Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 327 / 3 / a / Solution

For $\omega \ne 0$, insert the Gaussian damping factor $e^{-\epsilon x^2/2}$ and use the Gaussian integral:Taking $\epsilon \downarrow 0$ in ${\mathcal{S}}^{\prime }$ with the continuous square-root branch gives the Fresnel integralFor $\omega =0$, the function is constant and its Fourier transform is $\boxed{2\pi {\delta }_0}$.