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

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

# References

- 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.

Defines:

bidisc, distinguished boundary

Synonym:

open polydisc

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

32A07*no label found*32-00

*no label found*

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new correction: Closure properties on languages: DCFL not closed under reversal by babou

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