Gaussian Function
المؤلف:
MacTutor History of Mathematics Archive. "Frequency Curve." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Frequency.html.
المصدر:
MacTutor History of Mathematics Archive. "Frequency Curve." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Frequency.html.
الجزء والصفحة:
...
2-5-2019
3089
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 
,
![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.gif) |
(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.gif) |
(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.gif) |
(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.
REFERENCES:
MacTutor History of Mathematics Archive. "Frequency Curve." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Frequency.html.
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