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# 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

#### Abstract

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.