polydisc

Definition.

We denote the set

D^{n}(z,r):=\{w\in{\mathbb{C}}^{n}\mid\lvert z_{k}-w_{k}\rvert<r\text{ for all% }k=1,\ldots,n\}

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<r\}.

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
Canonical name	Polydisc
Date of creation	2013-03-22 14:29:41
Last modified on	2013-03-22 14:29:41
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	9
Author	jirka (4157)
Entry type	Definition
Classification	msc 32A07
Classification	msc 32-00
Synonym	open polydisc
Defines	bidisc
Defines	distinguished boundary