I've been trying to work around the integral $$H(-zz')\int_{-\infty}^{\infty}dk\frac{k}{k-(\tilde{\theta}/2)i}e^{ik|z-z'|}+H(zz')\frac{\tilde{\theta}}{2}i\int_{-\infty}^{\infty}dk \frac{1}{k-(\tilde{\theta}/2)i}e^{ik|z+z'|},$$ where $\tilde{\theta}>0$ and $H(x)$ is the Heaviside step function.
According to Mathematica, the solution is (for $\tilde{\theta}\in \mathbb{R}$) $$\frac{\tilde{\theta}}{2}\left(e^{\frac{\tilde{\theta}}{2}|z-z'|}H(-zz')+e^{\frac{\tilde{\theta}}{2}|z+z'|}H(zz')\right)\left[i \log \left(\frac{2i}{\tilde{\theta}}\right)-i \log \left(-\frac{2i}{\tilde{\theta}}\right)+\pi\right], $$ So if $\tilde{\theta}>0$, then the integral is zero, which is what I'm expecting.
I tried to apply the residue theorem by using a semicircular contour, but I don't know how to deal with the arc part (although the result given by Mathematica is somehow what I would expect from such theorem). I also tried to relate these integrals with the exponential integral function $Ei(x)$ by means of a change of variable, but had no luck. It seems both integrals have to vanish independently. Unfortunately, I'm not an expert at integrating complex variable functions.
If you could give me a hint it would be great!