Gaussian function

A Gaussian function has the form of a constant times an exponential of a negative quadratic. Its graph is bell-shaped, with its width, height, and center determined by its parameters.

Exponential function: A function in which the variable appears in the exponent. A Gaussian is built by applying an exponential to a negative quadratic.
Derivative: The instantaneous rate at which a function changes with its input. Differentiation of a Gaussian produces a linear factor times the same exponential.
Normal distribution: A continuous probability distribution whose density has a symmetric bell-shaped Gaussian form. It is the normalized probabilistic use of the Gaussian function.
Lorentzian function: A peaked function with tails that decrease quadratically with distance from its center. Its heavier tails produce very different peak and far-field behavior.
Quadratic function: A polynomial function of degree two, with a graph that is a parabola. The quadratic in the exponent determines the Gaussian's symmetry and decay.
Gaussian integral: The integral of an exponential of a negative squared variable, evaluated using a two-dimensional integral argument. It gives the exact total area beneath the Gaussian curve.
Gaussian blur: An image-processing filter that smooths pixels by weighting neighbors with a Gaussian kernel. It applies the Gaussian as a spatial smoothing weight.
Laplace distribution: A continuous probability distribution with a symmetric density that decreases exponentially with absolute distance from its center. Its sharper central peak and heavier tails contrast with a Gaussian density.
Standard deviation: A measure of spread around the mean, defined as the square root of variance. In the standard Gaussian form, it sets the curve's horizontal width.
Convolution: An operation that combines two functions by integrating products of shifted values. Convolving Gaussian functions produces another Gaussian, with variances adding.

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