 Solution

Jordan's lemma states that if $f$ is holomorphic in the upper half-plane apart from finitely many poles and $|f(z)|\le M/|z|$ on sufficiently large upper semicircles, then for $a>0$ the integral of $e^{iaz}f(z)$ over those arcs tends to zero. Indeed, split the arc away from its endpoints, where exponential decay is uniform, and bound the two short endpoint arcs using $|e^{iaR{e}^{i\theta }}|=e^{-aR\sin \theta }$ and $\sin \theta \ge 2\theta /\pi$ on $[0,\pi /2]$.

Apply the upper semicircle to $z{e}^{iz}/(1+z^2)$. The sole enclosed pole is $i$, with residue $e^{-1}/2$. Therefore the contour integral is $\pi i/e$; taking imaginary parts gives