Reinhardt domain

multiple-circled domain

A domain $ D $ in the complex space $ \mathbf C ^ {n} $, $ n \geq 1 $, with centre at a point $ a = ( a _ {1} \dots a _ {n} ) \in \mathbf C ^ {n} $, with the following property: Together with any point $ z ^ {0} = ( z _ {1} ^ {0} \dots z _ {n} ^ {0} ) \in D $, the domain also contains the set

$$ \{ z = ( z _ {1} \dots z _ {n} ): | z _ \nu - a _ \nu | = | z _ \nu ^ {0} - a _ \nu |,\ \nu = 1 \dots n \} . $$

A Reinhardt domain $ D $ with $ a = 0 $ is invariant under the transformations $ \{ z ^ {0} \} \rightarrow \{ z _ \nu ^ {0} e ^ {i \theta _ \nu } \} $, $ 0 \leq \theta _ \nu < 2 \pi $, $ \nu = 1 \dots n $. The Reinhardt domains constitute a subclass of the Hartogs domains (cf. Hartogs domain) and a subclass of the circular domains, which are defined by the following condition: Together with any $ z ^ {0} \in D $, the domain contains the set

$$ \{ {z = ( z _ {1} \dots z _ {n} ) } : { z = a + ( z ^ {0} - a) e ^ {i \theta } ,\ 0 \leq \theta < 2 \pi } \} , $$

i.e. all points of the circle with centre $ a $ and radius $ | z ^ {0} - a | = ( \sum _ {\nu = 1 } ^ {n} | z _ \nu ^ {0} - a _ \nu | ^ {2} ) ^ {1/2} $ that lie on the complex line through $ a $ and $ z ^ {0} $.

A Reinhardt domain $ D $ is called a complete Reinhardt domain if together with any point $ z ^ {0} \in D $ it also contains the polydisc

$$ \{ {z = ( z _ {1} \dots z _ {n} ) } : { | z _ \nu - a _ \nu | \leq | z _ \nu ^ {0} - a _ \nu |,\ \nu = 1 \dots n } \} . $$

A complete Reinhardt domain is star-like with respect to its centre $ a $( cf. Star-like domain).

Examples of complete Reinhardt domains are balls and polydiscs in $ \mathbf C ^ {n} $. A circular domain $ D $ is called a complete circular domain if together with any pont $ z ^ {0} \in D $ it also contains the entire disc $ \{ {z = a + ( z ^ {0} - a) \zeta } : {| \zeta | \leq 1 } \} $.

A Reinhardt domain $ D $ is called logarithmically convex if the image $ \lambda ( D ^ {*} ) $ of the set

$$ D ^ {*} = \ \{ {z= ( z _ {1} \dots z _ {n} ) \in D } : { z _ {1} \dots z _ {n} \neq 0 } \} $$

under the mapping

$$ \lambda : z \rightarrow \lambda ( z) = \ ( \mathop{\rm ln} | z _ {1} | \dots \mathop{\rm ln} | z _ {n} | ) $$

is a convex set in the real space $ \mathbf R ^ {n} $. An important property of logarithmically-convex Reinhardt domains is the following: Every such domain in $ \mathbf C ^ {n} $ is the interior of the set of points of absolute convergence (i.e. the domain of convergence) of some power series in $ z _ {1} - a _ {1} \dots z _ {n} - a _ {n} $, and conversely: The domain of convergence of any power series in $ z _ {1} \dots z _ {n} $ is a logarithmically-convex Reinhardt domain with centre $ a = 0 $.

References

[1]	V.S. Vladimirov, "Methods of the theory of functions of many complex variables" , M.I.T. (1966) (Translated from Russian)
[2]	B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1985) (In Russian)

Comments

The final paragraph reduces to: A Reinhardt domain is a domain of holomorphy if and only if it is logarithmically convex.

References

[a1]	L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973)
[a2]	R.M. Range, "Holomorphic functions and integral representation in several complex variables" , Springer (1986)