Winding number from complex analysis and differential geometry

Question

I showed that for a differentiable function $f:S^1 \rightarrow S^1$, the winding number is given by

$\frac{1}{2 \pi i } \int_{S^1} \frac{f'(z)}{f(z)} dz$. Now I want to show that the winding number can also be expressed in the following way:

Let $z \in S^1 $ be a regular value and $f^{-1}(z) = \{w_0 ,...,w_n\}$, then we have that the winding number is given by $\sum_{i=1}^k \varepsilon(w_i)$, where $\epsilon(w_i)= \pm 1$, depending on whether the differential keeps the orientation at this point as it is or whether the orientation changes.

I mean, I think that the integral above may be useful, but it does not need to be used.

Answer 1

This appears in complete generality in Guillemin and Pollack's Differential Topology. In addition, the $n$-dimensional version of winding numbers is used to give an elegant inductive proof of the Borsuk-Ulam Theorem.