Gaussian integral

The Gaussian integral, also known as the Euler–Poisson integral is the integral of the Gaussian function e^−x2 over the entire real line. It is named after the German mathematician and physicist Carl Friedrich Gauss. The integral is:

This 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 both to 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, also in the path integral formulation, and to find the propagator of the harmonic oscillator, we make use of this integral.

Although no elementary function exists for the error function, as can be proven by the Risch algorithm, the Gaussian integral can be solved analytically through the methods of multivariable calculus. That is, there is no elementary indefinite integral for

but the definite integral

can be evaluated.

The Gaussian integral is encountered very often in physics and numerous generalizations of the integral are encountered in quantum field theory.

A standard way to compute the Gaussian integral, the idea of which goes back to Poisson, is to make use of the property that:

${\displaystyle \left(\int _{-\infty }^{\infty }e^{-x^{2}}\,dx\right)^{2}=\int _{-\infty }^{\infty }e^{-x^{2}}\,dx\int _{-\infty }^{\infty }e^{-y^{2}}\,dy=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }e^{-(x^{2}+y^{2})}\,dx\,dy}$

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