Solving a class of multivariate integration problems via Laplace techniques

We consider the problem of calculating a closed form expression for the integral of a real-valued function f : Rn → R on a set S. We specialize to the particular cases when S is a convex polyhedron or an ellipsoid, and the function f is either a generalized polynomial, an exponential of a linear form (including trigonometric polynomials) or an exponential of a quadratic form. Laplace transform techniques allow us to obtain either a closed form expression, or a series representation that can be handled numerically. Finally, a methodology is proposed for multivariate functions f which have a (multidimensional) Laplace transform.