This question is based on some Physics motivation. Define a distance function $f(\mathbf{r})=\int\_{\Omega }d^2k\int\_{\Omega }d^2q \cos[(\mathbf{k}-\mathbf{q})\cdot\mathbf{r}]$, where $\mathbf{r},\mathbf{k},\mathbf{q}\in \mathbb{R}^2$ and $\Omega \subset\mathbb{R}^2$ is some finite region. 


The question is: At large distance $\left | \mathbf{r} \right |\rightarrow \infty $, what's the asymptotic behavior of this function $f(\mathbf{r})$? Does it decay like $e^{-\left | \mathbf{r} \right |/L}$ or $\left | \mathbf{r} \right |^{-a}$(with some characteristic length $L$ and some positive number $a$)? Or does the large distance behavior depend on the geometry of the region $\Omega$?


Thank you very much.



What's the asymptotic behavior of this function at large distance?
As OP seems to be stuck, I give here the few lines needed to solve this problem as sugested in the comments. Given
$$
f(\mathbf{r})=\int\_{\Omega }d^2k\int\_{\Omega }d^2q \cos[(\mathbf{k}-\mathbf{q})\cdot\mathbf{r}]=
\frac{1}{2}\int\_{\Omega }d^2ke^{i\mathbf{k}\cdot\mathbf{r}}\int\_{\Omega }d^2qe^{-i\mathbf{q}\cdot\mathbf{r}}
+\frac{1}{2}\int\_{\Omega }d^2ke^{-i\mathbf{k}\cdot\mathbf{r}}\int\_{\Omega }d^2qe^{i\mathbf{q}\cdot\mathbf{r}}.
$$
These integrals are easily done to give
$$
 f(\mathbf{r})=4\frac{(1-\cos(\pi x))(1-\cos(\pi y))}{x^2y^2}
$$
from which the asymptotic behavior can be readily gotten. This makes clear that the question is not appropriate here.


