Riemann Mapping Theorem

(Theory)

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Theory since 1900
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What is Riemann mapping theorem?

In complex analysis, 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 This mapping is known as a Riemann mapping. Intuitively, the condition that U be simply connected means that U does not contain any “holes”. The fact that f is biholomorphic implies that it is a conformal map and therefore angle-preserving. Intuitively, such a map preserves the shape of any sufficiently small figure, while possibly rotating and scaling (but not reflecting) it. Henri Poincaré proved that the map f is essentially unique: if z0 is an element of U and φ is an arbitrary angle, then there exists precisely one f as above such that f(z0) = 0 and such that the argument of the derivative of f at the point z0 is equal to φ. This is an easy consequence of the Schwarz lemma. As a corollary of the theorem, any two simply connected open subsets of the Riemann sphere which both lack at least two points of the sphere can be conformally mapped into each other.

Technology Types

bernhard riemannmathematical theorempropositionstatementtheoremtheorems in analysistheorems in complex analysistheory

Synonyms

Reimann mapping theoremRiemann mappingRiemann's Mapping TheoremRiemann's theorem on conformal mappingsSmooth Riemann mapping theorem

Translations

Afbeeldingstelling van Riemann (nl)Riemannscher Abbildungssatz (de)Teorema de representación conforme de Riemann (es)Teorema della mappa di Riemann (it)Teorema do mapeamento conforme de Riemann (pt)Théorème de l'application conforme (fr)Теорема Рімана про відображення (uk)리만 사상 정리 (ko)リーマンの写像定理 (ja)黎曼映射定理 (zh)

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