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 $[-2,2]$.

The best solution was submitted by Choi, Daebeom (최대범, 2016학번). Congratulations!

Here is his solution of problem 2016-17. (There are a few typos.)

No alternative solutions were submitted.