Use the Green's theorem (complex form) to show that
$$\frac{1}{2\pi i}\int_\gamma \frac{dz}{z-p}=\begin{cases} 0 & \text{if $p$ is outside $\gamma$} \\ 1& \text{if $p$ is inside $\gamma$}\\ \end{cases}$$
I proved this one by taking $z-p=re^{i\theta}$, but how could I prove it using Green's theorem?
The double integral vanishes since the integrand is zero. If $p$ is outside then just apply the Green's result. If $p$ is inside then you may consider the integration domain to be inside $\gamma$ without a small circle around $p$ and replace integration over $\gamma$ to integration over this circle. Then you can apply what you have proved.