polydisc
Polydisc
2004-07-26 13:24:53
2005-11-03 12:28:58
Definition
bidisc
distinguished boundary
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\begin{defn}
We denote the set
\begin{equation*}
D^n(z,r) := \{ w \in {\mathbb{C}}^n \mid \lvert z_k - w_k \rvert < r
\text{ for all } k = 1,\ldots,n \}
\end{equation*}
an {\em open polydisc}. We can also have {\em polydiscs} of the form
\begin{equation*}
D^1(z_1,r_1) \times \ldots \times D^1(z_n,r_n) .
\end{equation*}
The set $\partial D^1(z_1,r_1) \times \ldots \times \partial D^1(z_n,r_n)$ is called the {\em distinguished boundary} of the poydisc.
\end{defn}
Be careful not to confuse this with the open ball in ${\mathbb{C}}^n$
as that is defined as
\begin{equation*}
B(z,r) := \{ w \in {\mathbb{C}}^n \mid \lvert z - w \rvert < r \} .
\end{equation*}
When $n > 1$ then open balls and open polydiscs are not biholomorphically
equivalent (there is no 1-1 biholomorphic mapping between the two).
It is common to write $\bar{D}^n(z,r)$ for the closure of the polydisc.
Be careful with this notation however as some texts outside of
complex analysis use $D(x,r)$ and
the \PMlinkescapetext{term} ``disc'' to represent a closed ball in two real dimensions.
Also note that when $n=2$ the \PMlinkescapetext{term} {\em bidisc} is sometimes used.
\begin{thebibliography}{9}
\bibitem{Hormander:several}
Lars H\"ormander.
{\em \PMlinkescapetext{An Introduction to Complex Analysis in Several
Variables}},
North-Holland Publishing Company, New York, New York, 1973.
\bibitem{Krantz:several}
Steven~G.\@ Krantz.
{\em \PMlinkescapetext{Function Theory of Several Complex Variables}},
AMS Chelsea Publishing, Providence, Rhode Island, 1992.
\end{thebibliography}