ASYMPTOTIC EXPANSIONS OF EXPONENTIAL INTEGRALS AND NEWTON DIAGRAMS

We study the asymptotic expansion, as λ → 0⁺, of integrals of the form J_H,Χ(λ) =∫exp(H(χ)/λ). Χ(χ)dχ, where H and Χare smooth from R^p to R, H has a unique (degenerate) maximum at 0, Χ has compact support a neighborhood of 0.

If p = 2 or if the Newton Diagram of H contains only one facet, we give an algorithm to compute explicitely the complete asymptotic expansion of J_H,Χ(λ). In the general case, we show how to write J_H,Χ(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,Χ(λ), and specify the exact form of this first term under a simple additional condition.

Full Text (PDF format)

Published 1 January 2003