FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA
(FUNDAMENTAL AND APPLIED MATHEMATICS)
2002, VOLUME 8, NUMBER 1, PAGES 97-115
A. V. Latyshev
A. V. Moiseev
Abstract
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The theory of the solution of half-space boundary-value
problems for Chandrasekhar's equations describing the scattering
of polarized light in the case of a combination of Rayleigh and
isotropic scattering with arbitrary photon survival probability
in an elementary scattering is constructed. A theorem on
the expansion of the solution in terms of eigenvectors of
discrete and continuous spectra is proved. The proof reduces to
solving the Riemann--Hilbert vector boundary-value problem
with a matrix coefficient. The matrix that reduces the coefficient
to diagonal form has eight branch points in the complex plain.
The definition of an analytical branch of a diagonalizing matrix
gives us the opportunity to reduce the Riemann--Hilbert
vector boundary-value problem to two scalar boundary-value
problems on the major cut $[0,1]$ and two vector boundary-value problems on the supplementary cut.
The solution of the Riemann--Hilbert boundary-value
problem is given in the class of meromorphic vectors.
The solvability conditions enable
unique determination of the unknown coefficients of
the expansion and the free parameters of the solution.
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Last modified: July 5, 2002