Asymptotic solutions to multidimensional rapidly-oscillating integrals

Derived are asymptotic solutions to rapidly-oscillating d-dimensional integrals that have superior convergence properties than conventional numerical-quadrature techniques. The approach is analogous to the theory of steepest descent which utilises the fact that the first derivatives of a function vanish at certain critical points. The approach is demonstrated by solving the electromagnetic scattering integral for a curved surface.