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Zh. Mat. Fiz. Anal. Geom., 2018, Volume 14, Number 4, paper published in the English version journal (Mi jmag708)  

Asymptotic properties of integrals of quotients when the numerator oscillates and the denominator degenerates

Sergei Kuksinabc

a Institut de Mathémathiques de Jussieu–Paris Rive Gauche, CNRS, Université Paris Diderot, UMR 7586, Sorbonne Paris Cité, F-75013, Paris, France
b School of Mathematics, Shandong University, Shanda Nanlu, 27, 250100, PRC
c Saint Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg, Russia

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

Key words and phrases: asymptotic of integrals, oscillating integrals, four-waves interaction.

Funding Agency Grant Number
Russian Science Foundation 18-11-00032
Centre National de la Recherche Scientifique PRC CNRS/RFBR 2017-2019 No 1556
We acknowledge the support from the Centre National de la Recherche Scientifique (France) through the grant PRC CNRS/RFBR 2017-2019 No 1556, and from the Russian Science Foundation through the project 18-11-00032.


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

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MSC: 34E05, 34E10
Received: 01.02.2018
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Citation in format AMSBIB
\Bibitem{Kuk18}
\mathnet{http://mi.mathnet.ru/jmag708}
\crossref{https://doi.org/10.15407/mag14.04.510}


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