# Calculate using residues $\int_0^\infty\int_0^\infty{\cos\frac{\pi}2\Big(nx^2-\frac{y^2}n\Big)\cos\pi xy\over\cosh\pi x\cosh\pi y}dxdy,n\in\mathbb{N}$

Q: Is it possible to calculate the integral $$\int\limits_0^\infty \int\limits_0^\infty\frac{\cos\frac{\pi}2 \left(nx^2-\frac{y^2}n\right)\cos \pi xy}{\cosh \pi x\cosh \pi y}dxdy,~n\in\mathbb{N}\tag{1}$$ using residue theory?

For example, when $n=3$ $$\int\limits_0^\infty \int\limits_0^\infty\frac{\cos\frac{\pi}{2} \left(3x^2-\frac{y^2}{3}\right)\cos \pi xy}{\cosh \pi x\cosh \pi y}dxdy=\frac{\sqrt{3}-1}{2\sqrt{6}}.$$ There is a closed form formula to calculate (1) for arbitrary natural $n$, but I don't know how to do it by residue theory. Maybe it is possible in principle, but is residue theory practical in this particular case? It seems such an approach would lead to a sum with $O(n^2)$ terms. Any hints would be appreciated.

• can you give a reference for the amazing formula $n=3$? i bet it is due to ramanujan – tired Jan 19 '17 at 10:43
• @tired it is not due to Ramanujan. – Nemo Jan 19 '17 at 10:46
• thx, that is cool stuff...are do you doing this on a recreational basis or is this for professional purposes? – tired Jan 19 '17 at 11:15
• @ Nemo: would you mind if I can offer a bounty for this question? – Nicco Oct 11 '17 at 1:12
• This is very interesting. So you would be content if the integrals $I_1$ and $I_2$ just below (26) in this paper would be evaluated with residue theory? arxiv.org/pdf/1712.10324.pdf – Alfred Yerger Jan 8 at 16:47