Gaussian function

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In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form $${\displaystyle f(x)=\exp(-x^{2})}$$ and with parametric extension $${\displaystyle f(x)=a\exp \left(-{\frac {(x-b)^{2}}{2c^{2}}}\right)}$$ 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 σ² = c². In this case, the Gaussian is of the form^[1]

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

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.