Let $\gamma_1 = S_1 + L - S_2 - L$ and $\gamma_2 = S_1 + L + S_2 - L$, $$S_1(t) = e^{it} , t\in [0,2\pi] $$ $$S_2(t) = 2e^{it} , t\in [0,2\pi] $$ $$ L = [1,2] $$ Let $f(z) = (\cos z)/z$. By writing cos z as a power series and considering $f(z) = (1/z)+g(z)$, compute $\int_{\gamma_1} f$ and $\int_{\gamma_2} f$.
Computing the integration seems long so I thought about using the Generalized Version of Cauchy Theorem. Since the winding numbers about the origin add up to 0, the integral is equaled to 0. Am I going about this the right way? (I have not learned the Residue Theorem yet)
Cauchy's Integral formula says $\frac 1 {2\pi i}\int_{\gamma} \frac {f{(z)}} {z-0} dz=f(0)$. Taking $f(z)=\cos z$ we get the value of the integral immediately as $2\pi i$.