Gaussian function

In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the form: In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the form: 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: 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. Gaussian functions arise by composing the exponential function with a concave quadratic function. The Gaussian functions are thus those functions whose logarithm is a concave quadratic function. The parameter c is related to the full width at half maximum (FWHM) of the peak according to The function may then be expressed in terms of the FWHM, represented by w: Alternatively, the parameter c can be interpreted by saying that the two inflection points of the function occur at x = b − c and x = b + c. The full width at tenth of maximum (FWTM) for a Gaussian could be of interest and is