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| [[Image:Erays.png|right|thumb|[[simply connected]] [[Julia set]] as an image of [[unit circle]] under [[Riemann mapping theorem|Riemann map]]
| | Wilber Berryhill is what his wife loves to call him and he totally loves this name. For a while I've been in Alaska but I will have to move in a year or two. Office supervising is my profession. To perform lacross is some thing I really enjoy doing.<br><br>my page spirit messages - [http://ustanford.com/index.php?do=/profile-38218/info/ Highly recommended Reading] - |
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| In mathematical complex analysis, '''[[Carathéodory]]'s theorem''', proved by {{harvtxt|Carathéodory|1913}},<ref>{{Citation | last1=Carathéodory | first1=C. | title=Über die gegenseitige Beziehung der Ränder bei der konformen Abbildung des Inneren einer Jordanschen Kurve auf einen Kreis | publisher=Springer Berlin / Heidelberg | doi=10.1007/BF01456720 | jfm=44.0757.01 | year=1913 | journal=[[Mathematische Annalen]] | issn=0025-5831 | volume=73 | pages=305–320 | issue=2}}</ref> states that if ''U'' is a [[simply connected]] open subset of the [[complex plane]] '''C''', whose [[boundary (topology)|boundary]] is a [[Jordan curve]] Γ then the [[Riemann mapping theorem|Riemann map]]
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| :''f'': ''U'' → ''D''
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| from ''U'' to the [[unit disk]] ''D'' extends [[continuity (mathematics)|continuously]] to the boundary, giving a [[homeomorphism]]
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| :''F'' : Γ → ''S''<sup>1</sup>''
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| from ''Γ'' to the [[unit circle]] ''S<sup>1</sup>''.
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| Such a region is called a ''Jordan domain''. Equivalently, this theorem states that for such sets ''U'' there is a [[homeomorphism]]
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| :''F'' : cl(''U'') → cl(''D'')
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| from the [[closure (mathematics)|closure]] of ''U'' to the closed unit disk ''cl(D)'' whose [[function (mathematics)|restriction]] to the [[interior (topology)|interior]] is a Riemann map, i.e. it is a [[bijective]] [[holomorphic function|holomorphic]] [[conformal map]].
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| Another standard formulation of Carathéodory's theorem states that for any pair of [[fundamental group|simply connected]] open sets ''U'' and ''V'' bounded by Jordan curves Γ<sub>1</sub> and Γ<sub>2</sub>, a conformal map
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| :''f'' : ''U''→ ''V''
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| extends to a homeomorphism
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| :''F'': Γ<sub>1</sub> → Γ<sub>2</sub>''.
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| This version can be derived from the one stated above by composing the inverse of one Riemann map with the other.
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| A more general version of the theorem is the following. Let
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| :''g'' : ''D''<math>\to</math> ''U''
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| be 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 to
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| :''G'' : cl(''D'') → cl(''U'')
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| 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,<ref>{{Citation | last1=Torhorst | first1= Marie | title=Über den Rand der einfach zusammenhängenden ebenen Gebiete | journal=Mathematische Zeitschrift | year = 1921 | volume=9 | issue=1-2 |
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| pages = 44–65 | doi = 10.1007/BF01378335}}</ref> under the supervision of [[Hans Hahn (mathematician)|Hans Hahn]], using Carathéodory's theory of [[prime end]]s.
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| ==Context==
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| Intuitively, Carathéodory's theorem says that compared to general simply connected open sets in the complex plane '''C''', those bounded by Jordan curves are particularly [[well-behaved]].
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| Carathéodory's theorem is a basic result in the study of ''boundary behavior of conformal maps'', a classical part of complex analysis. In general it is very difficult to decide whether or not the Riemann map from an open set ''U'' to the unit disk ''D'' extends continuously to the boundary, and how and why it may fail to do so at certain points.
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| While having a Jordan curve boundary is ''sufficient'' for such an extension to exist, it is by no means ''necessary ''. For example, the map
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| :''f''(''z'') = ''z''<sup>2</sup>''
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| from the [[upper half-plane]] '''H''' to the open set ''G'' that is the [[complement (set theory)|complement]] of the positive real [[coordinate axis|axis]] is holomorphic and conformal, and it extends to a continuous map from the real line '''R''' to the positive real axis '''R<sup>+</sup>'''; however, the set ''G'' is not [[boundary (topology)|bounded]] by a Jordan curve.
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| ==References==
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| {{Reflist}}
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| ===Further References===
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| *{{Citation | last1=Markushevich | first1=A. I. | title=Theory of functions of a complex variable. Vol. III | mr=0444912 | year=1977 | publisher=Chelsea Publishing Co. | isbn=0-8284-0296-5}}
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| *{{Citation | last1=Shields | first1=Allen | title=Carathéodory and conformal mapping | doi=10.1007/BF03023846 | mr=918659 | year=1988 | journal=[[The Mathematical Intelligencer]] | issn=0343-6993 | volume=10 | issue=1 | pages=18–22}}
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| {{DEFAULTSORT:Caratheodory's Theorem (Conformal Mapping)}}
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| [[Category:Conformal mapping]]
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| [[Category:Homeomorphisms]]
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| [[Category:Theorems in complex analysis]]
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Wilber Berryhill is what his wife loves to call him and he totally loves this name. For a while I've been in Alaska but I will have to move in a year or two. Office supervising is my profession. To perform lacross is some thing I really enjoy doing.
my page spirit messages - Highly recommended Reading -