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 \(\gamma\), and let \(Z_f\) and \(P_f\) be the numbers of zeros and poles of \(f\) in \(\Omega\), counted with multiplicity. Then \[ \frac{1}{2\pi i}\int_\gamma\frac{f'(z)}{f(z)}\dz=Z_f-P_f. \]