Can you solve the integral $\int_{\mathbb{R}} e^{i\omega t}\dfrac{1}{(\omega-i\epsilon)^2 - m^2} d\omega$ using residues?

Question

The integral is $$\int_{\mathbb{R}} e^{i\omega t}\dfrac{1}{(\omega-i\epsilon)^2 - m^2} d\omega$$ where $t$ and $m$ are constants and $(\omega-i\epsilon)$ is the $i\epsilon$ prescription. I think you have to use the residue theorem or Jordan's lemma. Thanks in advance.

Answer 1

If I didn’t make any mistake, the integral equals($\epsilon >0$) $$2\pi i(\text{Res}_{\omega=m+i\epsilon}f(\omega)+ \text{Res}_{\omega=-m+i\epsilon}f(\omega))$$ where $f(\omega)$ is the integrand.

I will elaborate later.