# Cauchy-Riemann Equations

Let . The Cauchy-Riemann equations are:

## Alternative Form (Wirtinger Derivative)

The Cauchy-Riemann equations can be written as a single equation where is the Wirtinger derivative with respect to the conjugate variable.

## Goursat’s Theorem

Suppose is a complex-valued function which is differentiable as a function . Then is analytic in an open complex domain iff it satisfies the Cauchy-Riemann equations in the domain.