Copying comments from a source file is totally the same as writing a blog post, right? Has anyone really considered the full-width N-max of a gaussian before, and then extended that parameter to non gaussian data? Here's how they would start, if they did:
/* Discussion on FWHM for Gaussian lines:
Given a gaussian distribution defined by PDF = 1 / sqrt(2 * pi) * exp(-0.5 * u^2),
then you can define a fractional max location by:
PDF / PDF_MAX = exp(-0.5 * u^2) / exp(-0.5 * 0^2);
PDF / PDF_MAX = exp(-0.5 * u^2)
sqrt(-2.0 * log(R)) = u
So, for a given ratio of the maximum R, the half-width R-max is located at u.
This leads to the full-width R-max location being at
FWRM = 2.0 * sqrt(-2.0 * log(R))
Since we've used u, we can add the sigma dependence back in by noting that u = x / s:
FWRM = 2.0 * sqrt(-2.0 * log(R)) * sigma
The standard value is the FWHM, or R = 0.5:
FWHM = 2.0 * sqrt(-2.0 * log(0.5)) = 2.0 * sqrt(2.0 * log(2.0))
Do this for quarter max, R = 0.25:
FWQM = 2.0 * sqrt(-2.0 * log(0.25)) = 2.0 * sqrt(2.0 * log(4))
= 2.0 * sqrt(2.0 * 2.0 * log(2.0)) = sqrt(2) * FWHM
And for 3/4 max, R = 0.75:
FW3QM = 2.0 * sqrt(-2.0 * log(0.75)) = 2.0 * sqrt(2.0 * log(4/3))
= 2.0 * sqrt(2.0 * (log(4) - log(3)))
= 2.0 * sqrt(2.0 * (2.0 * log(2.0) - log(3)))
=~ 0.64423 * FWHM
If a distribution is "wingier" than a Gaussian, then FWQM will be inflated, such that
FWQM_wingy > FWQM_Gaussian
*/
