= Solution
For $t>0$, close the inverse-transform <contour> in the upper half-plane. There $|e^{i\omega t}|\leq1$; the rational factor is $O(|\omega|^{-2})$, so the arc integral vanishes. For $\Omega>\kappa$, write $\beta=\sqrt{\Omega^2-\kappa^2}>0$. The two <simple poles> are $i\kappa\pm\beta$, and the <residue theorem> yields
$$
\begin{aligned}
G(t)&=-i\left[\frac{e^{i\omega_+t}}{\omega_+-\omega_-}
+\frac{e^{i\omega_-t}}{\omega_--\omega_+}\right]\\
&=-i e^{-\kappa t}\frac{e^{i\beta t}-e^{-i\beta t}}{2\beta}
=\boxed{e^{-\kappa t}\frac{\sin(\beta t)}\beta}.
\end{aligned}
$$
This <causal Green function of a damped oscillator> has damped oscillations with envelope $e^{-\kappa t}$ and tends to zero as $t\to\infty$.
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