  An exact representation of the Euclidean fermion determinant in two
dimensions for centrally symmetric, finite-ranged Abelian background fields is
derived. Input data are the wave function inside the field's range and the
scattering phase shift with their momenta rotated to the positive imaginary
axis and fixed at the fermion mass for each partial-wave. The determinant's
asymptotic limit for strong coupling and small fermion mass for
square-integrable, unidirecitonal magnetic fields is shown to depend only on
the chiral anomaly. The concept of duality is extended from one to two-variable
fields, thereby relating the two-dimensional Euclidean determinant for a class
of background magnetic fields to the pair production probability in four
dimensions for a related class of electric pulses. Additionally, the
``diamagnetic'' bound on the two-dimensional Euclidean determinant is related
to the negative sign of dImS_eff/dm^2 in four dimensions in the strong
coupling, small mass limit, where S_eff is the one-loop effective action.
