Formally closed solution for a crystal with spatial dispersion

A formally closed solution is exhibited for a semi-infinite spatially dispersive crystal, with point-dipole harmonic oscillators located at the crystal sites, and $s$-polarized incident light. The dipole interactions, including retardation, are taken into account exactly, and observable quantities may be calculated with arbitrary accuracy without great computational effort. The method requires the summing of a rapidly converging series, followed by a contour integral around the unit circle. Some numerical results are presented, and a comparison made with standard approximate methods. The case of $p$-polarized light is postponed to a future publication.