---
tag: GVVG201079
kind: definition
name: Several complex variables
msc: [32A10]
defines: BH57D2FBNR
uses: [09FTAET5Z4, 0XHDTE5YMZ]
sources:
  - url: https://en.wikipedia.org/wiki/Function_of_several_complex_variables
---

## Definition

Let $n \ge 1$ be an integer.
The space $\C^n$ is the set of $n$-tuples $z = (z_1, \dots, z_n)$ of [[complex numbers|09FTAET5Z4]], with coordinatewise addition and scalar multiplication.
Its norm is $\abs{z} = \left( \abs{z_1}^2 + \cdots + \abs{z_n}^2 \right)^{1/2}$, which makes it the Euclidean space $\R^{2n}$ through $z_j = x_j + i y_j$.
A *function of several complex variables* is a [[function|0XHDTE5YMZ]] $f \colon D \to \C$, where $D$ is a subset of $\C^n$.

## Remarks

Example: $f(z_1, z_2) = e^{z_1} \sin z_2$ is a function on $D = \C^2$.
Non-example: $g(x, y) = x^2 + y^2$ on $\R^2$ is a function of two real variables, not of complex variables.
It becomes a function of the complex variable $z = x + iy$ only when it is rewritten as $\abs{z}^2$ on $\C$, with $n = 1$.
A map $F \colon D \to \C^m$ is the same as $m$ such functions $F_1, \dots, F_m$, its components.
The case $n = 1$ is the function theory of one complex variable.