Integral of the Gaussian function, equal to sqrt(π)
This integral from statistics and physics is not to be confused with Gaussian quadrature, a method of numerical integration.
The Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussian function over the entire real line. Named after the German mathematician Carl Friedrich Gauss, the integral is
Abraham de Moivre originally discovered this type of integral in 1733, while Gauss published the precise integral in 1809.[1] 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.
Although no elementary function exists for the error function, as can be proven by the Risch algorithm,[2] 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 definite integral of an arbitrary Gaussian function is
^Stahl, Saul (April 2006). "The Evolution of the Normal Distribution" (PDF). MAA.org. Retrieved May 25, 2018.
^Cherry, G. W. (1985). "Integration in Finite Terms with Special Functions: the Error Function". Journal of Symbolic Computation. 1 (3): 283–302. doi:10.1016/S0747-7171(85)80037-7.
The Gaussianintegral, also known as the Euler–Poisson integral, is the integral of the Gaussian function f ( x ) = e − x 2 {\displaystyle f(x)=e^{-x^{2}}}...
In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form f ( x ) = exp ( − x 2 ) {\displaystyle f(x)=\exp(-x^{2})}...
Common integrals in quantum field theory are all variations and generalizations of Gaussianintegrals to the complex plane and to multiple dimensions.: 13–15 ...
below for other intervals). An integral over [a, b] must be changed into an integral over [−1, 1] before applying the Gaussian quadrature rule. This change...
probability distribution. This follows from a change of variables in the Gaussianintegral: ∫ − ∞ ∞ e − u 2 d u = π {\displaystyle \int _{-\infty }^{\infty }e^{-u^{2}}\...
with domain coloring. The error function at +∞ is exactly 1 (see Gaussianintegral). At the real axis, erf z approaches unity at z → +∞ and −1 at z →...
used for. Gaussian surfaces are usually carefully chosen to exploit symmetries of a situation to simplify the calculation of the surface integral. If the...
theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional...
hyperbolic functions List of integrals of exponential functions List of integrals of logarithmic functions List of integrals of Gaussian functions Gradshteyn...
(hx)}{1+x^{2}}}\,dx} is Owen's T function. Owen has an extensive list of Gaussian-type integrals; only a subset is given below. ∫ φ ( x ) d x = Φ ( x ) + C {\displaystyle...
an infinite sum. Occasionally, an integral can be evaluated by a trick; for an example of this, see Gaussianintegral. Computations of volumes of solids...
In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued...
The Fresnel integrals S(x) and C(x) are two transcendental functions named after Augustin-Jean Fresnel that are used in optics and are closely related...
form an integral domain, usually written as Z [ i ] {\displaystyle \mathbf {Z} [i]} or Z [ i ] . {\displaystyle \mathbb {Z} [i].} Gaussian integers share...
list of integrals of exponential functions. For a complete list of integral functions, please see the list of integrals. Indefinite integrals are antiderivative...
these integrals is infinite, or both if they have the same sign. An example of an improper integral where both endpoints are infinite is the Gaussian integral...
In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that...
product is a Gaussian as a function of x(t + ε) centered at x(t) with variance ε. The multiple integrals are a repeated convolution of this Gaussian Gε with...
processing, a Gaussian filter is a filter whose impulse response is a Gaussian function (or an approximation to it, since a true Gaussian response would...
Gaussian integralGaussian variogram model Gaussian mixture model Gaussian network model Gaussian noise Gaussian smoothing The inverse Gaussian distribution...
still be finite. The functional integrals that can be evaluated exactly usually start with the following Gaussianintegral: ∫ exp { − 1 2 ∫ R [ ∫ R f (...
Gaussian units constitute a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs...
for the Gaussian curvature is Liouville's equation in terms of the Laplacian in isothermal coordinates. The surface integral of the Gaussian curvature...
formula, graphing quadratic functions, evaluating integrals in calculus, such as Gaussianintegrals with a linear term in the exponent, finding Laplace...
directly from the path integral. The factor of i disappears in the Euclidean theory. Because each field mode is an independent Gaussian, the expectation values...
H {\displaystyle H} , then the above integral can be interpreted as integration against a well-defined (Gaussian) measure on B {\displaystyle B} . In...