The Fourier transform of exp(-cx^2)

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 way: \[\int_{-\infty}^{+\infty}f_c(x)dx=\frac{1}{\sqrt{c}}\int_{-\infty}^{+\infty}\exp(-(\sqrt{c}x)^2)d\sqrt{c}x=\sqrt\frac{\pi}{c}.\] Now we dive into this integral and see what we can get.Computing ...

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