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Date: 3-11-2020
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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.
REFERENCES:
MacTutor History of Mathematics Archive. "Frequency Curve." https://www-groups.dcs.st-and.ac.uk/~history/Curves/Frequency.html.
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