# 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)