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

{| width = "100%"

|- ! colspan="3" align="center" | Mathematics desk |- ! width="20%" align="left" | < June 23 ! width="25%" align="center"|<< May | June | Jul >> ! width="20%" align="right" |Current desk > |}

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

June 24

Integral identity

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\underset{0}{\overset{\mathrm{\infty }}{\int }}\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{xyz}{xyz(x+y+z)+xy+xz+yz}dxdydz=\frac{i}{\sqrt{3}}({\mathrm{Li}}_2(e^{-i\pi /3})-{\mathrm{Li}}_2(e^{i\pi /3}))$

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 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).

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)