While reading a physics paper I bumped into the following integral:
$\int_0^{\infty}\mathrm{d}x\, e^{- x}\int_{-\infty}^{+\infty}\mathrm{d}y\,e^{iy}\,\text{cos}(\frac{\sqrt{x+ ay}}{a} + \frac{\sqrt{x- ay}}{a}) $,
where "a" is a parameter.
The authors claim that in the limit $a \to 0$ the integral vanishes because of the highly oscillating nature of the integrand function involved in the Fourier Transform. This closely reminds me of the Riemann-Lebesgue lemma, however I can not understand how I should apply this lemma in the present case, especially because of the functional form of the argument of the cosine.
Is it the correct approach to invoke the Rieman-Lebesgue lemma or should I follow another path to prove their claim?