- #1
clumps tim
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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
$$\Bigl[\frac {d^2}{dt^2}+ \omega_{0}^{2}\Bigl] G(t, t') = \delta(t-t')$$
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