Any simply connected open subset of the complex plane that is not the whole plane can be conformally mapped, in a bijective and angle-preserving way, onto the open unit disk. Stated by Bernhard Riemann, it is a foundational result of complex analysis and conformal geometry.
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StatementThe Riemann mapping theorem states that if U is a non-empty simply connected open subset of the complex number plane C which is not all of C, then there exists a biholomorphic mapping f from U onto the open unit disk D. 1 Classification
Statement Form Riemann Mapping Theorem
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1. Riemann Mapping Theorem (Wikipedia)
Wikimedia Foundationlead section, first paragraph
the Riemann mapping theorem states that if U is a non-empty simply connected open subset of the complex number plane C which is not all of C, then there exists a biholomorphic mapping f (i.e. a bijective holomorphic mapping whose inverse is also holomorphic) from U onto the open unit disk D
History section, Osgood proof sentence
The first rigorous proof of the theorem was given by William Fogg Osgood in 1900.
View the Source Wikipedia: Bernhard Riemann
WikipediaPrincipal Result Of: Bernhard Riemann, Article textQuote, Principal Result Of: Bernhard Riemann, Article text
The famous Riemann mapping theorem says that a simply connected domain in the complex plane is "biholomorphically equivalent" (i.e.
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