Adelaide preprint number: ADP-97-10/T248
We apply the perturbation theory integral representation (PTIR) to solve for the bound state Bethe-Salpeter (BS) vertex for an arbitrary scattering kernel, without the need for any Wick rotation. The results derived are applicable to any scalar field theory (without derivative coupling). It is shown that solving directly for the BS vertex, rather than the BS amplitude, has several major advantages, notably its relative simplicity and superior numerical accuracy. In order to illustrate the generality of this approach we obtain numerical solutions using this formalism for a number of scattering kernels, including cases where the Wick rotation is not possible. Those interested in applications of this technique may request a copy of the computer code from the authors.