SOME PROPERTIES OF AN INTEGRAL THAT OCCURS IN PROBLEMS OF RADIATIVE-TRANSFER

The integral S(r, n, tau) = integral(0)(1) mu(r)e(-tau/mu)P(n)(mu)d mu where r and n are non-negative integers, and P-n, a Legendre polynomial, occurs in certain treatments of radiative transfer problems, particularly if the radiance field is expanded in a series of spherical harmonics and reflection occurs. This function can be expressed exactly as a series of exponential integrals of differing order, but it will be shown that this representation is inappropriate for its calculation for even moderate values of n because of heavy cancellation. In this note a recurrence relation is derived which can be used to evaluate S for a wide range of n, at fixed r and tau. A simple approximation for this integral is also developed, which gives the asymptotic behaviour for large n and fixed r, tau. The asymptotic behaviour of S for fixed r, n and large tau (tau greater than or equal to n(2)) is also given.