## Problem of the week

### 2016-17 Integral with two variables

Set $L(z,w)=\int_{-2}^2\int_{-2}^2 ( \log(z-x)-\log(z-y))( \log(w-x)-\log(w-y))Q(x,y) dx dy,$ for $$z,w\in \mathbb{C}\setminus(-\infty, 2]$$, where $Q(x,y)= \frac{4-xy}{(x-y)^2\sqrt{4-x^2}\sqrt{4-y^2}}.$ Prove that $L(z,w)=2\pi^2 \log \left[ \frac{(z+R(z))(w+R(w))}{2(zw-4+R(z)R(w))} \right],$ where $$R(z)=\sqrt{z^2-4}$$ with branch cut $$[-2,2]$$.