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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})} and with parametric extension f (x) = a exp (− (x − b) 2 2 c 2) {\displaystyle f(x)=a\exp \left(-{\frac {(x-b)^{2}}{2c^{2}}}\right)} 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 σ = c. In this case, the Gaussian is of the form

g (x) = 1 σ 2 π exp (− 1 2 (x − μ) 2 σ 2). {\displaystyle g(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}\exp \left(-{\frac {1}{2}}{\frac {(x-\mu)^{2}}{\sigma ^{2}}}\right).}

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.