Argument principle states: If $f$ is a meromorphic function inside and on some closed contour $C$, and $f$ has no zeros or poles on $C$, then $$\frac{1}{2\pi i}\oint_C \frac{f'(z)}{f(z)}\,dz=Z-P$$ where $Z$ and $P$ denote the number of zeros and poles in $C$ (counting multiplicity). The central...