  We present two novel additions to the semi-analytic solution of Lyman
$\alpha$ (Ly$\alpha$) radiative transfer in spherical geometry: (1)
implementation of the correct boundary condition for a steady source, and (2)
solution of the time-dependent problem for an impulsive source. For the
steady-state problem, the solution can be represented as a sum of two terms: a
previously-known analytic solution of the equation with mean intensity $J=0$ at
the surface, and a novel, semi-analytic solution which enforces the correct
boundary condition of zero-ingoing intensity at the surface. This solution is
compared to that of the Monte Carlo method, which is valid at arbitrary optical
depth. It is shown that the size of the correction is of order unity when the
spectral peaks approach the Doppler core and decreases slowly with line center
optical depth, specifically as $(a \tau_0)^{-1/3}$, which may explain
discrepancies seen in previous studies. For the impulsive problem, the time,
spatial, and frequency dependence of the solution are expressed using an
eigenfunction expansion in order to characterize the escape time distribution
and emergent spectra of photons. It is shown that the lowest-order
eigenfrequency agrees well with the decay rate found in the Monte Carlo escape
time distribution at sufficiently large line-center optical depths. The
characterization of the escape-time distribution highlights the potential for a
Monte Carlo acceleration method, which would sample photon escape properties
from distributions rather than calculating every photon scattering, thereby
reducing computational demand.
