Analytic verification of Kramers-Kronig Relations

That means that the integral along the contour is zero. This in turn means that the residue at the pole at ω is equal to the integral along the contour.In summary, the Kramers-Kronig relations can be used to show that the real and imaginary parts of the susceptibility function satisfy the K-K relationships. This can be done using the residue theorem and locating the two poles of the function. By choosing a contour that does not enclose these poles, the integral along the contour is zero, and the residue at the pole at ω is equal to the integral along the contour.
  • #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,
 
  • #3
Septim said:
$$ \chi(\omega) = \frac{\omega_{p}^2}{(\omega_0^2-\omega^2)+i\gamma\omega} $$
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.
 

What are the Kramers-Kronig relations?

The Kramers-Kronig relations are a set of mathematical equations that describe the relationship between the real and imaginary parts of a complex function. They are commonly used in physics and engineering to analyze the behavior of systems in the frequency domain.

Why is it important to verify the Kramers-Kronig relations?

Verifying the Kramers-Kronig relations is important because it ensures the accuracy and validity of experimental data. These relations are based on fundamental principles of causality and are essential for understanding the physical properties of materials and systems.

How are the Kramers-Kronig relations used in scientific research?

The Kramers-Kronig relations are used in a variety of scientific fields, including optics, materials science, and signal processing. They are often used to analyze the dielectric properties of materials, study the behavior of light in different media, and correct for measurement errors in experimental data.

What are some common methods used to verify the Kramers-Kronig relations?

There are several methods used to verify the Kramers-Kronig relations, including the graphical method, the Cauchy integral method, and the Hilbert transform method. Each method has its own advantages and limitations, and the choice of method depends on the specific application and available data.

Can the Kramers-Kronig relations be violated?

Yes, the Kramers-Kronig relations can be violated in certain cases, such as when dealing with non-linear systems or in the presence of noise in the data. However, these violations are typically small and can be corrected for using advanced techniques. In general, the Kramers-Kronig relations are considered to be a robust and reliable tool for analyzing physical systems.

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