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In the science paper labelled "Effect of Fermi surface geometry on electron-electron scattering", by Hodges, Smith and Wilkins, there is a following identity:

$$ \int_{0}^{ \infty}dx\int_{0}^{ \infty}dz f(z)\left[ 1- f(x) \right]\left[ 1- f(t+z-x)\right] = \frac{1}{2}(\pi ^2 + t^2)\left[ 1- f(t)\right] $$


$$f(x) = \frac{1}{e^x + 1}$$

Now, can anyone tell me is there some fancy way to prove it, without the "brute- force" method.


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Would you please include a description of the "brute-force" method? – Glen Wheeler Jul 1 '11 at 12:32
I don't think the result is correct as stated. I would suppose that the integration region should be $[-\infty,\infty]$ for both integrals... – Fabian Jul 1 '11 at 15:56

Hint: Differentiate the right hand side.

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