By applying the matrix-exponential operator technique to the radiative-transfer equation in discrete form, new analytical solutions are obtained for the transmission and reflection matrices in the limiting cases <i>x</i> ≪ 1 and <i>x</i> ≫ 1, where <i>x</i> is the optical depth of the layer. Orthogonality of the eigenvectors of the matrix exponential apparently yields new conditions for determining Chandrasekhar’s characteristic roots. The exact law of reflection for the discrete eigenfunctions is also obtained. Finally, when used in conjunction with the doubling method, the matrix exponential should result in reductions in both computation time and loss of precision.
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