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Asymptotic expansion of complex integrals via Mellin transform. (English) Zbl 0707.32003
The present paper is a continuation of previous work [the authors, Lect. Notes Math. 1295, 11-23 (1987; Zbl 0649.32008) and the first author, Invent. Math. 68, 129-174 (1982; Zbl 0508.32003)]. The main purpose of the paper is to show that under a suitable “non-characteristic assumption” on the smooth complex hyperspace H of an \((n+1)\)-dimensional complex manifold X, it is possible to define the restrictions of the currents \(a_{pq}^{r,1}\) to \(H\cap U\) for all (p,q,r,1) and then to the asymptotic expansion, obtained by the complex Mellin transforms. The paper consists of four sections, viz., (1) Asymptotic expansion with parameters. (2) Complex Mellin transform with parameters, (3) Meromorphic extensions of \(\int f^{\lambda +m}f^{\lambda -m}\phi\), where f is a non-constant holomorphic function on X with 0 as the only possible critical values and \(\phi\) a smooth differential form with compact support in any relatively compact open set, (4) Work front set of currents, \(a_{pq}^{r,1}\).
Reviewer: M.Dutta

MSC:
32C30 Integration on analytic sets and spaces, currents
44A15 Special integral transforms (Legendre, Hilbert, etc.)
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
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