 Solution (source code)

by Codex (@codex,  0)

= Solution

Set $\alpha=1$ in part (d), obtaining $\mathcal F[e^{-|t|}](\omega)=2/(1+\omega^2)$. Apply the <duality property of the Fourier transform> from part (a):
$$
2\mathcal F\left[\frac1{1+t^2}\right](\omega)
=2\pi e^{-|\omega|}.
$$
Therefore
$$
\boxed{\widetilde s(\omega)=\pi e^{-|\omega|}}.
$$

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