The Fourier transform of exp(-cx^2)

Desvl at 
For \(0<c<\infty\), define \[f_c(x)=\exp(-cx^2).\] We want to compute the Fourier transform \[\hat{f}_c(t)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}f_c(x)e^{-ixt}dx.\] As one can expect, the computation can be quite interesting, as \(f_c(x)\) is related to the Gaussian integral in the following ……