Prove Complex Integral: $\int_m^\infty\sqrt{x^2-m^2}e^{-ixt}dx$

[Silviu]

Hello! I found a proof in my physics books and at a step it says that: $\int_m^\infty\sqrt{x^2-m^2}e^{-ixt}dx \sim_{t \to \infty} e^{-imt}$. Any advice on how to prove this?

 

[Buzz Bloom]

Hi @Silviu:

Here is a thought that might help.

Substitute x= y+m. This will produce e^-imt F(t)
where
F(t) = Inverse Fourier Transform of f(x)
where
f(x) = √(y (y+2m)).
Therefore, this reduces the problem to prove lim[t→∞] F(t) = 1.

Good luck.

Regards,
Buzz