## You are here

Homepolydisc

## Primary tabs

# 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*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

Apr 15

new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz

Mar 26

new correction: Misspelled name by DavidSteinsaltz

Mar 21

new correction: underline-typo by Filipe

Mar 19

new correction: cocycle pro cocyle by pahio

Mar 7

new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier

new image: expected waiting time by robert_dodier

new image: plot W(t) = P(waiting time <= t) by robert_dodier

new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz

Mar 26

new correction: Misspelled name by DavidSteinsaltz

Mar 21

new correction: underline-typo by Filipe

Mar 19

new correction: cocycle pro cocyle by pahio

Mar 7

new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier

new image: expected waiting time by robert_dodier

new image: plot W(t) = P(waiting time <= t) by robert_dodier