Gaussian Function

In one dimension, the Gaussian function is the probability density function of the normal distribution,

(1)

sometimes also called the frequency curve. The full width at half maximum (FWHM) for a Gaussian is found by finding the half-maximum points . The constant scaling factor can be ignored, so we must solve

(2)

But occurs at , so

(3)

Solving,

(4)

(5)

(6)

(7)

The full width at half maximum is therefore given by

(8)

In two dimensions, the circular Gaussian function is the distribution function for uncorrelated variates and having a bivariate normal distribution and equal standard deviation ,

(9)

The corresponding elliptical Gaussian function corresponding to is given by

(10)

The Gaussian function can also be used as an apodization function

(11)

shown above with the corresponding instrument function. The instrument function is

(12)

which has maximum

(13)

As , equation (12) reduces to

(14)

The hypergeometric function is also sometimes known as the Gaussian function.