The Argument principle:
If $f$ is meromorphic in an open connected set $\Omega$, with zeros $a_j$ and poles $b_k$ then $$\frac{1}{2 \pi i}\int_{\gamma}\frac{f'(z)}{f(z)} dz = \sum_j n(\gamma , a_j) – \sum_k n(\gamma , b_k)$$
Where the sums include multiplicities and $\gamma$ is a cycle homologous to zero in $\Omega$ and does not pass through any of the poles and zeros.
Here is am quite confused by the naming of the theorem. The proof does not seem to give me light on the naming either.
Thank you for the insight!
Best Answer
"Formally" $$\int_\gamma \frac{f'(z)}{f(z)}\,dz=i\int_\gamma d(\arg f(z)) =i\int_{f(\gamma)}d(\arg w) $$ is the overall change in the argument of $f(z)$ as $z$ moves around $\gamma$ (that is $2\pi$ times the winding number of the image contour $f(\gamma)$ about zero).