Riemann mapping theorem - Revision history

Vipul: 6 revisions

2008-05-18T19:18:10Z

6 revisions

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

Vipul at 20:07, 27 April 2008

2008-04-27T20:07:08Z

← Older revision		Revision as of 20:07, 27 April 2008
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	==Statement==		==Statement==

	Any [[simply connected domain]] in <math>\mathbb{C}</math> (i.e. any open, connected and simply connected subset of <math>\mathbb{C}</math>) that is not the whole of <math>\mathbb{C}</math>, is [[biholomorphically equivalent domains\|bihomolorphically equivalent]] to the [[open unit disk]].		Any [[simply connected domain]] in <math>\mathbb{C}</math> (i.e. any open, connected and simply connected subset of <math>\mathbb{C}</math>) that is not the whole of <math>\mathbb{C}</math>, is [[biholomorphically equivalent domains\|bihomolorphically equivalent]] to the [[open unit disk]]. A biholomorphic mapping from the simply connected domain to the open unit disk is termed a [[Riemann mapping]].

	==Corollaries==		==Corollaries==

Vipul at 00:46, 27 April 2008

2008-04-27T00:46:48Z

← Older revision		Revision as of 00:46, 27 April 2008
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	Any two simply connected domains are homeomorphic (follows from the Riemann mapping theorem and the fact that the unit disc is homeomorphic to <math>\mathbb{C}</math>).		Any two simply connected domains are homeomorphic (follows from the Riemann mapping theorem and the fact that the unit disc is homeomorphic to <math>\mathbb{C}</math>).

			==Facts used==

			* [[Maximum modulus principle]]
			* [[Open mapping theorem]]
	==Proof==		==Proof==

Vipul at 00:45, 27 April 2008

2008-04-27T00:45:57Z

← Older revision		Revision as of 00:45, 27 April 2008
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			{{basic fact}}
	==Statement==		==Statement==

Vipul at 00:45, 27 April 2008

2008-04-27T00:45:33Z

← Older revision		Revision as of 00:45, 27 April 2008
Line 1:		Line 1:
	==Statement==		==Statement==

	Any [[simply connected domain]] in <math>\mathbb{C}</math> (i.e. any open, connected and simply connected subset of <math>\mathbb{C}</math>) that is not the whole of <math>\mathbb{C}</math>, is [[biholomorphically equivalent domains\|bihomolorphically equivalent]] to the open unit ~~disc~~.		Any [[simply connected domain]] in <math>\mathbb{C}</math> (i.e. any open, connected and simply connected subset of <math>\mathbb{C}</math>) that is not the whole of <math>\mathbb{C}</math>, is [[biholomorphically equivalent domains\|bihomolorphically equivalent]] to the [[open unit disk]].

	==Corollaries==		==Corollaries==

	Any two simply connected domains are homeomorphic (follows from the Riemann mapping theorem and the fact that the unit disc is homeomorphic to <math>\mathbb{C}</math>).		Any two simply connected domains are homeomorphic (follows from the Riemann mapping theorem and the fact that the unit disc is homeomorphic to <math>\mathbb{C}</math>).

			==Proof==

			===Proof outline===

			The standard proof of the Riemann mapping theorem has the following three steps:

			# Obtain a conformal mapping from the simply connected domain ''into'' the [[open unit disk]] (we do this by first obtaining a map into the upper half plane, and then using a [[fractional linear transformation]] to pass from the upper half plane to the open unit disk)
			# Maximize among all such mappings with respect to certain criteria. Prove that a maximal mapping must exist.
			# Prove that this maximal conformal mapping is also surjective, and hence, must be a biholomorphic mapping

Vipul at 15:19, 20 April 2008

2008-04-20T15:19:20Z

← Older revision		Revision as of 15:19, 20 April 2008
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	Any [[simply connected domain]] in <math>\mathbb{C}</math> (i.e. any open, connected and simply connected subset of <math>\mathbb{C}</math>) that is not the whole of <math>\mathbb{C}</math>, is [[biholomorphically equivalent domains\|bihomolorphically equivalent]] to the open unit disc.		Any [[simply connected domain]] in <math>\mathbb{C}</math> (i.e. any open, connected and simply connected subset of <math>\mathbb{C}</math>) that is not the whole of <math>\mathbb{C}</math>, is [[biholomorphically equivalent domains\|bihomolorphically equivalent]] to the open unit disc.

			==Corollaries==

			Any two simply connected domains are homeomorphic (follows from the Riemann mapping theorem and the fact that the unit disc is homeomorphic to <math>\mathbb{C}</math>).

Vipul: New page: ==Statement== Any simply connected domain in $\mathbb{C}$ (i.e. any open, connected and simply connected subset of $\mathbb{C}$ ) that is not the whole of $\...$

2008-04-20T15:18:34Z

New page: ==Statement== Any simply connected domain in <math>\mathbb{C}</math> (i.e. any open, connected and simply connected subset of <math>\mathbb{C}</math>) that is not the whole of <math>\...

New page

==Statement==

Any [[simply connected domain]] in <math>\mathbb{C}</math> (i.e. any open, connected and simply connected subset of <math>\mathbb{C}</math>) that is not the whole of <math>\mathbb{C}</math>, is [[biholomorphically equivalent domains|bihomolorphically equivalent]] to the open unit disc.

Riemann mapping theorem - Revision history

Vipul: 6 revisions

Vipul at 20:07, 27 April 2008

Vipul at 00:46, 27 April 2008

Vipul at 00:45, 27 April 2008

Vipul at 00:45, 27 April 2008

Vipul at 15:19, 20 April 2008

Vipul: New page: ==Statement== Any simply connected domain in \mathbb{C} (i.e. any open, connected and simply connected subset of \mathbb{C}) that is not the whole of \...

Vipul: New page: ==Statement== Any simply connected domain in $\mathbb{C}$ (i.e. any open, connected and simply connected subset of $\mathbb{C}$ ) that is not the whole of $\...$