Generalized Sobolev's phi function for the resolvent of a Milne-type integral equation with a degenerate kernel |
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Authors: | Sueo Ueno |
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Institution: | Department of Electrical Engineering, University of Southern California, Los Angeles, California 90007 USA |
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Abstract: | It is well known in the field of radiative transfer that Sobolev was the first to introduce the resolvent into Milne's integral equation with a displacement kernel. Thereafter it was shown that the resolvent plays an important role in the theory of formation of spectral lines. In the theory of line-transfer problems, the kernel representation in Milne's integral equation has been used to provide an approximate solution in a manner similar to that given by the discrete ordinales method.In this paper, by means of invariant imbedding we show how to determine an exact solution of a Milne-type integral equation with a degenerate kernel, whose form is more general than the Pincherle-Gourast kernel. A Cauchy system for the resolvent is expressed in terms of generalized Sobolev's Φ- and Ψ-functions, which are computed by solving a system of differential equations for auxiliary functions. Furthermore, these functions are expressed in terms of components of the kernel representation. |
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