Inverse Laplace of function cos(a s)

Question

In a physical problem, I need to calculate the inverse Laplace of function cos(a s), in which a is a real non-negative value. Is there an analytical or a numerical way to calculate the inversion?

Answer 1

Formally you would get $(\delta(t + a i) + \delta(t - a i))/2$. This can be made rigorous by defining the Laplace transform in terms of the Fourier transform as $$\mathcal L_{t \to s}[f(t)] = \mathcal F_{t \to \omega}[f(t) e^{-\sigma t}], \\ \mathcal L_{s \to t}^{-1}[F(s)] = e^{\sigma t} \mathcal F_{\omega \to t}^{-1}[F(s)],$$ where $s = \sigma + i \omega$, and working with analytic functionals defined on complex lines, as is done in Gelfand and Shilov (so that $e^t$ or even $e^{t^2}$ has a Fourier transform). Then $$e^{\sigma t} \mathcal F^{-1}[\cos a (\sigma + i \omega)] = \frac {e^{\sigma t}} 2 (e^{i a \sigma} \delta(t + i a) + e^{-i a \sigma} \delta(t - i a)) = \\ \frac 1 2 \delta(t + i a) + \frac 1 2 \delta(t - i a).$$