Cauchy's Integral Formula and Green's Theorem

We may safely assume $a=0$ since translation will not influence the integral in any ways. We want to prove that $\oint_{C}\frac{f(z)}{z}dz=2\pi if(0)$. Consider $z=re^{i\theta}$ we have $$\int_{|r|=R}\frac{f(re^{i\theta})}{re^{i\theta}}(dre^{i\theta}+re^{i\theta}id\theta)=\int_{|r|=R}\frac{f(re^{i\theta})}{r}dr+if(re^{i\theta})d\theta$$

We cannot apply Green's theorem directly because it assumes the area is simply connected. But we can calculate the integral directly via taking limit $R\rightarrow 0$. This can be done by noticing that since $r=R$ the left term actually vanishes; Thus we are left with $$\oint_{|r|=R}if(Re^{i\theta})d\theta=i\int^{2\pi}_{\theta=0}f(Re^{i\theta})d\theta$$

Taking the limit of this integral when $R\rightarrow 0$ should give you $2\pi if(0)$.

This small trivial calculation is associated with Poincare's lemma and DeRham Cohomology. You may venture to read some reference books if you are interested.