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I am looking for a possible closed-form for the following double sum which is not tractable directly with Mathematica.

$f(x,y)=\sum_{n=1}^\infty \sum_{m=1}^\infty 2e^{\frac{-\pi}{2} \sqrt{m^2+n^2}}\cos(my) \cos(nx)$.

Any suggestions or ideas are welcomed.

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    This sum does not seems to converge uniformly. What makes you think that this sum is indeed convergent? – 2017-01-05
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    Actually I am not sure about this. When mathematica just return the expression, I think we can not say anything about the convergence, It can be convergent as it can not. What makes you think it is not ? perhaps I can prove that. – 2017-01-05
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    Isn't it a double fourier series? Can't you try to solve an integral equation for the coefficients (assuming this approach make sense). – 2017-01-05
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    Well, if you put $x=y=2 \pi$, then you get a divergent sum (terms do not converge to $0$). In general the terms do not converge to $0$ almost for any $x,y$. – 2017-01-05
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    Thank you for making the question more clear. :D Now I can tackle it for real! – 2017-01-05
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    Sorry for the misunderstanding, I modified it. – 2017-01-05
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    I don't believe there is any point in having large $x,y$, since $\cos$ is periodic. – 2017-01-05

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