Polydisc

Remarks

So $P(a, r) = D(a_1, r_1) \times \cdots \times D(a_n, r_n)$, a product of open discs in $\C$. The unit polydisc is $\Delta^n = P(0, (1, \dots, 1))$, the $n$-fold product of the unit disc. Example: $(1/2, i/2) \in \Delta^2$, since both coordinates have modulus $1/2 < 1$. Non-example: $(0, 1) \notin \Delta^2$, since the second coordinate has modulus $1$, which fails the strict inequality. The open polydisc is the open ball of radius $1$ about $a$ for the norm $\max_j \abs{z_j - a_j}/r_j$. For $n = 1$ a polydisc is an ordinary disc.