The Gaussian integral, 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 Gaussian integrals to the complex plane and to multiple dimensions.: 13–15 ...

}{(2p-1)!!}}\sim {\sqrt {\pi \,p}}.} The Gaussian integral can be evaluated through the use of Wallis' integrals. We first prove the following inequalities:...

probability distribution. This follows from a change of variables in the Gaussian integral: ∫ − ∞ ∞ e − u 2 d u = π {\displaystyle \int _{-\infty }^{\infty }e^{-u^{2}}\...

(elliptic integral) 1 ln ⁡ x {\displaystyle {\frac {1}{\ln x}}} (logarithmic integral) e − x 2 {\displaystyle e^{-x^{2}}} (error function, Gaussian integral) sin...

an infinite sum. Occasionally, an integral can be evaluated by a trick; for an example of this, see Gaussian integral. Computations of volumes of solids...

work in a basis of Cartesian Gaussians even when spherical Gaussians are requested, as integral evaluation is much easier in the Cartesian basis, and the...

with domain coloring. The error function at +∞ is exactly 1 (see Gaussian integral). At the real axis, erf z approaches unity at z → +∞ and −1 at z →...

list of integrals of exponential functions. For a complete list of integral functions, please see the list of integrals. Indefinite integrals are antiderivative...

In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that...

form an integral domain, usually written as Z [ i ] {\displaystyle \mathbf {Z} [i]} or Z [ i ] . {\displaystyle \mathbb {Z} [i].} Gaussian integers share...

each. Using the fact that it is a product and the formula for the Gaussian integral gives: ∫ R n f d V = ∏ i = 1 n ( ∫ − ∞ ∞ exp ⁡ ( − 1 2 x i 2 ) d x...

A normal distribution or Gaussian distribution (also known as the "bell-shaped curve") is a concept used in probability theory and statistics. The normal...

theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional...

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...

Gaussian integral Gaussian variogram model Gaussian mixture model Gaussian network model Gaussian noise Gaussian smoothing The inverse Gaussian distribution...

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...

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...

hyperbolic functions List of integrals of exponential functions List of integrals of logarithmic functions List of integrals of Gaussian functions Gradshteyn...

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...

for the Gaussian curvature is Liouville's equation in terms of the Laplacian in isothermal coordinates. The surface integral of the Gaussian curvature...

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...

Dawson function or Dawson integral (named after H. G. Dawson) is the one-sided Fourier–Laplace sine transform of the Gaussian function. The Dawson function...

used for. Gaussian surfaces are usually carefully chosen to exploit symmetries of a situation to simplify the calculation of the surface integral. If the...

(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...

Gaussian units constitute a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs...

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...

integral. The last integral in particular makes clear the connection between the gamma function at half integer arguments and the Gaussian integral:...

p(x)=e^{-x^{2}/2},\quad x\in (-\infty ,\infty )} we have the corresponding Gaussian integral ∫ − ∞ ∞ p ( x ) d x = ∫ − ∞ ∞ e − x 2 / 2 d x = 2 π , {\displaystyle...