---
tag: BH57D2FBNR
kind: concept
name: Several complex variables
msc: [32A10]
sources:
  - url: https://en.wikipedia.org/wiki/Function_of_several_complex_variables
same_as:
  wikidata: Q2052951
  wikipedia: Function of 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|507C34W0ZT]] 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|FG5448NZ18]].
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|PDQ9KW9FAP]] and the [[polydisc|DJBWP7D3M8]] are not biholomorphic.

## Explanations

### L1 human

In one complex variable a function takes one complex number $z$ and returns a complex number.
In several complex variables it takes a list $(z_1, \dots, z_n)$ of complex numbers.
A picture: $\C^n$ is like $\R^{2n}$, with each coordinate $z_j = x_j + i y_j$ giving two real directions.
For example, $f(z_1, z_2) = z_1 z_2 + z_2^2$ is a function of two complex variables.
It is a polynomial in each variable, and such functions are the simplest examples of the theory.