A cauchy integral equation method for analytic solutions of half-space convolution equations

Abstract

This paper is devoted to a method of Cauchy integral equation for the solution of half-space convolution equations. It was introduced by Frisch and Frisch to solve Wiener-Hopf integral equations with algebraically decreasing kernels, arising in non-coherent transfer with complete frequency redistribution. The standard Wiener-Hopf technique, which requires exponentially decreasing kernels, is not directly applicable to those equations. We show here that coherent transfer may also be treated by the Cauchy integral method, which draws its name from the fact that the first step is to perform an inverse Laplace transformation on the convolution equation, in order to recast it into a singular integral equation of the Cauchy-type. For complete redistribution, this Cauchy integral equation may be solved by the classical reduction technique to a boundary value problem in the complex plane. For monochromatic transfer, the situation is somewhat more complicated, because of the presence of discrete eigenvalues in the spectrum of the transport operator. Dirac distributions localized at the isolated eigenvalues must be introduced before the complex plane method of solution becomes applicable again. The analytic solution readily yields the behavior of the radiation field at infinity. The emergent radiation field is also easily obtained. Various examples chosen among standard problems are used to illustrate the method.