I was looking at the Power Round of Stanford Math Tournament and I came across this formula for a winding number of a curve: $$w(\gamma)=\frac{1}{2πi}\int_0^{2π}\frac{\gamma’(s)}{\gamma(s)} \mathrm {ds}$$ Where $w(\gamma)$ is the winding number of the curve $\gamma(s)$, which is defined to be $\gamma : [0,2π] \to \mathbb C -\{0\}$, and $\gamma(0)=\gamma(2π)$
I evaluated the integral and found out that the winding number is always 0: By letting $\gamma(s)=u$,
$$\int\frac{\gamma’(s)}{\gamma(s)} \mathrm {ds} =\int\frac1u \mathrm{du}=\ln(u)+C$$
So in this case, $$\int_0^{2π}\frac{\gamma’(s)}{\gamma(s)}=\ln(\gamma(2π))-\ln(\gamma(0))=0$$ Because $\gamma(2π)=\gamma(0)$
I know that this is clearly wrong by trying examples like $\gamma(s)=e^{is}$, because the integral would simply evaluate to 2πi, but where did my calculation go wrong?
Link to the problems pdf: https://sumo.stanford.edu/pdfs/smt2021/power-problems.pdf. (Pages 5-6)