Let $\phi\in C_c^{\infty}(\mathbb{C}).$ Prove that $\displaystyle \int_{|z-w|>\epsilon} \log|z-w|\Delta\phi(z)dA(z)=\int_0^{2\pi}(\phi(w+re^{it})-r\log r\frac{\partial \phi}{\partial r}(w+re^{it})|_{r=\epsilon}dt$ using Green's Theorem.
So, $\displaystyle \int_{|z-w|>\epsilon} \log|z-w|\Delta\phi(z)dA(z)=\displaystyle \int_{|z-w|>\epsilon} \phi \Delta\log|z-w|dA(z)$ by Greens Theorem and since $\phi$ has compact support. Then, again by Green's theorem, $\displaystyle \displaystyle \int_{|z-w|>\epsilon} \phi \Delta\log|z-w|dA(z)=\int_0^{2\pi} (\phi(w+re^{it}) \frac{\partial \log r}{\partial n})|_{r=\epsilon} dt$ where $\hat{n}$ is the unit inner normal vector to the circle $|z|=\epsilon$. Now, how do I proceed towards the final result? I am stuck here. Can somebody please help me?