  The vacuum-polarization correction for bound electrons or muons is examined.
The objective is to formulate a framework for calculating the correction from
bound-state quantum electrodynamics entirely in coordinate space, including the
Uehling potential which is usually isolated and treated separately.
Pauli-Villars regularization is applied to the coordinate-space calculation and
the most singular terms are shown to be eliminated, leaving the physical
correction after charge renormalization. The conventional derivation of the
Uehling potential in momentum space is reviewed and compared to the
coordinate-space derivation.
