  The Eliashberg theory of superconductivity is based on a dynamical
electron-phonon interaction as opposed to a static interaction present in BCS
theory. The standard derivation of Eliashberg theory is based on an equation of
motion approach, which incorporates certain approximations such as Migdal's
approximation for the pairing vertex. In this paper we provide a
functional-integral-based derivation of Eliashberg theory and we also consider
its Gaussian-fluctuation extension. The functional approach enables a
self-consistent method of computing the mean-field equations, which arise as
saddle-point conditions, and here we observe that the conventional Eliashberg
self energy and pairing function both appear as Hubbard-Stratonovich
transformations. An important consequence of this fact is that it provides a
systematic derivation of the Cooper and density-channel interactions in the
Gaussian fluctuation response. We also investigate the strong-coupling
fluctuation diamagnetic susceptibility near the critical temperature.
