Why is the winding number defined as $\frac{1}{2\pi i}\oint_C \frac{f'(z)}{f(z)}dz$?

Question

I'd appreciate some clear explanation as to why the number is defined as such. I think in my book, in the proof of the argument principle, it seems like this integral pops out of the blue, without providing motivation. In Wikipedia, I think they are using a special case, and also without much motivation for this division of $\frac{dz}{z}$.

Answer 1

The anti-derivative of $\frac{1}{z}$ is the logarithm. Thus, we can view $\int_C\frac{1}{z}$ as measuring the difference in the logarithm of the curve $C$ at its starting point before and after completing the circuit. Writing this point as $re^{i\theta}$ we see that the imaginary part of the logarithm will measure something to do with the angle of the function. Because the angle of the curve varies continuously, it must pickup an additional $2\pi$ radians every time it returns to its starting point, or more generally winds about the origin. Taking the difference of the logarithm evaluated at the beginning and end of this path exactly tracks this difference in angle.