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I am trying to evaluate the following integral involving the Bessel function, power, exponential and a Airy function:

$$ \int_{0}^\infty \text{Ai}(-x)J_0(\alpha x) \exp(-\beta x^2) \, x \, dx $$

where $\alpha =kx'/z$ and $\beta = ik/2z$. I tried a method that is to rewrite the airy function $\text{Ai}(x)$ with Bessel functions. In that case, the function above will become

$$ \int_{0}^\infty \frac { x^{\frac 3 2}} {3} J_{1/3} \left(\frac {3} {2} x^ \frac {3} {2}\right)J_0(\alpha x)\, \exp(-\beta x^2) \, dx + \int_{0}^\infty \frac { x^{\frac 3 2}} {3} J_{-1/3} \left(\frac {3} {2} x^ \frac {3} {2}\right)J_0(\alpha x)\, \exp(-\beta x^2)\, dx,$$

and two Bessel equations appear.

Could anyone please help?

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