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Analytic verification of Kramers-Kronig Relations

  • #1
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Homework Statement


Show that the real and imaginary parts of the following susceptibility function satisfy the K-K relationships. Use the residue theorem.
$$ \chi(\omega) = \frac{\omega_{p}^2}{(\omega_0^2-\omega^2)+i\gamma\omega} $$

Homework Equations



The Kramers-Kronig relations are

$$ \chi_r(\omega) = \frac{1}{\pi} P \int_{-\infty}^{\infty} d\bar{\omega} \frac{\chi_i(\bar{\omega})}{\bar{\omega}-\omega} $$
$$ \chi_i(\omega) = -\frac{1}{\pi} P \int_{-\infty}^{\infty} d\bar{\omega} \frac{\chi_r(\bar{\omega})}{\bar{\omega}-\omega} $$

The Attempt at a Solution


The problem is that my complex calculus is pretty rusty and I do not know which poles contribute exactly. There are 5 poles in total 4 from the susceptibility function and 1 from the denominator(see the expressions please). The poles are
$$\pm \gamma \pm i\frac{\sqrt{4\omega_0^2-\gamma}}{2} $$

I calculated the residues that are on the negative imaginary $\omega$ plane on Mathematica and they turned out to be zero except the pole at $\omega$ which is the value of the integral using the principal rule in Mathematica. Is this analytically tractable easily? I appreciate any assistance you provide.

Many thanks,
 

Answers and Replies

  • #2
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Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
 
  • #3
TSny
Homework Helper
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$$ \chi(\omega) = \frac{\omega_{p}^2}{(\omega_0^2-\omega^2)+i\gamma\omega} $$
There are 5 poles in total 4 from the susceptibility function and 1 from the denominator(see the expressions please).
## \chi(\omega)## has only two poles. If you locate these poles, you will see that you can choose the contour so that it doesn't enclose either of these two poles.
 

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