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.
Best Answer
Here is a proof of the conjecture, a proof that also shows how to compute the integrals explicitly.
The proof is somewhat similar to David Speyer's approach, but instead of using multivariable residues, I will just shift a one-variable contour. Without loss of generality, $m>0$. Eliminating the trigonometric functions yields $I(m,n)=(-1/4\pi^2) J(m,n)$, where $$ J(m,n) := \int_{|w|=1} \int_{|z|=1} \log(4+z+z^{-1}+w+w^{-1}) \, z^{m-1} w^{n-1} \, dz \, dw.$$ For fixed $w \ne -1$ on the unit circle, the values of $z$ such that $4+z+z^{-1}+w+w^{-1}$ are a negative real number and its inverse; let $g(w)$ be the value in $(-1,0)$. The function $\log(4+z+z^{-1}+w+w^{-1})$ of $z$ extends to the complement of the closed interval $[g(w),0]$ in a disk $|z|<1+\epsilon$. Shrink the contour $|z|=1$ like a rubber band so that it hugs the closed interval. The upper and lower parts nearly cancel, leaving $$ J(m,n) = \int_{|w|=1} \int_{z=0}^{g(w)} -2\pi i \, z^{m-1} w^{n-1} \,dz \,dw,$$ with the $-2\pi i$ coming from the discrepancy in the values of $\log$ on either side of the closed interval. Thus $$ J(m,n) = -\frac{2 \pi i}{m} \int_{|w|=1} g(w)^m w^{n-1} \, dw.$$ Next use the rational parametrization $(g(w),w) = \left( (1-u)/(u+u^2), (u^2-u)/(1+u) \right)$ that David Speyer wrote out. There is a path $\gamma$ from $-i$ to $i$ in the right half of the $u$-plane that maps to $|w|=1$ (counterclockwise from $-1$ to itself) and gives the correct $g(w)$. Integrating the resulting rational function of $u$ gives $$ J(m,n) = -\frac{2 \pi i}{m} \left.(f(u) + r \log u + s \log(1+u))\right|_\gamma = -\frac{2 \pi i}{m} (f(i)-f(-i) + r \pi i + s \pi i/2),$$ for some $f \in \mathbf{Q}(u)$ and $r,s \in \mathbf{Q}$. This shows that $I(m,n)=a + b/\pi$ for some $a,b \in \mathbf{Q}$.
Examples: Mathematica got the answers wrong, presumably because it doesn't understand homotopy very well! Here are some actual values, computed using the approach above: $$I(1,1) = \frac{1}{2} - \frac{2}{\pi}$$ $$I(2,1) = -1 + \frac{10}{3\pi}$$ $$I(20,10) = -\frac{14826977006}{5} + \frac{56979906453888224582}{6116306625 \pi}.$$