Radiative transfer in plane-parallel media and Cauchy integral equations III. The finite case

TL;DR

This paper revisits the solution of Cauchy integral equations in finite plane-parallel radiative transfer media, introducing auxiliary functions with notable analytical properties for accurate computation.

Contribution

It introduces new auxiliary functions, $ta_+$ and $ta_-$, with remarkable properties that generalize existing factorization relations for finite media.

Findings

Auxiliary functions $ta_+$ and $ta_-$ are regular and computable in the complex plane.

The paper extends Sobouti's functions into the complex plane.

X- and Y-functions are explicitly calculated across the complex plane.

Abstract

We come back to the Cauchy integral equations occurring in radiative transfer problems posed in finite, plane-parallel media with light scattering taken as monochromatic and isotropic. Their solution is calculated following the classical scheme where a Cauchy integral equation is reduced to a couple of Fredholm integral equations. It is expressed in terms of two auxiliary functions $\zeta_+$ and $\zeta_-$ we introduce in this paper. These functions show remarkable analytical properties in the complex plane. They satisfy a simple algebraic relation which generalizes the factorization relation of semi-infinite media. They are regular in the domain of the Fredholm integral equations they satisfy, and thus can be computed accurately. As an illustration, the X- and Y-functions are calculated in the whole complex plane, together with the extension in this plane of the so-called Sobouti's functions.