## Encyclopedia > Cauchy-Riemann equations

Article Content

# 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:

${ \partial u \over \partial x } = { \partial v \over \partial y}$

and

${ \partial u \over \partial y } = -{ \partial v \over \partial x}$

All Wikipedia text is available under the terms of the GNU Free Documentation License

Search Encyclopedia
 Search over one million articles, find something about almost anything!

Featured Article
 North Haven, New York ... (274.2/mi²). There are 578 housing units at an average density of 82.3/km² (213.3/mi²). The racial makeup of the village is 98.38% White, 0.40% ...