The singular eigenfunction technique of Case for solving one-dimension
al planar symmetry linear transport problems is extended to a restrict
ed class of three-dimensional problems. This class involves planar geo
metry, but with forcing terms (either boundary conditions or internal
sources) which are weakly dependent upon the transverse spatial variab
les. Our analysis involves a singular perturbation about the classic p
lanar analysis, and leads to the usual Case discrete and continuum mod
es, but modulated by weakly dependent three-dimensional spatial functi
ons. These functions satisfy parabolic differential equations, with a
different diffusion coefficient for each mode. Representative one-spee
d time-independent transport problems are solved in terms of these gen
eralized Case eigenfunctions. Our treatment is very heuristic, but may
provide an impetus for more rigorous analysis.