# polydisc

###### Definition.

We denote the set

 $D^{n}(z,r):=\{w\in{\mathbb{C}}^{n}\mid\lvert z_{k}-w_{k}\rvert

an open polydisc. We can also have polydiscs of the form

 $D^{1}(z_{1},r_{1})\times\ldots\times D^{1}(z_{n},r_{n}).$

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 polydisc.

Be careful not to confuse this with the open ball in ${\mathbb{C}}^{n}$ as that is defined as

 $B(z,r):=\{w\in{\mathbb{C}}^{n}\mid\lvert z-w\rvert

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 “disc” to represent a closed ball in two real dimensions.

Also note that when $n=2$ the bidisc is sometimes used.

## References

• 1 Lars Hörmander. , North-Holland Publishing Company, New York, New York, 1973.
• 2 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Title polydisc Polydisc 2013-03-22 14:29:41 2013-03-22 14:29:41 jirka (4157) jirka (4157) 9 jirka (4157) Definition msc 32A07 msc 32-00 open polydisc bidisc distinguished boundary