A new representation of the H -functions of radiative transfer

TLDR: In this paper, a new representation of Chandrasekhar's H-functions corresponding to the dispersion function was obtained in the form of a Fredholm type integral equation, which has proved to be very useful in solving coupled integral equations involving X-,Y -functions of transport problems; a closed form approximation ofH(z) to a sufficiently high degree of accuracy is then readily available by term integrations.

Abstract: We obtain a new representation of Chandrasekhar'sH-functionsH(z) corresponding to the dispersion functionT(z) = |δ rs −f rs (z)|, [f rs (z)] is of rank one.H(z) is obtained in the form $$H\left( z \right) = \left( {A_0 + A_1 z} \right)/\left( {K + z} \right) - \sum\limits_1^n {\int\limits_{E_r } {P_r (x) dx/(x + z),} }$$ WhereP r x(=o r (x)/H(x)) is continuous onE r which are subsets of [0, 1].A o ,A 1 are determinable constants andK is the positive root ofT(z),o r (x) are known functions. From this formH(z) is then obtained in terms of a Fredholm type integral equation. This new form ofH(z) has proved to be very useful in solving coupled integral equations involvingX-,Y-functions of transport problems.P r(x) can be replaced by approximating polynomials whose coefficients can be determined as functions of the moments of known functions; a closed form approximation ofH(z) to a sufficiently high degree of accuracy is then readily available by term integrations.