Wikipedia:Reference desk/Archives/Mathematics/2016 June 24

= June 24 =

Integral identity
$$\int_0^\infty \int_0^\infty \int_0^\infty \frac{x y z} {x y z (x + y + z) + xy + xz + yz} dx dy dz = \frac{i}{\sqrt{3}} (\operatorname{Li}_2(e^{-i \pi / 3}) - \operatorname{Li}_2(e^{i \pi / 3}))$$

Is this statement true? I numerically integrated the expression and then googled the result (from here), and I don't know how to prove it. 24.255.17.182 (talk) 23:25, 24 June 2016 (UTC)
 * One first step is to notice that as the integrand is a symmetric function of (x, y, z) it can be expressed in terms of the power sums si=xi+yi+zi like this
 * $$\frac{x y z} {x y z (x + y + z) + xy + xz + yz} = (s_1+s_{-1})^{-1}$$
 * Bo Jacoby (talk) 00:31, 25 June 2016 (UTC).


 * Also, I get that the RHS is 1/1-1/4+1/16-1/25+1/49-1/64+... . This is from the series given in Spence's function. --RDBury (talk) 03:35, 25 June 2016 (UTC)