# polydisc

###### Definition.

We denote the set

$$ |

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

$${D}^{1}({z}_{1},{r}_{1})\times \mathrm{\dots}\times {D}^{1}({z}_{n},{r}_{n}).$$ |

The set $\partial {D}^{1}({z}_{1},{r}_{1})\times \mathrm{\dots}\times \partial {D}^{1}({z}_{n},{r}_{n})$ is called the distinguished boundary of the polydisc.

Be careful not to confuse this with the open ball in ${\u2102}^{n}$ as that is defined as

$$ |

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 ${\overline{D}}^{n}(z,r)$ 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 |