Order of zero equals residue of logarithmic derivative

Statement

Suppose $U\subset \mathbb{C}$ is an open subset and $f:U\to \mathbb{C}$ is a meromorphic function. Suppose $z_0\in U$ is a point. Define the order of zero of $f$ at $z_0$ as the unique $n$ for which the function $z\mapsto (z-z_0{)}^{-n}f(z)$ is holomorphic and nonzero at $z_0$.

Then, the order of zero of $f$ at $z_0$ equals the residue of its logarithmic derivative at $z_0$.

Note that if $f$ has a pole at $z_0$, then the order of the pole is the negative of the residue for its logarithmic derivative (that's because the order as a pole is the negative of order as a zero).