Argument principle counting zeros and poles

Let \(\Omega\subseteq\CC\) be a bounded open set whose boundary \(\gamma\coloneqq\partial\Omega\) consists of finitely many piecewise smooth closed curves, oriented positively with respect to \(\Omega\). Let \(f\) be meromorphic on an open set containing \(\overline{\Omega}\), with no zeros or poles on \(\gamma\). Let \(Z_f\) and \(P_f\) be the numbers of zeros and poles of \(f\) in \(\Omega\), counted with multiplicity. Then \[ \frac{1}{2\pi i}\int_\gamma\frac{f'(z)}{f(z)}\dz=Z_f-P_f. \] If moreover \(f\) is holomorphic with zeros \(z_1,\ldots,z_m\) in \(\Omega\), listed with multiplicity, then \[ \frac{1}{2\pi i}\int_\gamma\frac{f'(z)}{f(z)}\dz=m \quad\text{and}\quad \frac{1}{2\pi i}\int_\gamma\frac{zf'(z)}{f(z)}\dz=\sum_{k=1}^mz_k. \]