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 )
https://doi.org/10.15407/mag14.04.510

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

Received February 1, 2018.

Dedicated to V.A. Marchenko on the occasion of his 95th birthday

Abstract

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.

Mathematics Subject Classification 2000: 34E05, 34E10.
Key words: asymptotic of integrals, oscillating integrals, four-waves interaction

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