Reinhardt domain

Definition.

We call an open set $G\\subset{\\mathbb{C}}^{n}$ a Reinhardt domainif $(z_{1},\\ldots,z_{n})\\in G$ implies that $(e^{i\\theta_{1}}z_{1},\\ldots,e^{i\\theta_{n}}z_{n})\\in G$ for all real $\\theta_{1},\\ldots,\\theta_{n}$ .

The reason for studying these kinds of domains is thatlogarithmically convex (http://planetmath.org/LogarithmicallyConvexSet)Reinhardt domain are the domains of convergence of power series inseveral complex variables. Note that in one complex variable, aReinhardt domain is just a disc.

Note that the intersection ofReinhardt domains is still aReinhardt domain, so for every Reinhardt domain, there is a smallestReinhardt domain which contains it.

Theorem.

Suppose that $G$ is a Reinhardt domain which contains 0 andthat $\\tilde{G}$ is the smallestReinhardt domain such that $G\\subset\\tilde{G}$ . Thenany function holomorphic on $G$ has a holomorphic to $\\tilde{G}$ .

It actually turns out that aReinhardt domain is a domain of convergence.

examples ofReinhardt domains in ${\\mathbb{C}}^{n}$ are polydiscs such as $\\underbrace{{\\mathbb{D}}\\times\\cdots\\times{\\mathbb{D}}}_{n}$ where ${\\mathbb{D}}\\subset{\\mathbb{C}}$ is the unit disc.

References

1 Lars Hörmander.,North-Holland Publishing Company, New York, New York, 1973.
2 Steven G. Krantz.,AMS Chelsea Publishing, Providence, Rhode Island, 1992.

单词	ReinhardtDomain
释义	Reinhardt domain Definition. We call an open set $G\\subset{\\mathbb{C}}^{n}$ a Reinhardt domainif $(z_{1},\\ldots,z_{n})\\in G$ implies that $(e^{i\\theta_{1}}z_{1},\\ldots,e^{i\\theta_{n}}z_{n})\\in G$ for all real $\\theta_{1},\\ldots,\\theta_{n}$ . The reason for studying these kinds of domains is thatlogarithmically convex (http://planetmath.org/LogarithmicallyConvexSet)Reinhardt domain are the domains of convergence of power series inseveral complex variables. Note that in one complex variable, aReinhardt domain is just a disc. Note that the intersection ofReinhardt domains is still aReinhardt domain, so for every Reinhardt domain, there is a smallestReinhardt domain which contains it. Theorem. Suppose that $G$ is a Reinhardt domain which contains 0 andthat $\\tilde{G}$ is the smallestReinhardt domain such that $G\\subset\\tilde{G}$ . Thenany function holomorphic on $G$ has a holomorphic to $\\tilde{G}$ . It actually turns out that aReinhardt domain is a domain of convergence. examples ofReinhardt domains in ${\\mathbb{C}}^{n}$ are polydiscs such as $\\underbrace{{\\mathbb{D}}\\times\\cdots\\times{\\mathbb{D}}}_{n}$ where ${\\mathbb{D}}\\subset{\\mathbb{C}}$ is the unit disc. References 1 Lars Hörmander.,North-Holland Publishing Company, New York, New York, 1973. 2 Steven G. Krantz.,AMS Chelsea Publishing, Providence, Rhode Island, 1992.
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