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Definition 1 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 open polydisc. We can also have 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 distinguished boundary of the poydisc.
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 term ``disc'' to represent a closed ball in two real dimensions.
Also note that when $n=2$ the term bidisc is sometimes used.
- 1
- Lars Hörmander. An Introduction to Complex Analysis in Several Variables, North-Holland Publishing Company, New York, New York, 1973.
- 2
- Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.
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