  We present a semi-analytic approach to solving the Boltzmann equation
describing the comptonisation of low frequency input photons by a thermal
distribution of electrons in the Thomson limit. Our work is based on the
formulation of the problem by Titarchuk & Lyubarskij (ApJ 450 (1995) 876), but
extends their treatment by accommodating an arbitrary anisotropy of the source
function. To achieve this, we expand the eigenfunctions of the
integro/differential eigenvalue problem defining the spectral index of
comptonised radiation in terms of Legendre polynomials and Chebyshev
polynomials. The resulting algebraic eigenvalue problem is then solved by
numerical means, yielding the spectral index and the full angular and spatial
dependence of the specific intensity of radiation. For a thin (\tau_0 < 1)
plasma disk, the radiation is strongly collimated along the disk surface - for
an optical thickness of \tau_0 = 0.05, the radiation intensity along the
surface is roughly ten times that along the direction of the normal, and varies
only slightly with the electron temperature. Our results for the spectral index
confirm those of Titarchuk & Lyubarskij over a wide range of electron
temperature and optical depth; the largest difference we find is roughly 10%
and occurs at low optical depth.
