Editing Gaussian function (section)

{{Short description|Mathematical function}}
{{Redirect|Gaussian curve|the band|Gaussian Curve (band)}}
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In [[mathematics]], a '''Gaussian function''', often simply referred to as a '''Gaussian''', is a [[function (mathematics)|function]] of the base form
<math display="block">f(x) = \exp (-x^2)</math>
and with parametric extension
<math display="block">f(x) = a \exp\left( -\frac{(x - b)^2}{2c^2} \right)</math>
for arbitrary [[real number|real]] constants {{mvar|a}}, {{mvar|b}} and non-zero {{mvar|c}}. It is named after the mathematician [[Carl Friedrich Gauss]]. The [[graph of a function|graph]] of a Gaussian is a characteristic symmetric "[[Normal distribution|bell curve]]" shape. The parameter {{mvar|a}} is the height of the curve's peak, {{mvar|b}} is the position of the center of the peak, and {{mvar|c}} (the [[standard deviation]], sometimes called the Gaussian [[Root mean square|RMS]] width) controls the width of the "bell".

Gaussian functions are often used to represent the [[probability density function]] of a [[normal distribution|normally distributed]] [[random variable]] with [[expected value]] {{math|1=<var>μ</var> = <var>b</var>}} and [[variance]] {{math|1=<var>σ</var>{{sup|2}} = <var>c</var>{{sup|2}}}}. In this case, the Gaussian is of the form<ref>{{Cite book |last=Squires |first=G. L. |url=https://www.cambridge.org/core/product/identifier/9781139164498/type/book |title=Practical Physics |date=2001-08-30 |publisher=Cambridge University Press |isbn=978-0-521-77940-1 |edition=4 |doi=10.1017/cbo9781139164498}}</ref>

<math display="block">g(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\left( -\frac{1}{2} \frac{(x - \mu)^2}{\sigma^2} \right).</math>

Gaussian functions are widely used in [[statistics]] to describe the [[normal distribution]]s, in [[signal processing]] to define [[Gaussian filter]]s, in [[image processing]] where two-dimensional Gaussians are used for [[Gaussian blur]]s, and in mathematics to solve [[heat equation]]s and [[diffusion equation]]s and to define the [[Weierstrass transform]]. They are also abundantly used in [[quantum chemistry]] to form [[Basis set (chemistry)|basis sets]].