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Definition 1 We denote the set
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an open polydisc. We can also have polydiscs of the form
The set
 is called the distinguished boundary of the poydisc.
Be careful not to confuse this with the open ball in
 as that is defined as
When  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
 for the closure of the polydisc. Be careful with this notation however as some texts outside of complex analysis use  and the term “disc” to represent a closed ball in two real dimensions.
Also note that when  the term bidisc is sometimes used.
- 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.
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"polydisc" is owned by jirka.
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(view preamble)
| Other names: |
open polydisc |
| Also defines: |
bidisc, distinguished boundary |
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Cross-references: dimensions, real, represent, complex analysis, closure, biholomorphic mapping, biholomorphically equivalent, open ball
There are 7 references to this entry.
This is version 5 of polydisc, born on 2004-07-26, modified 2005-11-03.
Object id is 6030, canonical name is Polydisc.
Accessed 3719 times total.
Classification:
| AMS MSC: | 32-00 (Several complex variables and analytic spaces :: General reference works ) | | | 32A07 (Several complex variables and analytic spaces :: Holomorphic functions of several complex variables :: Special domains ) |
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Pending Errata and Addenda
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