The Cauchy-Riemann equations reference article from the English Wikipedia on 24-Jul-2004 (provided by Fixed Reference: snapshots of Wikipedia from wikipedia.org)

Cauchy-Riemann equations

In complex analysis, the Cauchy-Riemann differential equations are two partial differential equations which provide a necessary and sufficient condition for a function to be holomorphic.

Let f = u + iv be a function from an open subset of the complex numbers C to C, and regard u and v as real-valued functions defined on an open subset of R². Then f is holomorphic if and only if u and v are differentiable and their partial derivatives satisfy the Cauchy-Riemann equations, which are:

and

It follows from the equations that u and v must be harmonic functions. The equations can therefore be seen as the conditions on a given pair of harmonic functions to come as real and imaginary parts of a complex-analytic function.

Polar representation

Considering the polar representation , the equations take the form