  Two methods are developed for solving the steady-state spherically symmetric
radiative transfer equation for resonance line radiation emitted by a point
source in the Intergalactic Medium. One method is based on solving the ray and
moment equations using finite differences. The second uses a Monte Carlo
approach incorporating methods that greatly improve the accuracy compared with
previous approaches in this context. Several applications are presented serving
as test problems for both a static medium and an expanding medium, including
inhomogeneities in the density and velocity fields. Solutions are obtained in
the coherent scattering limit and for Doppler RII redistribution with and
without recoils. We find generally that the radiation intensity is linear in
the cosine of the azimuthal angle with respect to radius to high accuracy over
a broad frequency region across the line centre for both linear and perturbed
velocity fields, yielding the Eddington factors f(nu) = 1/3 and g(nu) = 3/5. We
show the radiation field produced by a point source divides into three spatial
regimes for a uniformly expanding homogeneous medium: at radii r small compared
with a characteristic radius r*, the mean intensity near line centre varies as
1/ r^(7/3), while at r > r* it approaches 1/ r^2; for r << r* it is modified by
frequency redistribution. Before the reionization epoch, r* takes on the
universal value 1.1 Mpc, independent of redshift. The mean intensity and
scattering rate are found to be very sensitive to the gradient of the velocity
field, growing exponentially with the amplitude of the perturbation as the
limit of a vanishing velocity gradient is approached near the source. We expect
the 21cm signal from the Epoch of Reionization to thus be a sensitive probe of
both the density and the peculiar velocity fields.
