Methods and Applications of Analysis

Volume 10 (2003)

Number 3

ASYMPTOTIC EXPANSIONS OF EXPONENTIAL INTEGRALS AND NEWTON DIAGRAMS

Pages: 413 – 456

DOI: http://dx.doi.org/10.4310/MAA.2003.v10.n3.a6

Author

STÉPHANE ROSSIGNOL

Abstract

We study the asymptotic expansion, as λ → 0+, of integrals of the form JH,Χ(λ) =∫exp(H(χ)/λ). Χ(χ)dχ, where H and Χare smooth from Rp 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 JH,Χ(λ). In the general case, we show how to write JH,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 JH,Χ(λ), and specify the exact form of this first term under a simple additional condition.

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