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 electrons energy loss.
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