# Multiple Poles at Negative Integers for ∫<sub><em>A</em></sub><em>f<sup>λ</sup></em><font size="+1">☐</font> in the Case of an Almost Isolated Singularity

### Daniel Barlet

Université Henri Poincaré, Vandoeuvre, France

## Abstract

We give a necessary and sufficient topological condition on *A* ∈ *H_0({ f ≠ 0},ℂ), for a real analytic germ f : (ℝ_n*+1,0) → (ℝ,0), whose complexification has an isolated singularity relatively to the eigenvalue 1 of the monodromy, in order that the meromorphic continuation of ∫_A__fλ_☐ has a multiple pole at sufficiently “large” negative integers. We show that if such a multiple pole exists, it occurs already at

*λ*= −(

*n*+ 1) with its maximal order which is computed topologically.

## Cite this article

Daniel Barlet, Multiple Poles at Negative Integers for ∫<sub><em>A</em></sub><em>f<sup>λ</sup></em><font size="+1">☐</font> in the Case of an Almost Isolated Singularity. Publ. Res. Inst. Math. Sci. 35 (1999), no. 4, pp. 571–584

DOI 10.2977/PRIMS/1195143493