A new representation of theH-functions of radiative transfer

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(=ø _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),ø _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.