Residue of function at point - Revision history

Vipul: 6 revisions

2008-05-18T19:18:02Z

6 revisions

← Older revision	Revision as of 19:18, 18 May 2008
(No difference)

Vipul: /* Definition */

2008-04-28T23:46:02Z

Definition

← Older revision		Revision as of 23:46, 28 April 2008
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	<math>\lim_{z \to z_0} (z - z_0)f(z)</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>).

Vipul: /* Definition */

2008-04-28T23:43:48Z

Definition

← Older revision		Revision as of 23:43, 28 April 2008
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	* 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 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>:		* 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>		<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>

Vipul at 23:41, 28 April 2008

2008-04-28T23:41:56Z

← Older revision		Revision as of 23:41, 28 April 2008
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	* 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 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>:		* 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>		<math>res(f;z_0) := \frac{1}{2 \pi i} \oint_\gamma f(z) \, dz</math>

Vipul: /* Definition */

2008-04-20T15:04:18Z

Definition

← Older revision		Revision as of 15:04, 20 April 2008
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	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:		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>		* 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 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>:		* 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>		<math>res(f;z_0) := \frac{1}{2 \pi i} \oint_\gamma f(z) \, dz</math>

Vipul: /* Definition */

2008-04-20T15:03:01Z

Definition

← Older revision		Revision as of 15:03, 20 April 2008
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	==Definition==		==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>		* 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>

Vipul: New page: ==Definition== Suppose $U \subset \mathbb{C}$ is an open subset and $f:U \to \mathbb{C}$ be a holomorphic function. Suppose $z_0$ is a point in $...$

2008-04-20T15:02:37Z

New page: ==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>...

New page

==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:

* 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>
* 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>

Residue of function at point - Revision history

Vipul: 6 revisions

Vipul: /* Definition */

Vipul: /* Definition */

Vipul at 23:41, 28 April 2008

Vipul: /* Definition */

Vipul: /* Definition */

Vipul: New page: ==Definition== Suppose U \subset \mathbb{C} is an open subset and f:U \to \mathbb{C} be a holomorphic function. Suppose z_0 is a point in ...

Vipul: New page: ==Definition== Suppose $U \subset \mathbb{C}$ is an open subset and $f:U \to \mathbb{C}$ be a holomorphic function. Suppose $z_0$ is a point in $...$