Bandwidth of Gamma-Distribution-Shaped Functions via Lambert W Function
Abstract
The full width at half maximum (FWHM) is a useful quantity for characterizing the bandwidth of unimodal functions. However, a closed-form expression for the FWHM of gamma-shaped functions-i.e. functions that are shaped like the gamma distribution probability density function (PDF)-is not widely available. Here, we derive and present just such an expression. To do so, we use the Lambert W function to compute the inverse of the gamma PDF. We use this inverse to derive an exact analytic expression for the width of a gamma distribution at an arbitrary proportion of the maximum, from which the FWHM follows trivially. (An expression for the octave bandwidth of gamma-shaped functions is also provided.) The FWHM is then compared to the Gaussian approximation of gamma-shaped functions. A few other related issues are discussed.
Cite
@article{arxiv.2509.19307,
title = {Bandwidth of Gamma-Distribution-Shaped Functions via Lambert W Function},
author = {Anthony LoPrete and Johannes Burge},
journal= {arXiv preprint arXiv:2509.19307},
year = {2025}
}