The Gaussian integral, or probability 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 wide applications. When normalized so that its value is 1, it is the density function of the normal distribution (see also error function). It is an eigenfunction of the continuous Fourier transform.
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 tools of calculus. That is, there is no elementary indefinite integral for , but the definite integral can be evaluated.
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