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| In [[mathematics]], a '''Fatou–Bieberbach domain''' is a proper subdomain of <math>\mathbb{C}^n</math>, [[biholomorphic]]ally equivalent to <math>\mathbb{C}^n</math>. That is, an open set <math>\Omega \subsetneq \mathbb{C}^n</math> is called a Fatou–Bieberbach domain if there exists a [[bijective]] [[holomorphic]] [[function (mathematics)|function]] <math>f:\Omega \rightarrow \mathbb{C}^n</math> whose [[inverse function]] <math>f^{-1}:\mathbb{C}^n \rightarrow \Omega</math> is holomorphic.
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| == History ==
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| As a consequence of the [[Riemann mapping theorem]], there are no Fatou–Bieberbach domains in the case ''n'' = 1.
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| [[Pierre Fatou]] and [[Ludwig Bieberbach]] first explored such domains in higher dimensions in the 1920s, hence the name given to them later. Since the 1980s, Fatou–Bieberbach domains have again become the subject of mathematical research.
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| == References ==
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| * Fatou, Pierre: "Sur les fonctions méromorphs de deux variables. Sur certains fonctions uniformes de deux variables." ''C.R.'' Paris 175 (1922)
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| * Bieberbach, Ludwig: "Beispiel zweier ganzer Funktionen zweier komplexer Variablen, welche eine schlichte volumtreue Abbildung des <math>\mathcal{R}_4</math> auf einen Teil seiner selbst vermitteln". Preussische Akademie der Wissenschaften. ''Sitzungsberichte'' (1933)
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| * Rosay, J.-P. and Rudin, W: "Holomorphic maps from <math>\mathbb{C}^n</math> to <math>\mathbb{C}^n</math>". ''Trans. A.M.S.'' 310 (1988) [http://www.jstor.org/stable/2001110]
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| {{DEFAULTSORT:Fatou-Bieberbach domain}}
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| [[Category:Several complex variables]]
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| [[Category:Inverse functions]]
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Latest revision as of 19:30, 7 September 2014
Nice to meet you, I am Marvella Shryock. Playing baseball is the pastime he will never stop doing. Her family members life in Minnesota. Hiring is his profession.
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