Gaussian function

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 σ^2 = c^2. 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 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: where (Note: in , not to be confused with ) 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. The full width at tenth of maximum (FWTM) for a Gaussian could be of interest and is Gaussian functions are analytic, and their limit as x → ∞ is 0 (for the above case of b = 0). Gaussian functions are among those functions that are elementary but lack elementary antiderivatives; the integral of the Gaussian function is the error function: Nonetheless, their improper integrals over the whole real line can be evaluated exactly, using the Gaussian integral and one obtains This integral is 1 if and only if (the normalizing constant), and in this case the Gaussian is the probability density function of a normally distributed random variable with expected value μ = b and variance σ^2 = c^2: These Gaussians are plotted in the accompanying figure.