I have to calculate this integral with limit:
$$ G(m,n; E) = \lim_{\epsilon \rightarrow 0^+} \iint_{-\pi}^{+\pi} d k_x d k_y \frac{e^{i(k_x m + k_y n)}}{E+ i \epsilon + 2\cos k_x + 2\cos k_y } . $$
Here $m$, $n$ are integers and $E$ is some real number.
The problem is that the integrand can diverge if $-4< E < 4$.
My idea is to first set $\epsilon $ to some finite value, and then somehow extrapolate the value to the limit $\lim_{\epsilon \rightarrow 0^+}$.
I do not need to calculate the integral analytically. Actually, numerical approach is preferred.