Geodesic
Asymptotic properties of integrals of quotients when the numerator oscillates and the denominator degenerates
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 14 (2018) no. 4 
Sergei Kuksin. Asymptotic properties of integrals of quotients when the numerator oscillates and the denominator degenerates. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 14 (2018) no. 4. http://geodesic.mathdoc.fr/item/JMAG_2018_14_4_a3/
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     title = {Asymptotic properties of integrals of quotients when the numerator oscillates and the denominator degenerates},
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     url = {http://geodesic.mathdoc.fr/item/JMAG_2018_14_4_a3/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Math. Notes, 103 (2018), 713–723 | DOI | DOI | MR | Zbl

[2] S. Kuksin, “Asymptotic expansions for some integrals of quotients with degenerated divisors”, Russ. J. Math. Phys., 24 (2017), 476–487 | DOI | MR | Zbl

[3] S. Nazarenko, Wave Turbulence, Lecture Notes in Physics, 825, Springer, Heidelberg, 2011 | DOI | MR | Zbl