Elliptic integrals in diffraction theory

Maliuzhinets' technique (1958) remains today the most general approach for solving diffraction problems in wedge-shaped regions typically characterized by impedance boundary conditfons. The technique leads to a pair of first order difference equations for the spectra of the total fields and their period is related to the open angle of the wedge. For all solutions completed thus far, the equation pair is decoupled or can be made so by choosing a suitable linear combination of the spectral functions. In general, however, the equation pair cannot be decoupled and we are left with having to solve a second order difference equation with functional coefficients. There is no established method to solve such equations but a technique has recently been developed by Senior and Legault. The proposed approach is conceptually simple and requires the use of elliptic integrals of the first and third kind to construct solutions with the desired analyticity requirements. The generality of the technique is examined here by considering an extension of the equation solved by Senior and Legault.