  Eliashberg theory provides a theoretical framework for understanding the
phenomenon of superconductivity when pairing between two electrons is mediated
by phonons, and retardation effects are fully accounted for. BCS theory is
often viewed as the weak coupling limit of Eliashberg theory, in spite of a
handful of papers that have pointed out that this is not so. Here we present
very accurate numerical solutions in the weak coupling limit to complement the
existing analytical results, and demonstrate more convincingly the validity of
this limit by extending the analytical results to first order in the coupling
constant.
