  We have recently derived a gapless theory of the linear response of a
Bose-condensed gas to external perturbations at finite temperature and used it
to explain quantitatively the measurements of condensate excitations and decay
rates made at JILA [D. S. Jin et.al., Phys. Rev. Lett. 78, 764 (1997)]. The
theory describes the dynamic coupling between the condensate and non-condensate
via a full quasiparticle description of the time-dependent normal and anomalous
averages and includes all Beliaev and Landau processes. In this paper we
provide a full discussion of the numerical calculations and a detailed analysis
of the theoretical results in the context of the JILA experiment. We provide
unambiguous proof that the dipole modes are obtained accurately within our
calculations and present quantitative results for the relative phase of the
oscillations of the condensed and uncondensed atom clouds. One of the main
difficulties in the implementation of the theory is obtaining results which are
not sensitive to basis cutoff effects and we have therefore developed a novel
asymmetric summation method which solves this problem and dramatically improves
the numerical convergence. This new technique should make the implementation of
the theory and its possible future extensions feasible for a wide range of
condensate populations and trap geometries.
