Simply connected domain - Revision history

Vipul: 3 revisions

2008-05-18T19:18:38Z

3 revisions

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

Vipul at 19:12, 26 April 2008

2008-04-26T19:12:47Z

← Older revision		Revision as of 19:12, 26 April 2008
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	* Every holomorphic function is integrable, i.e. has a global primitive		* Every holomorphic function is integrable, i.e. has a global primitive
	* The complement of the domain in the [[Riemann sphere]] is a connected set		* The complement of the domain in the [[Riemann sphere]] is a connected set

			The [[Riemann mapping theorem]] states that any simply connected domain that is not the whole of <math>\mathbb{C}</math> is [[biholomorphically equivalent domains\|biholomorphically equivalent]] to the [[open unit disk]].

Vipul at 20:02, 20 April 2008

2008-04-20T20:02:29Z

← Older revision		Revision as of 20:02, 20 April 2008
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	* Its first homology group is trivial		* Its first homology group is trivial
	* Any cycle in it is zero-homologous: it does not wind around any point in the complement of the domain		* Any cycle in it is zero-homologous: it does not wind around any point in the complement of the domain
			* Every holomorphic function is integrable, i.e. has a global primitive
	* The complement of the domain in the [[Riemann sphere]] is a connected set		* The complement of the domain in the [[Riemann sphere]] is a connected set

Vipul: New page: ==Definition== A domain (open connected subset) in $\mathbb{C}$ is termed a '''simply connected domain''' if it satisfies the following equivalent conditions: * It is simp...

2008-04-20T14:22:55Z

New page: ==Definition== A domain (open connected subset) in <math>\mathbb{C}</math> is termed a '''simply connected domain''' if it satisfies the following equivalent conditions: * It is simp...

New page

==Definition==

A [[domain]] (open connected subset) in <math>\mathbb{C}</math> is termed a '''simply connected domain''' if it satisfies the following equivalent conditions:

* It is simply connected as a topological space i.e. its fundamental group is trivial
* Its first homology group is trivial
* Any cycle in it is zero-homologous: it does not wind around any point in the complement of the domain
* The complement of the domain in the [[Riemann sphere]] is a connected set

Simply connected domain - Revision history

Vipul: 3 revisions

Vipul at 19:12, 26 April 2008

Vipul at 20:02, 20 April 2008

Vipul: New page: ==Definition== A domain (open connected subset) in \mathbb{C} is termed a '''simply connected domain''' if it satisfies the following equivalent conditions: * It is simp...

Vipul: New page: ==Definition== A domain (open connected subset) in $\mathbb{C}$ is termed a '''simply connected domain''' if it satisfies the following equivalent conditions: * It is simp...