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June 24

Integral identity

$\int _{0}^{\infty }\int _{0}^{\infty }\int _{0}^{\infty }{\frac {xyz}{xyz(x+y+z)+xy+xz+yz}}dxdydz={\frac {i}{\sqrt {3}}}(\operatorname {Li} _{2}(e^{-i\pi /3})-\operatorname {Li} _{2}(e^{i\pi /3}))$

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)[reply]

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 s_i=xⁱ+yⁱ+zⁱ like this

{\frac {xyz}{xyz(x+y+z)+xy+xz+yz}}=(s_{1}+s_{-1})^{-1}

Bo Jacoby (talk) 00:31, 25 June 2016 (UTC).[reply]

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)[reply]