  In this work, the paraxial approximation of the free Dirac equation is
examined. The results are first obtained by constructing superpositions of
exact solutions with suitable profiles, which are borrowed from paraxial
optics. In this manner, the paraxial Dirac beams are obtained in four cases: as
Gaussian, Bessel-Gaussian, modified Bessel-Gaussian and elegant
Laguerre-Gaussian beams. In the second part of the work, the paraxial Dirac
equation is derived, and then its solutions in the aforementioned cases are
directly obtained. All the resulting wave functions conform to those derived
formerly by virtue of superpositions, except for terms, that are negligible
upon the assumption that the paraxial functions along the propagation axis vary
only slightly over a distance equal to the de Broglie wavelength, which is the
standard paraxial requirement.
