  Relativistic impulse approximation (RIA) has been widely used in atomic,
condensed matter, nuclear, and elementary particle physics. In former
treatments of RIA formulation, differential cross sections for Compton
scattering processes were factorized into atomic Compton profiles by performing
further simplified approximations in the integration. In this study, we develop
an ``exact'' numerical method without using any further simplified
approximations or factorization treatments. The validity of the approximations
and factorizations used in former RIA treatments can be tested using our
approach. Calculations for C, Cu, Ge, and Xe atomic systems are carried out
using Dirac-Fock wavefunctions, and comparisons between the proposed approach
and former treatments of RIA are performed and discussed in detail. Numerical
results indicate that these simplified approximations work reasonably in the
Compton peak region, and our results have little difference with the best of
the former RIA treatments in the entire energy region. While in regions far
from the Compton peak, the RIA results become inaccurate, even when our
``exact'' numerical treatment is used.
