Cauchy–Riemann equations in several variables

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, 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 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.