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Développements asymptotiques, transformation de Mellin complexe et intégration sur les fibres. (Asymptotic expansions, complex Mellin transformation and integration on fibers). (French) Zbl 0649.32008
Sémin. d’analyse P. Lelong - P. Dolbeault - H. Skoda, Paris 1985/86, Lect. Notes Math. 1295, 11-23 (1987).
[For the entire collection see Zbl 0623.00006.]
Let X be an analytic complex manifold of dimension $$n+1$$ and f a holomorphic function on X, such that $$f^{-1}(s)$$ is a submanifold for $$s\neq 0$$; if u is an (n,n)-differential form with compact support, it was proved by the first author [Invent. Math. 68, 129-174 (1982; Zbl 0508.32003)] that the fiber integral $$F(s)=\int_{f^{-1}(s)}u$$ has an asymptotic expansion of the form $F(s)\sim \sum a^ i_{jkm}| s|^{2r_ i} s^ j \bar s^ k(\log | s|)^ m\quad (s\to 0),$ with $$r_ i\in {\mathbb{Q}}$$, $$0\leq r_ 1<...<r_ p<1$$, j,k,m$$\in {\mathbb{N}}$$, $$m\leq n.$$
This paper gives a simplified proof of this result, using relations between the poles of the meromorphic continuation of the “complex Mellin transform” $M_ qF(t)=\int_{C}| z|^{2t} z^ q \bar z^{-q} F(z,\bar z)dz d\bar z$ and the asymptotic expansion of F.
Reviewer: G.Roos

##### MSC:
 32C30 Integration on analytic sets and spaces, currents 32Sxx Complex singularities 32S45 Modifications; resolution of singularities (complex-analytic aspects)