Cauchy–Riemann equations in several variables

Remarks

With the Wirtinger derivatives $\frac{\partial}{\partial \bar z_j} = \frac12 \left( \frac{\partial}{\partial x_j} + i \frac{\partial}{\partial y_j} \right)$, the $2n$ real equations are the $n$ complex equations $\partial f / \partial \bar z_j = 0$, $j = 1, \dots, n$. For $n = 1$ they are the classical Cauchy–Riemann equations. A function $f$ of class $C^1$ on $D$ is holomorphic if and only if it satisfies the Cauchy–Riemann equations on $D$. Example: $f(z_1, z_2) = z_1 z_2$ satisfies them, since $\partial f / \partial \bar z_1 = 0$ and $\partial f / \partial \bar z_2 = 0$. Non-example: $f(z_1, z_2) = z_1 \bar z_2$ satisfies the pair for $j = 1$ but not the pair for $j = 2$, since $\partial f / \partial \bar z_2 = z_1$, which is not zero when $z_1 \ne 0$.