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| In [[mathematics]], especially [[several complex variables]], an open subset <math>G</math> of '''C'''<sup>n</sup> is called '''Reinhardt domain''' if <math>(z_1, \dots, z_n) \in G</math> implies <math>(e^{i\theta_1} z_1, \dots, e^{i\theta_n} z_n) \in G</math> for all real numbers <math>\theta_1, \dots, \theta_n</math>.
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| The reason for studying these kinds of domains is that [[logarithmically convex set|logarithmically convex]] Reinhardt domain are the [[domain of convergence|domains of convergence]] of [[power series]] in several complex variables. Note that in one complex variable, a logarithmically convex Reinhardt domain is simply a [[disk (mathematics)|disc]].
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| The intersection of logarithmically convex Reinhardt domains is still a logarithmically convex Reinhardt domain, so for every Reinhardt domain, there is a smallest logarithmically convex Reinhardt domain which contains it.
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| A simple example of logarithmically convex Reinhardt domains is a [[polydisc]], that is, a product of disks.
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| Thullen's classical result says that a 2-dimensional bounded Reinhard domain containing the origin is biholomorphic to one of the following domains provided that the orbit of the origin by the automophism group has positive dimension:
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| (1) <math>\{(z,w)\in \mathbf{C}^2;~|z|<1,~|w|<1\}</math> (polydisc);
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| (2) <math>\{(z,w)\in \mathbf{C}^2;~|z|^2+|w|^2<1\}</math> (unit ball);
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| (3) <math>\{(z,w)\in \mathbf{C}^2;~|z|^2+|w|^{2/p}<1\} (p>0,\neq 1)</math> (Thullen domain).
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| In 1978, [[Toshikazu Sunada]] established a generalization of Thullen's result, and proved that two <math>n</math>-dimensional bounded Reinhardt domains <math>G_1</math> and <math>G_2</math> are mutually biholomorphic if and only if there exists a transformation <math>\varphi:\mathbf{C}^n\longrightarrow \mathbf{C}^n</math> given by
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| <math>z_i\mapsto r_iz_{\sigma(i)} (r_i>0)</math>, <math>\sigma</math> being a
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| permutation of the indices), such that <math>\varphi(G_1)=G_2</math>.
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| == References ==
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| * {{PlanetMath attribution|id=6029|title=Reinhardt domain}}
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| * [[Lars Hörmander]]. ''An Introduction to Complex Analysis in Several Variables,'' North-Holland Publishing Company, New York, New York, 1973.
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| * T.Sunada, Holomorphic equivalence problem for bounded Reinhaldt domains, ''Math. Ann.'' '''235''' (1978), 111-128
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| [[Category:Several complex variables]]
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| {{mathanalysis-stub}}
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