# Bicylindrical domain

A domain $ D $
in the complex space $ \mathbf C ^ {2} $
that can be represented in the form of the Cartesian product of two planar domains $ D _ {1} $
and $ D _ {2} $,
i.e.

$$ D = \{ {(z _ {1} , z _ {2} ) } : { z _ {1} \in D _ {1} , z _ {2} \in D _ {2} } \} . $$

A special case of a bicylindrical domain is the bidisc (bicylinder) $ B(a, r) = \{ {(z _ {1} , z _ {2} ) } : {| z _ {1} - a _ {1} | < r _ {1} , | z _ {2} - a _ {2} | < r _ {2} } \} $ of radius $ r = (r _ {1} , r _ {2} ) $ with centre at $ a = (a _ {1} , a _ {2} ) $. The Cartesian product of $ n $( for $ n \geq 3 $) planar domains is said to be a polycylindrical domain. A polydisc (polycylinder) is defined in a similar way.

**How to Cite This Entry:**

Bicylindrical domain.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Bicylindrical_domain&oldid=46052