Math: Fourier Transform of Gauss Function (20 min)

The Gauss function

$$f(x):=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{x^2}{2\sigma^2}}$$ (2)

is a convenient example to discuss properties of the Fourier transform. Show that it can be decomposed into plane waves by

$$\tilde{f}(k) = \int_{-\infty}^{\infty}dx {f}(x) e^{-ikx}=e^{-\frac{1}{2}\sigma^2... ...x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}dk e^{-\frac{1}{2}\sigma^2k^2} e^{ikx}.$$ (3)

Draw $f(x)$ and $\tilde{f}(k)$ for different values of $\sigma$ and discuss their relation.