Math: Gauss function

The Gauss function

$$g(x):=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{x^2}{2\sigma^2}}$$ (45)

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

$$\tilde{g}(k) = \int_{-\infty}^{\infty}dx {g}(x) e^{-ikx}=e^{-\frac{1}{2}\sigma^2k^2}$$

$$g(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty}dk e^{-\frac{1}{2}\sigma^2k^2} e^{ikx}$$ (46)

An important integral in this context is

$$\int_{-\infty}^{\infty} dx e^{-ax^2+bx} = \sqrt{\frac{\pi}{a}}e^{\frac{b^2}{4a}}.$$ (47)

A very broad Gauss function $g(x)$ with large $\sigma$ is the result of a superposition of plane waves $e^{ikx}$ in a very small range of $k$-values around $k=0$: if $\sigma$ is very large, the weights $e^{-\frac{1}{2}\sigma^2k^2}$ in the Fourier decomposition become very small for large $\vert k\vert$. Discuss the opposite case of small $\sigma$!