MHB Pole shifting for Fourier transform

[clumps tim]

Hi, I have a simple harmonic oscillation problem whose Green function is given by

$$\Bigl[\frac {d^2}{dt^2}+ \omega_{0}^{2}\Bigl] G(t, t') = \delta(t-t')$$

Now I found out the Fourier transform of $G(t, t')$ to be $$G(\omega)= \frac{1}{2\pi} \frac{1}{\omega_{0}^{2}-\omega^2}$$ which has poles at $\omega=\pm\omega_{0}$

now how can i identify the way the poles can be shifted , like shifting $\omega_{0}\rightarrow \omega_{0}+ i\epsilon$
my instructor said there are four ways to shift. can you please guide me mathematically to pole shifting ?

also suggest me the reading materials to know about these shifting .
regards

 

[Joppy]

There are far more than 4 ways to have your system's poles be relocated to some desired location. In any case, did you happen to find out what these particular ways are?