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:

<math>{ \partial u \over \partial x } = { \partial v \over \partial y}</math>

and

<math>{ \partial u \over \partial y } = -{ \partial v \over \partial x}</math>