Suppose that $f$ is a complex function which is defined in some open set $D \subseteq \mathbb{C}$ which has a simple zero at some point $p \in D$ . Then we have $$ p = {1 \over 2 \pi i} \oint_C {z f'(z) \over f(z)} \, dz $$ where $C$ is a closed path in $D$ which encloses $p$ but does not enclose or pass through any other zeros of $f$ .

This follows from the Cauchy residue theorem. We have that the poles of $f'/f$ occur at the zeros of $f$ and that the residue of a pole of $f'/f$ is $1$ at a simple zero of $f$ . Hence, the residue of $z f'(z) / f(z)$ at $p$ is $p$ , so the above formula follows from the residue theorem.