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In one dimension, the Gaussian function is the probability density function of the normal distribution,
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(1)
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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
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(2)
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But 
occurs at 
,
so
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(3)
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Solving,
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(4)
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(5)
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(6)
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(7)
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The full width at half maximum is therefore given by
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(8)
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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)).](/images/equations/GaussianFunction/NumberedEquation9.svg) |
(9)
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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)]).](/images/equations/GaussianFunction/NumberedEquation10.svg) |
(10)
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The Gaussian function can also be used as an apodization function
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(11)
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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)))],](/images/equations/GaussianFunction/NumberedEquation12.svg) |
(12)
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which has maximum
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(13)
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As 
,
equation (12) reduces to
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(14)
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The hypergeometric function is also sometimes known as the Gaussian function.
