We study the asymptotic expansion, as A -* 0+, of integrals of the form J_(H,X)(λ) = ∫exp(H(x)/λ).x(x)dx , where H and x are smooth from R~p to R, H has a unique (degenerate) maximum at 0, x has compact support a neighborhood of 0. If p = 2 or if the Newton Diagram of II contains only one facet, we give an algorithm to compute explicitely the complete asymptotic expansion of J_(H,X)(λ). In the general case, we show how to write J_(H,X)(λ) as a linear combination of simpler integrals, involving only the fundamental part of H. We give an equivalent of the first term of the expansion of J_(H,X)(λ), and specify the exact form of this first term under a simple additional condition.
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