Cauchy–Riemann equations

In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which form a necessary and sufficient condition for a complex function of a complex variable to be complex differentiable.

These equations are

{\frac {\partial u}{\partial x}}={\frac {\partial v}{\partial y}}

1a

and

{\frac {\partial u}{\partial y}}=-{\frac {\partial v}{\partial x}},

1b

where $u (x, y)$ and $v (x, y)$ are real bivariate differentiable functions.

Typically, $u$ and $v$ are respectively the real and imaginary parts of a complex-valued function $f (x + iy) = f (x, y) = u (x, y) + iv (x, y)$ of a single complex variable $z = x + iy$ where $x$ and $y$ are real variables; $u$ and $v$ are real differentiable functions of the real variables. Then $f$ is complex differentiable at a complex point if and only if the partial derivatives of $u$ and $v$ satisfy the Cauchy–Riemann equations at that point.

A holomorphic function is a complex function that is differentiable at every point of some open subset of the complex plane $\mathbb {C}$ . It has been proved that holomorphic functions are analytic and analytic complex functions are complex-differentiable. In particular, holomorphic functions are infinitely complex-differentiable.

This equivalence between differentiability and analyticity is the starting point of all complex analysis.