Cauchy–Riemann equations

Statement

Let $U \subseteq \mathbb{C}$ be an open set and let $f : U \to \mathbb{C}$ be a function. Identify $\mathbb{C}$ with $\mathbb{R}^2$ by $x + i y \leftrightarrow (x, y)$. Let $u = \operatorname{Re} f$ and $v = \operatorname{Im} f$ be the real and imaginary parts of $f$, as real functions of $(x, y)$ on $U$. Let $z_0 = x_0 + i y_0 \in U$. Then $f$ is complex differentiable at $z_0$ if and only if the following two conditions hold. First, $u$ and $v$ are differentiable at $(x_0, y_0)$ as functions of two real variables. Second, they satisfy $$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \qquad \frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x}$$ at $(x_0, y_0)$. In this case $f'(z_0) = \frac{\partial u}{\partial x}(x_0, y_0) + i \frac{\partial v}{\partial x}(x_0, y_0)$.