Complex integral on real axis

Question

I need to show $\int_m^\infty{\sqrt{x^2-m^2}e^{-ixt}dx} \sim_{t \to \infty} e^{-imt}$. I tried to use the fact that the integral has branch points at $\pm m$, but I didn't get anything. Then I tried to do the substitution $z=m+x$ and the integral becomes $e^{-imt}\int_0^\infty\sqrt{z^2+2mz}e^{-izt}dz$. But I am not sure how to go from here. Any idea? Thank you!