Gaussian Function

In 1-D, the Gaussian function is the function from the Gaussian Distribution,

$$f(x) = {1\over\sigma\sqrt{2\pi}} e^{-(x-\mu)^2/2\sigma^2},$$ (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

$$e^{-(x_0-\mu)^2/2\sigma^2}={\textstyle{1\over 2}}f(x_{\rm max})$$ (2)

But $f(x_{\rm max})$ occurs at $x_{\rm max}=\mu$, so

$$e^{-(x_0-\mu)^2/2\sigma^2}={\textstyle{1\over 2}}f(\mu)={\textstyle{1\over 2}}.$$ (3)

Solving,

$$e^{-(x_0-\mu)^2/2\sigma^2} = 2^{-1}$$ (4)

$$-{(x_0-\mu)^2\over 2\sigma^2} =-\ln 2$$ (5)

$$(x_0-\mu)^2=2\sigma^2\ln 2$$ (6)

$$x_0 =\pm \sigma\sqrt{2\ln 2}+\mu.$$ (7)

The Full Width at Half Maximum is therefore given by

$${\rm FWHM} \equiv x_+-x_- = 2\sqrt{2\ln 2}\,\sigma \approx 2.3548\sigma.$$ (8)

In 2-D, the circular Gaussian function is the distribution function for uncorrelated variables $x$ and $y$ having a Gaussian Bivariate Distribution and equal Standard Deviation $\sigma=\sigma_x=\sigma_y$,

$$f(x,y)={1\over 2\pi\sigma^2} e^{-[(x-\mu_x)^2+(y-\mu_y)^2]/2\sigma^2}.$$ (9)

The corresponding elliptical Gaussian function corresponding to $\sigma_x\not=\sigma_y$ is given by

$$f(x,y)={1\over 2\pi\sigma_x\sigma_y}e^{-[(x-\mu_x)^2/2{\sigma_x}^2+[(y-\mu_y)^2/2{\sigma_y}^2]}.$$ (10)

The above plots show the real and imaginary parts of $(2\pi)^{-1/2}e^{-z^2}$ together with the complex absolute value $\vert(2\pi)^{-1/2}e^{-z^2}\vert$.

The Gaussian function can also be used as an Apodization Function, shown above with the corresponding Instrument Function.

The Hypergeometric Function is also sometimes known as the Gaussian function.

See also Erf, Erfc, Fourier Transform--Gaussian, Gaussian Bivariate Distribution, Gaussian Distribution, Normal Distribution

References

MacTutor History of Mathematics Archive. ``Frequency Curve.'' http://www-groups.dcs.st-and.ac.uk/~history/Curves/Frequency.html.