Editing Gaussian integral (section)

{{Use American English|date = January 2019}}
{{Short description|Integral of the Gaussian function, equal to sqrt(π)}}
{{hatnote|This integral from statistics and physics is not to be confused with [[Gaussian quadrature]], a method of numerical integration.}}
[[Image:Gaussian Integral.svg|thumb|right|A graph of the function <math>f(x) = e^{-x^2}</math> and the area between it and the <math>x</math>-axis, (i.e. the entire real line) which is equal to <math>\sqrt{\pi}</math>.]]

The '''Gaussian integral''', also known as the '''Euler–Poisson integral''', is the [[integral]] of the [[Gaussian function]] <math>f(x) = e^{-x^2}</math> over the entire real line. Named after the German mathematician [[Carl Friedrich Gauss]], the integral is
<math display="block">\int_{-\infty}^\infty e^{-x^2}\,dx = \sqrt{\pi}.</math>

[[Abraham de Moivre]] originally discovered this type of integral in 1733, while Gauss published the precise integral in 1809,<ref name="The Evolution of the Normal Distribution">{{cite web |url=https://www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/stahl96.pdf |title=The Evolution of the Normal Distribution |work=MAA.org |first=Saul|last=Stahl|date=April 2006|access-date=May 25, 2018}}</ref> attributing its discovery to [[Laplace]]. The integral has a wide range of applications.  For example, with a slight change of variables it is used to compute the [[normalizing constant]] of the [[normal distribution]].  The same integral with finite limits is closely related to both the [[error function]] and the [[cumulative distribution function]] of the [[normal distribution]]. In physics this type of integral appears frequently, for example, in [[quantum mechanics]], to find the probability density of the ground state of the harmonic oscillator. This integral is also used in the path integral formulation, to find the propagator of the harmonic oscillator, and in [[statistical mechanics]], to find its [[partition function (statistical mechanics)|partition function]].

Although no [[elementary function]] exists for the error function, as can be proven by the [[Risch algorithm]],<ref>{{cite journal |first=G. W. |last=Cherry |title=Integration in Finite Terms with Special Functions: the Error Function |journal=Journal of Symbolic Computation |volume=1 |issue=3 |year=1985 |pages=283–302 |doi=10.1016/S0747-7171(85)80037-7 |doi-access=free }}</ref> the Gaussian integral can be solved analytically through the methods of [[multivariable calculus]]. That is, there is no elementary ''[[indefinite integral]]'' for
<math display="block">\int e^{-x^2}\,dx,</math>
but the [[definite integral]]
<math display="block">\int_{-\infty}^\infty e^{-x^2}\,dx</math>
can be evaluated. The definite integral of an arbitrary [[Gaussian function]] is
<math display="block">\int_{-\infty}^{\infty}  e^{-a(x+b)^2}\,dx= \sqrt{\frac{\pi}{a}}.</math>