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 biholomorphicallyequivalent (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 ofcomplex analysis use $D(x,r)$ andthe “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.