I've come across this integral during a physics problem:
\begin{equation} \int_{-\infty}^{+\infty} dz \frac{i\Omega}{z^2-\Omega^2+i\epsilon} e^{z} K_{-i(a+z)}(b) \end{equation}
where $\Omega$, $a$ and $b$ are real numbers and $\epsilon\rightarrow 0^{+}$. $K$ is the Modified Bessel Function of the Second Kind.
The problem is the following: $K$ is divergent when the imaginary part of $z$ is very large so the common half-circle contour doesn't work here.