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.