Residue theorem: Difference between revisions
(New page: ==Statement== Suppose <math>U</math> is an open subset of <math>\mathbb{C}</math>. Let <math>z_1, z_2, \ldots, z_r</math> be points in <math>U</math> and <math>f:U \setminus \{ z_j\}_{1 \...) |
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==Statement== | ==Statement== | ||
Suppose <math>U</math> is an open subset of <math>\mathbb{C}</math>. Let <math>z_1, z_2, \ldots, z_r</math> be points in <math>U</math> and <math>f:U \setminus \{ z_j\}_{1 \le j \le r} \to \mathbb{C}</math> be a [[holomorphic function]]. Let <math>c</math> be a 0-homologous sum of loops in <math>U \setminus \{ z_j\}_{1 \le j \le r}</math> such that <math>c</math> is zero-homologous. Then, we have: | Suppose <math>U</math> is an open subset of <math>\mathbb{C}</math>. Let <math>z_1, z_2, \ldots, z_r</math> be points in <math>U</math> and <math>f:U \setminus \{ z_j\}_{1 \le j \le r} \to \mathbb{C}</math> be a [[holomorphic function]]. Let <math>c</math> be a 0-homologous sum of loops in <math>U \setminus \{ z_j\}_{1 \le j \le r}</math> such that <math>c</math> is zero-homologous. Then, we have: | ||
<math>\oint_c f(z) \, dz = \sum_{j=1}^r n(c;z_j)res(f; z_j)</math> | <math>\oint_c f(z) \, dz = \sum_{j=1}^r n(c;z_j)\operatorname{res}(f; z_j)</math> | ||
The residue <math>res</math> here denotes the coefficient of <math>1/(z - z_j)</math> in the laurent expansion about <math>z_j</math>. | The residue <math>\operatorname{res}</math> here denotes the coefficient of <math>1/(z - z_j)</math> in the laurent expansion about <math>z_j</math>. | ||
==Related facts== | ==Related facts== | ||
* [[Winding number version of Cauchy integral formula]] is a special case of this, where the function has precisely one pole of order one. | * [[Winding number version of Cauchy integral formula]] is a special case of this, where the function has precisely one pole of order one. However, we typically ''use'' the winding number version of Cauchy integral formula to establish the residue theorem. | ||
Latest revision as of 19:18, 18 May 2008
This article gives the statement, and possibly proof, of a basic fact in complex analysis.
View a complete list of basic facts in complex analysis
Statement
Suppose is an open subset of . Let be points in and be a holomorphic function. Let be a 0-homologous sum of loops in such that is zero-homologous. Then, we have:
The residue here denotes the coefficient of in the laurent expansion about .
Related facts
- Winding number version of Cauchy integral formula is a special case of this, where the function has precisely one pole of order one. However, we typically use the winding number version of Cauchy integral formula to establish the residue theorem.