APPLICATION OF THE HADAMARD FINITE-PART CONCEPT TO THE ASYMPTOTIC-EXPANSION OF A CLASS OF MULTIDIMENSIONAL INTEGRALS

The aim of this paper is to study the expansion with respect to the la rge and real parameter lambda of the integral I-n(f)(lambda) := integr al(0)(1)...integral(0)(1) x(1)(alpha 1)...x(n)(alpha n)log(l1)(x(1)).. .log(ln)(x(n)) xf(x(1),...,x(n))g(lambda x(1)(alpha 1)...x(n)(alpha n) )dx(1)... dx(n), where for i is an element of {1,...,n} : alpha(i) > 0 , l(i) is an element of N and alpha(i) is complex with Re(alpha(i)) > -1. Moreover, f is a smooth enough function and g belongs to B-r(]0,+i nfinity[), 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 Hadama rd.