  An eikonal expansion is developed in order to provide systematic corrections
to the eikonal approximation through order 1/k^2, where k is the wave number.
The expansion is applied to wave functions for the Klein-Gordon equation and
for the Dirac equation with a Coulomb potential. Convergence is rapid at
energies above about 250 MeV. Analytical results for the eikonal wave functions
are obtained for a simple analytical form of the Coulomb potential of a
nucleus. They are used to investigate distorted-wave matrix elements for
quasi-elastic electron scattering from a nucleus. Focusing factors are shown to
arise from the corrections to the eikonal approximation. A precise form of the
effective-momentum approximation is developed by use of a momentum shift that
depends on the electron's energy loss.
