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| [[Image:Biholomorphism illustration.svg|right|thumb|The complex [[exponential function]] mapping biholomorphically a rectangle to a quarter-[[annulus (mathematics)|annulus]].]]
| | Nice to satisfy you, my title is Numbers Held though I don't really like becoming called like that. Bookkeeping is what I do. It's not a typical factor but what she likes performing is base jumping and now she is attempting to make cash with it. Years ago we moved to North Dakota.<br><br>my web blog :: [http://Www.gaysphere.net/blog/167593 at home std testing] |
| In the [[mathematics|mathematical theory]] of functions of [[complex analysis|one]] or [[several complex variables|more complex variables]], and also in [[complex algebraic geometry]], a '''biholomorphism''' or '''biholomorphic function''' is a [[bijective]] [[holomorphic function]] whose [[inverse function|inverse]] is also [[holomorphic function| holomorphic]].
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| Formally, a ''biholomorphic function'' is a function <math>\phi</math> defined on an [[open subset]] ''U'' of the <math>n</math>-dimensional complex space '''C'''<sup>''n''</sup> with values in '''C'''<sup>''n''</sup> which is [[holomorphic function| holomorphic]] and [[injective function|one-to-one]], such that its [[image (mathematics)|image]] is an open set <math>V</math> in '''C'''<sup>''n''</sup> and the inverse <math>\phi^{-1}:V\to U</math> is also [[holomorphic function| holomorphic]]. More generally, ''U'' and ''V'' can be [[complex manifold]]s. By applying the chain rule, one may prove that it is enough for <math>\phi</math> to be holomorphic and one-to-one in order for it to be biholomorphic onto its image.
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| If there exists a biholomorphism <math>\phi \colon U \to V</math>, we say that ''U'' and ''V'' are '''biholomorphically equivalent''' or that they are '''biholomorphic'''.
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| If <math>n=1,</math> every [[simply connected]] open set other than the whole complex plane is biholomorphic to the [[unit disc]] (this is the [[Riemann mapping theorem]]). The situation is very different in higher dimensions. For example, open [[unit ball]]s and open unit [[polydisc]]s are not biholomorphically equivalent for <math>n>1.</math> In fact, there does not exist even a [[Proper map|proper]] holomorphic function from one to the other.
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| In the case of the complex plane '''C''', some authors (e.g. Freitag 2009, Definition IV.4.1) define [[conformal map]] as a synonym for ''biholomorphism''. The usual conditions for a function <math>\phi</math> to be conformal on '''C''' - namely, that <math>\phi</math> is holomorphic and that its derivative is nowhere zero - are equivalent to biholomorphism.
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| ==References==
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| * {{cite book | author=Steven G. Krantz | title=Function Theory of Several Complex Variables | publisher=American Mathematical Society | year=2002 | isbn= 0-8218-2724-3}}
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| * {{cite book | author=John P. D'Angelo | title=Several Complex Variables and the Geometry of Real Hypersurfaces | publisher=CRC Press | year=1993 | isbn= 0-8493-8272-6}}
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| * {{cite book | author=Eberhard Freitag and Rolf Busam | title=Complex Analysis | publisher=Springer-Verlag | year=2009 | isbn= 978-3-540-93982-5}}
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| {{PlanetMath attribution|id=6032|title=biholomorphically equivalent}}
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| [[Category:Several complex variables]]
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| [[Category:Algebraic geometry]]
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| [[Category:Complex manifolds]]
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| [[Category:Functions and mappings]]
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Nice to satisfy you, my title is Numbers Held though I don't really like becoming called like that. Bookkeeping is what I do. It's not a typical factor but what she likes performing is base jumping and now she is attempting to make cash with it. Years ago we moved to North Dakota.
my web blog :: at home std testing