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 **R**^{2}. Then *f* is holomorphic if and only if *u* and *v* are differentiable and their partial derivatives satisfy the Cauchy-Riemann equations, which are:

- .

*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.