MATHIEU INTEGRAL-TRANSFORMS

Orthogonality and completeness relations are presented for the quasiorthogonal (i.e., orthogonal with respect to a discontinuous weight function) eigenfunctions of a singular (in the sense of Sturm-Liouville theory) boundary-value problem involving the two-dimensional Helmholtz equation in elliptic-cylinder coordinates. These relations yield as special cases integral transforms whose kernels are products of periodic Mathieu functions and modified Mathieu functions of integral order. The new transforms are analogs of the Weber-Orr transform and of a recently published [J. Math. Phys. 30, 41 (1989)] generalized Hankel transform, and would be applicable to boundary-value problems with elliptical geometries. The proof of the orthogonality and completeness relations is surprisingly simple and is based on a novel application of the Sokhotski-Plemelj equations of distribution theory.