Given a smooth and nice function $f: \mathbb{R} \rightarrow \mathbb{C}$.
For a parameter range $t\in [0,1]$ is there a simple method to count the number of rotations the locus of $f$ performs around the origin? Mathematically that is roughly this measure:
$$R=\int_0^1 \left.\frac{d}{dt}(\angle f(t))\right|_{t=t'} dt'$$
(I integrate the phase change between two infinitesimally adjacent points on the curve.)
After the fundamental theorem of calculus, this is R=$\angle f(b)-\angle f(a)$. However, here the "modulo" part of the angle is problematic, as it cannot represent how many times the function went around the origin.
For a closed path $f$, the contour integral of $\frac1z$ along $f$ is $2\pi\mathrm i$ times the winding number.
If $f$ isn't a closed path, you can consider $\frac{f(t)}{|f(t)|}$ instead. This is a path on the unit circle (on which you can ignore the magnitude and are just integrating a phase factor), and the contour integral is proportional to the angle covered, even if you don't complete an integral number of revolutions about the origin.