Light transport in homogeneous tissue with m-dependent anisotropic scattering I: Case's singular eigenfunctions solution and Chandrasekhar polynomials.

This paper is the first of two deriving the analytical solutions for light transport in infinite homogeneous tissue with an azimuth-dependent (m-dependent) anisotropic scattering kernel by two approaches, Case's singular eigenfuncions expansion and Fourier transform, as well as proving the consistence of the two solutions. In this paper, Case's method was applied and extended to the general m-dependent anisotropic scattering case. The explicit Green's function of radiance distributions, which was regarded as the comparative standard for the equivalent solution via Fourier transform and inversion in our second accompanying paper, was expanded into a complete set of the discrete and continuous eigenfunctions. Considering that the two kinds of m-dependent Chandrasekhar orthogonal polynomials that play vital roles in these analytical solutions are very sensitive to the typical optical parameters of biological tissue as well as the degrees or orders, four numerical evaluation methods were benchmarked to find the stable, reliable and feasible numerical evaluation methods in high degrees and high orders.