Homotopy-invariance formulation of Cauchy's theorem - Revision history

Vipul: 3 revisions

2008-05-18T19:13:51Z

3 revisions

← Older revision	Revision as of 19:13, 18 May 2008
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Vipul at 18:49, 26 April 2008

2008-04-26T18:49:30Z

← Older revision		Revision as of 18:49, 26 April 2008
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			{{basic fact}}

	==Statement==		==Statement==

	Suppose <math>U \subset \mathbb{C}</math> is an open subset and <math>~~c_1~~, ~~c_2~~</math> are two ~~cycles (cycle being~~ a ~~sum of~~ smooth ~~simple closed curves) that are smoothly homotopic~~. Then, for any [[holomorphic function]] <math>f: U \to \mathbb{C}</math>, we have:		Suppose <math>U \subset \mathbb{C}</math> is an open subset and <math>\gamma_1,\gamma_2</math> are two smooth paths in <math>U</math> such that there is a smooth homotopy between <math>\gamma_1</math> and <math>\gamma_2</math> preserving endpoints. Then, for any [[holomorphic function]] <math>f: U \to \mathbb{C}</math>, we have:

	<math>\oint_{~~c_1~~} f(z) dz = \oint_{~~c_2~~} f(z) dz</math>		<math>\oint_{\gamma_1} f(z) dz = \oint_{\gamma_2} f(z) dz</math>

	In particular, if <math>c</math> is a zero-homologous cycle, we have <math>\oint_c f(z) dz = 0</math>.		In particular, if <math>c</math> is a zero-homologous cycle, we have <math>\oint_c f(z) dz = 0</math>.

Vipul: Homotopy invariance formulation of Cauchy's theorem moved to Homotopy-invariance formulation of Cauchy's theorem

2008-04-26T18:47:21Z

Homotopy invariance formulation of Cauchy's theorem moved to Homotopy-invariance formulation of Cauchy's theorem

← Older revision	Revision as of 18:47, 26 April 2008
(No difference)

Vipul: New page: ==Statement== Suppose $U \subset \mathbb{C}$ is an open subset and $c_1, c_2$ are two cycles (cycle being a sum of smooth simple closed curves) that are smoothly hom...

2008-04-19T20:53:34Z

New page: ==Statement== Suppose <math>U \subset \mathbb{C}</math> is an open subset and <math>c_1, c_2</math> are two cycles (cycle being a sum of smooth simple closed curves) that are smoothly hom...

New page

==Statement==

Suppose <math>U \subset \mathbb{C}</math> is an open subset and <math>c_1, c_2</math> are two cycles (cycle being a sum of smooth simple closed curves) that are smoothly homotopic. Then, for any [[holomorphic function]] <math>f: U \to \mathbb{C}</math>, we have:

<math>\oint_{c_1} f(z) dz = \oint_{c_2} f(z) dz</math>

In particular, if <math>c</math> is a zero-homologous cycle, we have <math>\oint_c f(z) dz = 0</math>.

==Related facts==

* [[Goursat's integral lemma]]: It states something very similar, albeit in a very special case: where <math>c</math> forms the smooth boundary of a region.

Homotopy-invariance formulation of Cauchy's theorem - Revision history

Vipul: 3 revisions

Vipul at 18:49, 26 April 2008

Vipul: Homotopy invariance formulation of Cauchy's theorem moved to Homotopy-invariance formulation of Cauchy's theorem

Vipul: New page: ==Statement== Suppose U \subset \mathbb{C} is an open subset and c_1, c_2 are two cycles (cycle being a sum of smooth simple closed curves) that are smoothly hom...

Vipul: New page: ==Statement== Suppose $U \subset \mathbb{C}$ is an open subset and $c_1, c_2$ are two cycles (cycle being a sum of smooth simple closed curves) that are smoothly hom...