Gaussian function

A Gaussian function (the first syllable rhymeshouse) isfunction ofform

for some real constants a > 0, b,c. The eponymthese functionsCarl Friedrich Gauss.

Gaussian functionsc²=2eigenfunctions ofFourier transform.

Gaussian functionsamong those functions that"elementary" but lack "elementary antiderivatives", i.e., their antiderivativesnot amongfunctions ordinarily consideredfirst-year calculus courses. Nonetheless their definite integrals overwhole real line can be evaluated exactly.

This calculation can be performed byresidue theoremcomplex analysis, but therealsosimpleinstructive waydocalculation. Callvaluethis integral I. Then,

Noterenaming ofvariableintegration from xy (see dummy variable). We now changeplane polar coordinates

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

The density function ofnormal probability distribution isGaussian function.

See also: Lorentzian function