# 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 ...