A spectrally accurate quadrature for resolving the logarithmic endpoint singularities of the Chandrasekhar H-function

Abstract

The Chandrasekhar H-function of scattering theory was computed by Chandrasekhar himself using a Legendre–Gauss quadrature. Through the explicit forms of the first three terms of a perturbation series (Neumann series) in the albedo ε , we show that the H-function has an x log ( x ) singularity at one endpoint. Consequently, the Legendre coefficients a n of the H-function fall algebraically rather than exponentially with degree, being asymptotically proportional to n - 7 / 2 . We show that by using a new quadrature, one can obtain very high accuracy from a moderate number of grid points. The method of successive substitution converges only inversely linear with iteration number when ε = 1 ; Richardson extrapolation allows moderate accuracy without an immoderate number of iterations.