 Solution

For $t>0$, use the Bromwich inversion formula with a vertical contour $\mathrm{Re}s=c>0$:Close the contour to the left, where the exponential decays. The large-arc contribution tends to zero by the usual exponential arc estimate, and the two simple poles are at $s=ia$ and $s=-ia$. The residues of the complete integrand are $e^{iat}/2$ and $e^{-iat}/2$. The residue theorem therefore givesThe right limit at zero is one. The sign of nonzero real $a$ has no effect on the answer.