A technique is presented to solve a class of second-order functional difference equations that arise in diffraction theory. Branch-free solutions are obtained by linearly combining branched functions satisfying first-order equations derived from the second-order difference equation. The approach used is conceptually simple, and the underlying analysis is relatively straightforward. Analysis and computations both demonstrate that the resulting solutions have the desired analytical properties and recover the known expressions in the proper limit. ¿ 2000 American Geophysical Union