I have been stuck with an integration of the following type $$\int\frac{2^{1/3}}{x^{1/3}}\operatorname{Ai}\left(\frac{2^{1/3}(x-y)}{x^{1/3}}\right)e^{-b\left(k+x\omega\right)^2}\,dx$$ I am not so sure about the limit of the integration. Is there any closed form possible for the above integral ?
The Airy function is coming from the asymptotic expansion of the Bessel function of first kind and the expansion is valid for $|x-y|\approx |x|^{9/15}$ and I think the limit of the integration has to be calculated obeying the constraint.
Main problem: I need to perform the following summation, $$\sum_{x=-\infty}^{\infty}J_{x}(y) e^{-b\left(k+x\omega\right)^2}$$ where $y\gg 1$. In such a situation the main contribution of the Bessel function comes from $y\approx x$. Further in such situation ($y\gg 1$) the peaks of the Bessel functions are very closely spaced which implies the summation can be transformed to an integration over $x$. Also when $y\approx x\gg 1$ the Bessel function can be written as $$\frac{2^{1/3}}{x^{1/3}}\operatorname{Ai}\left(\frac{2^{1/3}(x-y)}{x^{1/3}}\right)$$ which ultimately gives the required integration. The Airy function can be written in terms of $K_{1/3}$ and a combination of $J_{1/3}$, $J_{-1/3}$ for when $x-y>0$ and $y-x>0$ respectively.