Gaussian function

In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a of the form:

for arbitrary real constants a, b and c. It is named after the mathematician Carl Friedrich Gauss.

The graph of a Gaussian is a characteristic symmetric " 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 widely used in statistics where they describe the normal distributions, in signal processing where they serve to define Gaussian filters, in image processing where two-dimensional Gaussians are used for Gaussian blurs, and in mathematics where they are used 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

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.

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