Gaussian function

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

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

Gaussian functions with c²=2 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.

This calculation can be performed by the residue theorem of complex analysis, but there is also a simple and instructive way to do the calculation. Call the value of this integral I. Then,

Note the renaming of the variable of integration from x to y (see dummy variable). We now change to plane polar coordinates

(The substitution u = r², du = 2r dr was used.)

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