>\lim_{z \to z_0} (z - z_0)f(z)</math>
+(the limit is zero iff the function is [[holomorphic function|holomorphic]] at <math>z_0</math>, and is finite nonzero iff it has a [[simple pole]] at <math>z_0</math>).