TITLE: A two-variable Fourier series and a strange integral
QUESTION [11 upvotes]: I have recently had occasion to investigate the Fourier series of the function $f(x,y)=\log({2+\cos 2\pi x} +\cos{2\pi y})$.  Accordingly, define
$I(m,n)=\int_{0,0}^{1,1}f(x,y)\cos{2\pi mx}\cos{2\pi ny}dxdy$ 
which is the $(m,n)$th Fourier coefficient.  If there is some easy change of variables or flick of the wrist to compute $I$ in closed terms, I haven't been able to find it. One may compute $I(0,0)=\frac{4G}{\pi}-\log 2$, where $G$ is Catalan's constant.  As for other values, Mathematica's integration routine returns the values $I(1,1)=\frac{1}{2}+\frac{2}{\pi}$ and $I(2,1)=1-\frac{10}{3\pi}$ (and takes about 10 minutes per computation).  I have tried and failed to affirm the following conjecture, which seems reasonable to present here.
Conjecture. If $(m,n)\neq(0,0)$, then $I(m,n)=a+\pi^{-1}b$ with $a,b\in \mathbb{Q}$.
Of course, if you could prove this, it would be great to give a closed formula for $a,b$.
Full disclosure: My interest in this stems from a formula of Kasteleyn for the partition function of dimer coverings of squares; in fact the evaluation of $I(0,0)$ given above can be extracted from his original paper.

REPLY [5 votes]: Mathematica gets the (1,1) case right. 
Integrate the x-variable to obtain an answer 
that involves the absolute value of Cos[Pi y].
Then split the y-integral at y=1/2 to get the 
correct value.
The answer is 1/2 - 2/Pi