Several complex variables

Summary

A function of several complex variables is a function $f(z_1, \dots, z_n)$ whose arguments are $n$ complex numbers, that is, a function on a subset of $\C^n$. The theory of several complex variables studies such functions, mainly the holomorphic ones, and the domains and spaces on which they live. Many facts of one complex variable carry over, such as power series expansions and the Cauchy integral formula on a polydisc. Other facts change deeply when $n \ge 2$. Zeros are never isolated, some domains force holomorphic functions to extend to larger domains, and the unit ball and the polydisc are not biholomorphic.