Let $\beta>0$ and $\alpha$ real number fixed, $z$ is a positive parameter.
I'm looking for the asymptotic behavior (at least the leading term) as $z \to 0$ of the following integral:
$$ \int_0^{+\infty} \exp(-\alpha x) \,K_2\big(\beta\sqrt{x^2+z^2}\big)\,dx $$
where $K_2$ is the modified Bessel function of the second kind.