What does the function $n(\gamma , z_{0})$ denote in this version of Cauchy Integral Formula?

Question

In my lecture notes, the Cauchy Integral Formula for complex integrals is defined as $$ \int_{\gamma} \frac{f(z)}{z - z_{0}} dz = 2 \pi i \cdot n(\gamma , z_{0}) \cdot f(z_{0}) $$

What does the function $n(\gamma , z_{0})$ denote in this context, and how can I calculate it for specific values of $\gamma$ and $z_{0}$?

Thanks

Answer 1

What you are referring to is called the index of a contour or the winding number of a contour around some point $z_0$. Intuitively, it is the number of times a contour goes around this particular point. There are various ways of defining it all of which can be find on the associated Wikipedia page: https://en.m.wikipedia.org/wiki/Winding_number