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Polydisc

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In the theory of functions of several complex variables, a branch of mathematics, a polydisc is a Cartesian product of discs.

More specifically, if we denote by $D (z, r)$ the open disc of center z and radius r in the complex plane, then an open polydisc a set of the form

$D(z_1,r_1) \times \ldots \times D(z_n,r_n).$

It can be equivalently written as

$\{ w=(w_1, w_2, \dots, w_n) \in {\mathbf{C}}^n \mid | z_k - w_k | < r, \mbox{ for all} k = 1,\ldots,n \}.$

One should not confuse the polydisc with the open ball in Cⁿ, which is defined as

$\{ w \in \mathbf{C}^n \mid || z - w || < r \}.$

Here, the norm is the Euclidean distance in Cⁿ.

When $n > 1$ , open balls and open polydiscs are not biholomorphically equivalent, that is, there is no one-to-one biholomorphic mapping between the two.

When $n = 2$ the term bidisc is sometimes used.

References

Steven G. Krantz (1992). Function theory of several complex variables. AMS Chelsea Publishing, Providence, Rhode Island.

Categories: Several complex variables

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10-26-2009 08:16:03
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