Asymptotic expansion of a class of multi-dimensional integrals

The aim of this paper is to derive the expansion of the following class of multi-dimensional integrals I(λ) := fp (Latin small letter esh)_Ω K[x.λw(θ)r^a] dx. with respect to the large parameter λ when Ω is a subset of ℝⁿ, a > 0, w is a strictly positive and bounded function on Σ and fp means an integration in the finite part sense of Hadamard (see Section 2). This is performed for weak assumptions bearing on pseudofunction K and by extending to higher dimensional cases the tools developed in the one-dimensional context. The range of applications of the proposed results is outlined by the exhibition of several examples.