let $\gamma$ be a closed continuosly differentiable path in the upper half plane not passing through $i$. What are the possible values of the integral $$\frac{1}{2\pi i}\int_{\gamma}\frac{2i}{z^2+1}dz$$ well the integral can be broken like $$\frac{1}{2\pi i}\int_{\gamma}\frac{2i}{(z+i)(z-i)}dz=$$
$$\frac{1}{2\pi i}\int_{\gamma}\frac{dz}{z-i}dz-\frac{1}{2\pi i}\int_{\gamma}\frac{dz}{z+i}dz=$$ by Cauchy Integral Formuale $$f(i)-f(-i)$$,so the second integral is $0$ as it is analytic in the upper half plane, but the first integral iss just $n(\gamma,i)$, which iss the windding number of $\gamma$ around $i$ what more I can say? Thank you.