# Asymptotic expansions for some integrals of quotients with degenerated divisors

## TL;DR

This paper derives asymptotic expansions for certain integrals with degenerated divisors, relevant in four-wave interaction models, as the parameter approaches zero.

## Contribution

It provides new asymptotic expansion formulas for integrals with degenerate divisors, expanding understanding of their behavior in wave interaction contexts.

## Key findings

- Derived explicit asymptotic formulas as 0 for integrals with degenerate divisors.
- Applied results to analyze four-wave interaction integrals.
- Enhanced mathematical tools for studying wave interaction phenomena.

## Abstract

We study asymptotic expansion as $\nu\to0$ for integrals over ${ \mathbb{R} }^{2d}=\{(x,y)\}$ of quotients $F(x,y) \big/ \big( (x\cdot y)^2+(\nu \Gamma(x,y))^2\big)^{-1}$, where $\Gamma$ is strictly positive and $F$ decays at infinity sufficiently fast. Integrals of this kind appear in description of the four--waves interactions.