Jaqueline Kirby

2022-06-26

How do I show the following:
${\int }_{0}^{\alpha }dx{\int }_{0}^{\alpha }dy\frac{\mathrm{ln}|x-y|}{\sqrt{xy}}=4\alpha \left(\mathrm{ln}\left(\alpha \right)+2\mathrm{ln}\left(2\right)-3\right)$
where $\alpha >0$.

Harold Cantrell

Expert

Proceed as follows
$\begin{array}{rl}{\int }_{0}^{a}{\int }_{0}^{a}\frac{\mathrm{ln}|x-y|}{\sqrt{xy}}dy\phantom{\rule{mediummathspace}{0ex}}dx& =2{\int }_{0}^{a}{\int }_{0}^{x}\frac{\mathrm{ln}\left(x-y\right)}{\sqrt{xy}}dy\phantom{\rule{mediummathspace}{0ex}}dx\\ & \stackrel{y=x{t}^{2}}{=}4{\int }_{0}^{a}{\int }_{0}^{1}\left(\mathrm{ln}x+\mathrm{ln}\left(1-{t}^{2}\right)\right)dt\phantom{\rule{mediummathspace}{0ex}}dx\\ & =4{\int }_{0}^{a}\left(\mathrm{ln}x+2\mathrm{ln}2-2\right)\phantom{\rule{mediummathspace}{0ex}}dx\\ & =4a\phantom{\rule{mediummathspace}{0ex}}\left(\mathrm{ln}a+2\mathrm{ln}2-3\right)\end{array}$

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