Editing Gaussian function (section)

==Applications==
Gaussian functions appear in many contexts in the [[natural sciences]], the [[social sciences]], [[mathematics]], and [[engineering]].  Some examples include:
* In [[statistics]] and [[probability theory]], Gaussian functions appear as the density function of the [[normal distribution]], which is a limiting [[probability distribution]] of complicated sums, according to the [[central limit theorem]].
* Gaussian functions are the [[Green's function]] for the (homogeneous and isotropic) [[diffusion equation]] (and to the [[heat equation]], which is the same thing), a [[partial differential equation]] that describes the time evolution of a mass-density under [[diffusion]]. Specifically, if the mass-density at time ''t''=0 is given by a [[Dirac delta]], which essentially means that the mass is initially concentrated in a single point, then the mass-distribution at time ''t'' will be given by a Gaussian function, with the parameter '''''a''''' being linearly related to 1/{{radic|''t''}} and '''''c''''' being linearly related to {{radic|''t''}}; this time-varying Gaussian is described by the [[heat kernel]]. More generally, if the initial mass-density is &phi;(''x''), then the mass-density at later times is obtained by taking the [[convolution]] of &phi; with a Gaussian function. The convolution of a function with a Gaussian is also known as a [[Weierstrass transform]].
* A Gaussian function is the [[wave function]] of the [[ground state]] of the [[quantum harmonic oscillator]].
* The [[molecular orbital]]s used in [[computational chemistry]] can be [[linear combination]]s of Gaussian functions called [[Gaussian orbital]]s (see also [[basis set (chemistry)]]).
* Mathematically, the [[derivative]]s of the Gaussian function can be represented using [[Hermite functions]]. For unit variance, the ''n''-th derivative of the Gaussian is the Gaussian function itself multiplied by the ''n''-th [[Hermite polynomial]], up to scale.
* Consequently, Gaussian functions are also associated with the [[vacuum state]] in [[quantum field theory]].
* [[Gaussian beam]]s are used in optical systems, microwave systems and lasers.
* In [[scale space]] representation, Gaussian functions are used as smoothing kernels for generating multi-scale representations in [[computer vision]] and [[image processing]]. Specifically, derivatives of Gaussians ([[Hermite functions]]) are used as a basis for defining a large number of types of visual operations.
* Gaussian functions are used to define some types of [[artificial neural network]]s.
* In [[fluorescence microscopy]] a 2D Gaussian function is used to approximate the [[Airy disk]], describing the intensity distribution produced by a [[point source]].
* In [[signal processing]] they serve to define [[Gaussian filter]]s, such as in [[image processing]] where 2D Gaussians are used for [[Gaussian blur]]s. In [[digital signal processing]], one uses a [[discrete Gaussian kernel]], which may be approximated by the [[Binomial coefficient]]<ref>Haddad, R.A. and Akansu, A.N., 1991, ''[https://doi.org/10.1109/78.80892 A Class of Fast Gaussian Binomial Filters for Speech and Image processing]'', IEEE Trans. on Signal Processing, 39-3: 723–727</ref> or sampling a Gaussian.
* In [[geostatistics]] they have been used for understanding the variability between the patterns of a complex [[training image]]. They are used with kernel methods to cluster the patterns in the feature space.<ref>Honarkhah, M and Caers, J, 2010, ''[https://dx.doi.org/10.1007/s11004-010-9276-7 Stochastic Simulation of Patterns Using Distance-Based Pattern Modeling]'', Mathematical Geosciences, 42: 487–517</ref>