  The Dirac equation for an electron in two spatial dimensions in the Coulomb
and homogeneous magnetic fields is discussed. For weak magnetic fields, the
approximate energy values are obtained by semiclassical method. In the case
with strong magnetic fields, we present the exact recursion relations that
determine the coefficients of the series expansion of wave functions, the
possible energies and the magnetic fields. It is found that analytic solutions
are possible for a denumerably infinite set of magnetic field strengths. This
system thus furnishes an example of the so-called quasi-exactly solvable
models. A distinctive feature in the Dirac case is that, depending on the
strength of the Coulomb field, not all total angular momentum quantum number
allow exact solutions with wavefunctions in reasonable polynomial forms.
Solutions in the nonrelativistic limit with both attractive and repulsive
Coulomb fields are briefly discussed by means of the method of factorization.
