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GeezerVison is showing mathworld.wolfram.com — Gaussian Function -- from Wolfram MathWorld

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Gaussian Function

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A Gaussian function in one dimension, when normalized to have unit integral, is the probability density function of the normal distribution,

f(x)=1/(sigmasqrt(2pi))e^(-(x-mu)^2/(2sigma^2)),
(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 x_0. The constant scaling factor can be ignored, so we must solve

e^(-(x_0-mu)^2/(2sigma^2))=1/2f(x_(max))
(2)

But f(x_(max)) occurs at x_(max)=mu, so

e^(-(x_0-mu)^2/(2sigma^2))=1/2f(mu)=1/2.
(3)

Solving,

e^(-(x_0-mu)^2/(2sigma^2))=2^(-1)
(4)
-((x_0-mu)^2)/(2sigma^2)=-ln2
(5)
(x_0-mu)^2=2sigma^2ln2
(6)
x_0=+/-sigmasqrt(2ln2)+mu.
(7)

The full width at half maximum is therefore given by

FWHM=x_+-x_-=2sqrt(2ln2)sigma approx 2.3548sigma.
(8)
GaussianFunction2D

In two dimensions, the circular Gaussian function is the distribution function for uncorrelated variates X and Y having a bivariate normal distribution and equal standard deviation sigma=sigma_x=sigma_y,

f(x,y)=1/(2pisigma^2)e^(-[(x-mu_x)^2+(y-mu_y)^2]/(2sigma^2)).
(9)

The corresponding elliptical Gaussian function corresponding to sigma_x!=sigma_y 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)]).
(10)
GaussianApodization

The Gaussian function can also be used as an apodization function

A(x)=e^(-x^2/(2sigma^2)),
(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)))],
(12)

which has maximum

I_(max)=sigmasqrt(2pi)erf(a/(sigmasqrt(2))).
(13)

As a->infty, equation (12) reduces to

lim_(a->infty)I(k)=sigmasqrt(2pi)e^(-2pi^2k^2sigma^2).
(14)

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

See also

Bivariate Normal Distribution, Erf, Erfc, Fourier Transform--Gaussian, Hyperbolic Secant, Lorentzian Function, Normal Distribution, Owen T-Function, Witch of Agnesi

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References

MacTutor History of Mathematics Archive. "Frequency Curve." https://mathshistory.st-andrews.ac.uk/Curves/Frequency/.

Referenced on Wolfram|Alpha

Gaussian Function

Cite this as:

Weisstein, Eric W. "Gaussian Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GaussianFunction.html

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