  The Aharonov-Bohm effect is analyzed for a spin-1/2 particle in the case that
a $1/r$ potential is present. Scalar and vector couplings are each considered.
It is found that the approach in which the flux tube is given a finite radius
that is taken to zero only after a matching of boundary conditions does not
give physically meaningful results. Specifically, the operations of taking the
limit of zero flux tube radius and the Galilean limit do not commute. Thus
there appears to be no satisfactory solution of the relativistic
Aharonov-Bohm-Coulomb problem using the finite radius flux tube method.
