  An eikonal expansion is used to provide systematic corrections to the eikonal
approximation through order $1/k^2$, where $k$ is the wave number. Electron
wave functions are obtained for the Dirac equation with a Coulomb potential.
They are used to investigate distorted-wave matrix elements for quasi-elastic
electron scattering from a nucleus. A form of effective-momentum approximation
is obtained using trajectory-dependent eikonal phases and focusing factors.
Fixing the Coulomb distortion effects at the center of the nucleus, the
often-used ema approximation is recovered. Comparisons of these approximations
are made with full calculations using the electron eikonal wave functions. The
ema results are found to agree well with the full calculations.
