"Gaussian curve" redirects here. For the band, see Gaussian Curve (band).
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In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form
and with parametric extension
for arbitrary real constants a, b and non-zero c. It is named after the mathematician Carl Friedrich Gauss. The graph of a Gaussian is a characteristic symmetric "bell curve" shape. The parameter a is the height of the curve's peak, b is the position of the center of the peak, and c (the standard deviation, sometimes called the Gaussian RMS width) controls the width of the "bell".
Gaussian functions are often used to represent the probability density function of a normally distributed random variable with expected value μ = b and variance σ2 = c2. In this case, the Gaussian is of the form[1]
Gaussian functions are widely used in statistics to describe the normal distributions, in signal processing to define Gaussian filters, in image processing where two-dimensional Gaussians are used for Gaussian blurs, and in mathematics to solve heat equations and diffusion equations and to define the Weierstrass transform.
^Squires, G. L. (2001-08-30). Practical Physics (4 ed.). Cambridge University Press. doi:10.1017/cbo9781139164498. ISBN 978-0-521-77940-1.
In mathematics, a Gaussianfunction, 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})}...
the figures at right with domain coloring. The error function at +∞ is exactly 1 (see Gaussian integral). At the real axis, erf z approaches unity at...
a Gaussian beam is an idealized beam of electromagnetic radiation whose amplitude envelope in the transverse plane is given by a Gaussianfunction; this...
In image processing, a Gaussian blur (also known as Gaussian smoothing) is the result of blurring an image by a Gaussianfunction (named after mathematician...
processing, a Gaussian filter is a filter whose impulse response is a Gaussianfunction (or an approximation to it, since a true Gaussian response would...
The Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussianfunction f ( x ) = e − x 2 {\displaystyle f(x)=e^{-x^{2}}}...
_{0}^{a}{\frac {\varphi (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 (...
In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued...
coordinate axes. Only the Gaussianfunction is both separable and isotropic. The separable forms of all other window functions have corners that depend...
processing theory, Gaussian noise, named after Carl Friedrich Gauss, is a kind of signal noise that has a probability density function (pdf) equal to that...
In probability theory, an exponentially modified Gaussian distribution (EMG, also known as exGaussian distribution) describes the sum of independent normal...
modified Gaussian distribution or function, used for description of peak shape in many techniques Gauss error functionGaussian process Gaussian filter...
frequency within a bandwidth inversely proportional to that width; even a gaussianfunction is considered a wave packet because its Fourier transform is a "packet"...
generating function (logarithm of the characteristic function)[contradictory] is the inverse of the cumulant generating function of a Gaussian random variable...
of a Gaussian process is the joint distribution of all those (infinitely many) random variables, and as such, it is a distribution over functions with...
expected value x0; it is invariant under translations. If the FWHM of a Gaussianfunction is known, then it can be integrated by simple multiplication. In spectroscopy...
exponential function – Exponential function of an exponential function Exponential field – Mathematical field with an extra operation Gaussianfunction Half-exponential...
definite function. Such functions, including the Gaussian, inverse quadratic, and inverse multiquadric are often used as radial basis functions for this...
chemistry and molecular physics, Gaussian orbitals (also known as Gaussian type orbitals, GTOs or Gaussians) are functions used as atomic orbitals in the...
function or Dawson integral (named after H. G. Dawson) is the one-sided Fourier–Laplace sine transform of the Gaussianfunction. The Dawson function is...
uncertainty principle only for the Gaussianfunction. Equivalently, π is the unique constant making the Gaussian normal distribution e−πx2 equal to its...
normalized second derivative of a Gaussianfunction, i.e., up to scale and normalization, the second Hermite function. It is a special case of the family...
sample of the training set. Gaussianfunction Kernel (statistics) Polynomial kernel Radial basis function Radial basis function network Obst kernel network...
uncertainty principle. The critical case for this principle is the Gaussianfunction, of substantial importance in probability theory and statistics as...
a function f : R → R, named after Karl Weierstrass, is a "smoothed" version of f(x) obtained by averaging the values of f, weighted with a Gaussian centered...
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