Some properties of an integral that occurs in problems of radiative transfer

Details Some properties of an integral that occurs in problems of radiative transfer Some properties of an integral that occurs in problems of radiative transfer

Aug 1994

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1994jqsrt..52..195s&link_type=abstract

Journal of Quantitative Spectroscopy & Radiative Transfer (ISSN 0022-4073), vol. 52, no. 2, p. 195-206

Mathematics

1

Approximation, Asymptotic Properties, Integrals, Mathematical Models, Radiative Transfer, Bessel Functions, Legendre Functions, Reflection, Spherical Harmonics

Scientific paper

The integral S(r, n, tau) = integral from 0 to 1 of ((mu(exp r))(e(exp -tau/mu))(P(sub n)(nu))(d(mu))) where r and n are non-negative integers, and P(sub 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 if n because of heavy cancellation. In this note a recurrence relation is derived which can be used to evaluate S for wide range of n, at fixed r and tau. A simple approximation for this integral is also developed, which gives the asymptotic behavior for large n and fixed r, tau. The asymptotic behavior of S for fixed r, n and large tau (tau greater than or equal to n(exp 2) is also given.

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