MHB Pole shifting for Fourier transform

AI Thread Summary
The discussion centers on the mathematical concept of pole shifting in the context of a Green function for simple harmonic oscillation. The user has derived the Fourier transform of the Green function, which reveals poles at $\omega=\pm\omega_{0}$. They seek guidance on how to mathematically shift these poles, specifically in the form $\omega_{0} \rightarrow \omega_{0} + i\epsilon$. A response indicates that there are numerous methods for relocating poles, prompting the user to explore these techniques further. The conversation emphasizes the importance of understanding pole shifting in relation to system behavior in physics.
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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
 
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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?
 
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