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- Carathéodory's_theorem_(conformal_mapping) abstract "In mathematical complex analysis, Carathéodory's theorem, proved by Carathéodory (1913), states that if U is a simply connected open subset of the complex plane C, whose boundary is a Jordan curve Γ then the Riemann mapf: U → Dfrom U to the unit disk D extends continuously to the boundary, giving a homeomorphismF : Γ → S1from Γ to the unit circle S1.Such a region is called a Jordan domain. Equivalently, this theorem states that for such sets U there is a homeomorphismF : cl(U) → cl(D)from the closure of U to the closed unit disk cl(D) whose restriction to the interior is a Riemann map, i.e. it is a bijective holomorphic conformal map.Another standard formulation of Carathéodory's theorem states that for any pair of simply connected open sets U and V bounded by Jordan curves Γ1 and Γ2, a conformal mapf : U→ Vextends to a homeomorphismF: Γ1 → Γ2.This version can be derived from the one stated above by composing the inverse of one Riemann map with the other.A more general version of the theorem is the following. Letg : DUbe the inverse of the Riemann map, where D ⊂ C is the unit disk, and U ⊂ C is a simply connected domain. Then g extends continuously toG : cl(D) → cl(U)if and only if the boundary of U is locally connected. This result was first stated and proved by Marie Torhorst in her 1918 thesis, under the supervision of Hans Hahn, using Carathéodory's theory of prime ends.".
- Carathéodory's_theorem_(conformal_mapping) thumbnail Erays.png?width=300.
- Carathéodory's_theorem_(conformal_mapping) wikiPageID "2256844".
- Carathéodory's_theorem_(conformal_mapping) wikiPageRevisionID "592810010".
- Carathéodory's_theorem_(conformal_mapping) subject Category:Conformal_mapping.
- Carathéodory's_theorem_(conformal_mapping) subject Category:Homeomorphisms.
- Carathéodory's_theorem_(conformal_mapping) subject Category:Theorems_in_complex_analysis.
- Carathéodory's_theorem_(conformal_mapping) comment "In mathematical complex analysis, Carathéodory's theorem, proved by Carathéodory (1913), states that if U is a simply connected open subset of the complex plane C, whose boundary is a Jordan curve Γ then the Riemann mapf: U → Dfrom U to the unit disk D extends continuously to the boundary, giving a homeomorphismF : Γ → S1from Γ to the unit circle S1.Such a region is called a Jordan domain.".
- Carathéodory's_theorem_(conformal_mapping) label "Carathéodory's theorem (conformal mapping)".
- Carathéodory's_theorem_(conformal_mapping) label "Fortsetzungssatz von Carathéodory".
- Carathéodory's_theorem_(conformal_mapping) label "Принцип соответствия границ".
- Carathéodory's_theorem_(conformal_mapping) sameAs Carath%C3%A9odory's_theorem_(conformal_mapping).
- Carathéodory's_theorem_(conformal_mapping) sameAs Fortsetzungssatz_von_Carathéodory.
- Carathéodory's_theorem_(conformal_mapping) sameAs Q4378889.
- Carathéodory's_theorem_(conformal_mapping) sameAs Q4378889.
- Carathéodory's_theorem_(conformal_mapping) wasDerivedFrom Carathéodory's_theorem_(conformal_mapping)?oldid=592810010.
- Carathéodory's_theorem_(conformal_mapping) depiction Erays.png.
