Gaussian Function - Properties

Properties

Gaussian functions arise by applying the exponential function to a general quadratic function. The Gaussian functions are thus those functions whose logarithm is a 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.

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.

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 a = 1/(c√(2π)), and in this case the Gaussian is the probability density function of a normally distributed random variable with expected value μ = b and variance σ2 = c2. These Gaussians are graphed in the accompanying figure.

Gaussian functions centered at zero minimize the Fourier uncertainty principle.

The product of two Gaussian functions is a Gaussian, and the convolution of two Gaussian functions is again a Gaussian, with .

Taking the Fourier transform (unitary, angular frequency convention) of a Gaussian function with parameters a, b = 0 and c yields another Gaussian function, with parameters ac, b = 0 and 1/c. So in particular the Gaussian functions with b = 0 and c = 1 are kept fixed by the Fourier transform (they are eigenfunctions of the Fourier transform with eigenvalue 1).

The fact that the Gaussian function is an eigenfunction of the Continuous Fourier transform allows to derive the following interesting identity from the Poisson summation formula: