Cauchy-Riemann differential equations

Definition

Suppose $U$ is an open subset of $\mathbb{C}$ and $f:U\to \mathbb{C}$ is a function. Write $f$ in terms of its real and imaginary parts as follows:

$f(z):=u(z)+iv(z)$

We say that $f$ satisfies the Cauchy-Riemann differential equations at a point $z_0\in U$ if the partial derivatives of the $u,v$ in both the $x$ and $y$ directions exist, and satisfy the following conditions:

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

Using the subscript notation for partial derivatives, we can write this more compactly as:

$u_x=v_y,u_y=-v_x$

Note that if we are given that $u$ and $v$ are both continuously differentiable in the real sense, then satisfying the Cauchy-Riemann differential equations at $z_0$ is equivalent to being complex-differentiable at $z_0$.