Journal of Mathematical Physics, Analysis, Geometry 2018, vol. 14, No 4, pp. 510-518   https://doi.org/10.15407/mag14.04.510     ( to contents , go back )

### Asymptotic Properties of Integrals of Quotients when the Numerator Oscillates and the Denominator Degenerates

Sergei Kuksin

Institut de Mathémathiques de Jussieu–Paris Rive Gauche, CNRS, Université Paris Diderot, UMR 7586, Sorbonne Paris Cité, F-75013, Paris, France

School of Mathematics, Shandong University, Shanda Nanlu, 27, 250100, PRC; Saint Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg, Russia
E-mail: Sergei.Kuksin@imj-prg.fr

We study asymptotic expansion as $\nu\to0$ for integrals over ${ \mathbb{R} }^{2d}=\{(x,y)\}$ of quotients of the form $F(x,y) \cos(\lambda x\cdot y) \big/ \big( (x\cdot y)^2+\nu^2\big)$, where $\lambda\ge 0$ and $F$ decays at infinity sufficiently fast. Integrals of this kind appear in the theory of wave turbulence.