TITLE: A difficult double integral
QUESTION [5 upvotes]: How can the double integral
$$
\int_{0}^{\infty}\int_{0}^{\infty}e^ {{-\beta(x^2+y^2+xy)}}\cos(2\pi Mx)\cos(2\pi Ny)\,\text{d}x \,\text{d}y,
$$
where $\beta>0$ and $M$, $N\in\mathbb{Z}$, be evaluated? The exponent can be separated by completing the square, which seems a natural first step, but I can't seem to make much progress from there. 
For those who are interested, the integral arises when applying the Poisson summation formula to the partition function 
$$
Z(\beta) = \sum_{m,n=1}^{\infty}\exp(-\beta E_{m,n}).
$$
Here $E_{m,n}=m^{2}+n^{2}+mn$ gives the eigenvalue spectrum for the Helmholtz equation solved inside a planar domain shaped like an equilateral triangle.

REPLY [6 votes]: Write the cosines in complex form, that will cast the integral into the exponential of a quadratic form in $x,y$ with complex coefficients, then complete the square and the integral becomes a Gaussian integral in two dimensions, which can be evaluated directly.