Application of the Hadamard's finite part concept to the asymptotic expansion of a class of multidimensional integrals

Abstract

The aim of this paper is to study the expansion with respect to the large and real parameter λ of the integral I n f ( λ ) := ∫ 0 1 … ∫ 0 1 x 1 α 1 … x n α n log l 1 ( x 1 ) … log l n ( x n ) × f ( x 1 , … , x n ) g ( λ x 1 α 1 … x n α n ) d x 1 … d x n , where for i ∈ { 1 ,..., n} : α i > 0, l i Є N and α i is complex with Re(α i ) > -1 . Moreover, f a smooth enough function and g belongs to β r (]0, +∞[), a space defined below. The derivation of such an asymptotic expansion is established by induction on the integer n and makes use of a basic concept: the integration in the finite part sense of Hadamard.