We denote the set

Dn(z,r):={wn|zk-wk|<r for all k=1,,n}

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


The set D1(z1,r1)××D1(zn,rn) is called the distinguished boundary of the polydisc.

Be careful not to confuse this with the open ball in 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 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.


  • 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