= Solution
For $\kappa>\Omega$, put $\eta=\sqrt{\kappa^2-\Omega^2}$, so $0<\eta<\kappa$ and the <poles> are $i(\kappa\pm\eta)$. Closing in the upper half-plane for $t>0$ again includes both <simple poles>. The <residue theorem> gives
$$
\boxed{G(t)=e^{-\kappa t}\frac{\sinh(\eta t)}\eta
=\frac{e^{-(\kappa-\eta)t}-e^{-(\kappa+\eta)t}}{2\eta}}.
$$
This <overdamped> response has no oscillation. Its slow tail is $e^{-(\kappa-\eta)t}/(2\eta)$, and it tends to zero because $\eta<\kappa$.
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