Double integral $ \int_{0}^{\alpha}dx\int_{0}^{\alpha}dy\frac{\ln|x-y|}{\sqrt{xy}}$

Question

How do I show the following: $$ \int_{0}^{\alpha}dx\int_{0}^{\alpha}dy\frac{\ln|x-y|}{\sqrt{xy}}=4\alpha(\ln(\alpha)+2\ln(2)-3) $$ where $\alpha>0$. The integral has arisen as I've been studying the singularity of the Hankel function near the origin (which behaves as a logarithm).

Thanks in advance for any help.

Answer 1

Proceed as follows

\begin{align} \int_{0}^{a}\int_{0}^{a}\frac{\ln|x-y|}{\sqrt{xy}}dy \>dx &=2\int_{0}^{a}\int_{0}^{x}\frac{\ln(x-y)}{\sqrt{xy}}dy\>dx\\ &\stackrel{y=xt^2}{=}4\int_{0}^{a}\int_{0}^{1}(\ln x+\ln(1-t^2))dt\>dx\\ &=4\int_{0}^{a}(\ln x+2\ln2-2)\>dx\\ &=4a\>(\ln a+2\ln2-3)\end{align}