On certain multiple integrals occurring in a waveguide scattering problem

Closed-form results are presented for some $n$-fold integrals where the integrand contains the exponential of a specific quadratic form in $n$ variables. These integrals arise in the ray-optical analysis of reflection and diffraction problems for an open-ended parallel-plane waveguide. The results are obtained by three methods: the first method is elementary, the second method uses an,integral equation which is solved by the Wiener–Hopf technique, and the third method is based on a probabilistic interpretation of the integrals.