Solving a class of multivariate integration problems via Laplace techniques

Jean B. Lasserre¹ ; Eduardo S. Zeron²

¹ LAAS-CNRS 7 Avenue du Colonel Roche 31077 Toulouse Cedex 4, France
² Departamento de Matemáticas CINVESTAV-IPN Apdo. Postal 14740 México, D.F. 07000, Mexico

Applicationes Mathematicae, Tome 28 (2001) no. 4, pp. 391-405

Résumé

We consider the problem of calculating a closed form expression for the integral of a real-valued function $f:{\Bbb R}^n\rightarrow {\Bbb R}$ 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.

Détail

DOI : 10.4064/am28-4-2

Keywords: consider problem calculating closed form expression integral real valued function bbb rightarrow bbb set specialize particular cases convex polyhedron ellipsoid function either generalized polynomial exponential linear form including trigonometric polynomials exponential quadratic form laplace transform techniques allow obtain either closed form expression series representation handled numerically finally methodology proposed multivariate functions which have multidimensional laplace transform