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 
,
![f(x,y)=1/(2pisigma^2)e^(-[(x-mu_x)^2+(y-mu_y)^2]/(2sigma^2)).](http://mathworld.wolfram.com/images/equations/GaussianFunction/NumberedEquation9.svg) | (9) |
The corresponding elliptical Gaussian function corresponding to 
is given by
![f(x,y)=1/(2pisigma_xsigma_y)e^(-[(x-mu_x)^2/(2sigma_x^2)+(y-mu_y)^2/(2sigma_y^2)]).](http://mathworld.wolfram.com/images/equations/GaussianFunction/NumberedEquation10.svg) | (10) |
The Gaussian function can also be used as an apodization function
 | (11) |
shown above with the corresponding instrument function. The instrument function is
![I(k)=e^(-2pi^2k^2sigma^2)sigmasqrt(pi/2)[erf((a-2piiksigma^2)/(sigmasqrt(2)))+erf((a+2piiksigma^2)/(sigmasqrt(2)))],](http://mathworld.wolfram.com/images/equations/GaussianFunction/NumberedEquation12.svg) | (12) |
which has maximum
 | (13) |
As 
, equation (12) reduces to
 | (14) |
The hypergeometric function is also sometimes known as the Gaussian function.
