I want to analytically evaluate the 2D integral for some real $E< 1$
$I^\pm(E)=\lim_{\eta \rightarrow 0}\int_{-1}^{1} dx dy \frac{1}{E\pm i\eta -xy}$
In particular, I want to understand the behavior of the real and complex part of the integral when $E\rightarrow 0$.
Some comments: Numerical evaluations reveal that $Re[I(E)]$ is discontinuous across $E=0$ and $Im[I(E)]$ diverges logarithmically when $E\rightarrow 0$.
In physics, the integral corresponds to the integration over a retarded/advanced Green's function, where the 2D dispersion $xy$ has a saddle point at energy $E=0$.