---
tag: 9PT772R94A
kind: concept
name: Cauchy–Riemann equations in several variables
msc: [32W05, 32A10]
sources:
  - url: https://en.wikipedia.org/wiki/Cauchy%E2%80%93Riemann_equations
---

## Summary

The Cauchy–Riemann equations in several variables are a system of $2n$ real first-order partial differential equations.
They are $n$ pairs of [[Cauchy–Riemann equations|0T71SSYZ39]], one pair for each complex coordinate $z_j = x_j + i y_j$.
In complex form they read $\partial f / \partial \bar z_j = 0$ for $j = 1, \dots, n$.
A continuously differentiable function on an open subset of $\C^n$ is [[holomorphic|507C34W0ZT]] exactly when it satisfies them.
For $n \ge 2$ the system is overdetermined: $n$ complex equations for one unknown function, which is a source of the rigidity of the theory.

## Explanations

### L1 human

A function of one complex variable is holomorphic when its real and imaginary parts satisfy two linked equations.
In several variables, the same two equations must hold in each variable separately.
So there are two equations for each coordinate, $2n$ in all.
For example, $f(z_1, z_2) = z_1 z_2$ satisfies all of them, while $f(z_1, z_2) = z_1 \bar z_2$ fails the pair for $z_2$.