I have the following integral, $$\int_{-1}^{1}dx \int_{-1}^{1}d\mu J_{n}(\beta\sqrt{1-x^{2}}\sqrt{1-\mu^{2}})$$ where $\beta\gg 1$.
and I naively used the asymptotic form of the Bessel function,
$$J_{n}(\beta\sqrt{1-x^{2}}\sqrt{1-\mu^{2}})\approx\frac{{2^{1/3} }}{{n^{1/3} }} \operatorname{Ai} \left( {\frac{{2^{1/3} (n - \beta\sqrt{1-x^{2}}\sqrt{1-\mu^{2}} )}}{{n^{1/3} }}} \right)$$ and then tried to evaluate the integral.
Is this the correct way to proceed ?