What is the Fourier transform of the function $e^{-i\pi t^2}\chi_{[-1/2,1/2]}(t)$ where $\chi$ is the characteristic function?

Question

Consider the function $t \mapsto e^{-i\pi t^2}\chi_{[-1/2,1/2]}(t)$ where $\chi$ is the characteristic function. I'm trying to get a formula for the Fourier transform of this function, i.e. I nee to evaluate the integral $$ \int_{-1/2}^{1/2} e^{-i\pi t^2} e^{-2\pi i k t} \, dt $$ where $k \in \mathbb R$. I don't really have an approach to evaluate this integral. Someone has an idea?

Answer 1

Complete the square: $$ \begin{align} \int_{-1/2}^{1/2} e^{-i\pi t^2} e^{-2\pi i k t} \, dt &= \int_{-1/2}^{1/2} e^{-i\pi t^2 -2\pi i k t} \, dt \\ & = e^{i \pi k^2}\int_{-1/2}^{1/2} e^{-i\pi(t^2 + 2k t + k^2)} \, dt \\ & = e^{i \pi k^2}\int_{-1/2}^{1/2} e^{-i\pi(t+k)^2} \, dt \\ & = e^{i \pi k^2}\int_{-1/2 + k}^{1/2 + k} e^{-i\pi\tau^2} \, d\tau = e^{i \pi k^2}\int_{(-1/2 + k)/\sqrt{\pi}}^{(1/2 + k)/\sqrt{\pi}} e^{-i\tau^2} \, d\tau. \end{align} $$ This last integral can be expressed in terms of the error function.