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.
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.