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This one can be done pretty slick by using the residues. Consider: $$f(z)=\frac{\pi \cot(\pi z) z^{2}}{(4z^{2}-1)^{2}}$$ Note that $$\sum_{-\infty}^{\infty}f(n)=2\sum_{n=1}^{\infty}f(n)$$ $$\lim_{z\to \pm 1/2}\frac{d}{dz}\left[\frac{\pi \cot(\pi z)z^{2}}{4(2z\pm 1)^{2}}\right]=\frac{-\pi^{2}}{64}$$ ...

Good job with digamma-related approach, R. I will give it a go another way with residues. . Well, it still uses $\pi \cot(\pi z)$ like RD done. Using the $$\sum_{-\infty}^{\infty}f(z)=-\text{sum of residues of} \cot(\pi z) \text{at the poles of f(z)}$$ $$\sum_{k=-\infty}^{\infty}f(k)=\sum_{k=-\infty...

Good start, T.

Let's stew on it and see what we come up with.

But, I think it'll involve $erf$.

Here is a Poisson-like integral I ran across that has gotten no bites on other forums. I reckon this means it is not doable?. $$\int_{0}^{\infty}x^{2}e^{-x^{2}+x}\log(1-2a\cos(x)+a^{2})dx$$ I ran it through Maple and Wolfram, but neither would return anything. Maple returned numerical answers for sp...

Clever and consise, RD.

Evaluate:

$$\int_{0}^{\infty}\frac{(1-\sin(ax))(1-\cos(bx))}{x^{2}}dx$$