Bandwidth of gamma-distribution-shaped functions via Lambert W function

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 full-width at an arbitrary y -proportion of the maximum (FWyM) of a gamma distribution, 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.