  The exact wave functions that describe scattering of a charged particle by a
confined magnetic field (Aharonov-Bohm effect) and by a Coulomb field are
analyzed. It is well known that the usual procedure of finding asymptotic forms
of these functions which admit a separation into a superposition of an incident
plane wave and a circular or spherical scattered wave is problematic near the
forward direction. It thus appears to be impossible to express the conservation
of probability by means of an optical theorem of the usual kind. Both the total
cross section and the forward scattering amplitude appear to be infinite. To
address these difficulties we find a new representation for the asymptotic form
of the Aharonov-Bohm wave function that is valid for all angles. Rather than
try to define a cross section at forward angles, however, we work instead with
the probability current and find that it is quite well behaved. The same is
true for Coulomb scattering. We trace the usual difficulties to a nonuniformity
of limits.
