  The McMillan-Rowell tunneling inversion program, which extracts the
electron-phonon spectral function $\alpha^2F(\Omega)$ and the Coulomb
pseudopotential $\mu^*$ from experimental tunneling data, is generalized to
include the lowest-order vertex correction. We neglect the momentum dependence
of the electron-phonon matrix elements, which is equivalent to using a local
approximation. The perturbation theory is performed on the imaginary axis and
then an exact analytic continuation is employed to produce the density of
states on the real axis. Comparison is made with the experimental data for Pb.
