= Solution
Since $\gamma>0$, the causal damped sine has an absolutely convergent <Fourier transform> for real $\omega$. Resolving the sine into complex exponentials gives
$$
\widetilde f(\gamma,p,\omega)
=\frac1{2i}\int_0^\infty\left[e^{-[\gamma+i(\omega-p)]t}-e^{-[\gamma+i(\omega+p)]t}\right]dt
=\frac1{2i}\left[\frac1{\gamma+i(\omega-p)}-\frac1{\gamma+i(\omega+p)}\right].
$$
Combining the two fractions yields
$$
\boxed{\widetilde f(\gamma,p,\omega)=\frac{p}{(\gamma+i\omega)^2+p^2}.}
$$
The formula also includes $p=0$, for which both the original function and its transform vanish. Its sign reverses when the real parameter $p$ reverses.
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