Gaussian function

A Gaussian function (the first syllable rhymes with house) is a function of the form

$f(x)=ae^{-(x-b)^{2}/c^{2}}$

for some real constants a > 0, b, and c. The eponym of these functions is Carl Friedrich Gauss.

Gaussian functions are eigenfunctions of the Fourier transform.

Gaussian functions are among those functions that are "elementary" but lack "elementary antiderivatives", i.e., their antiderivatives are not among the functions ordinarily considered in first-year calculus courses. Nonetheless their definite integrals over the whole real line can be evaluated exactly.

$\int _{-\infty }^{\infty }e^{-x^{2}}\,dx={\sqrt {\pi }}.$

[Later I may add details showing how this integral is found.]

The density function of the normal probability distribution is a Gaussian function.