Fourier transform of electric susceptibility example

[schniefen]

Homework Statement

Evaluate $\chi (t)$ for the model function

$\tilde{\chi}(\omega)=\frac{Nq^2}{Vm\epsilon_0}\frac{1}{\omega_0^2-\omega^2-\mathrm{i}\omega\gamma}=\frac{Nq^2}{Vm\epsilon_0}\frac{1}{2\sqrt{\omega_0^2-\frac{\gamma^2}{4}}}\left(\frac{1}{\omega+\mathrm{i}\frac{\gamma}{2}+\sqrt{\omega_0^2-\frac{\gamma^2}{4}}}-\frac{1}{\omega+\mathrm{i}\frac{\gamma}{2}-\sqrt{\omega_0^2-\frac{\gamma^2}{4}}}\right) \ ,$

and interpret the result.

Relevant Equations

Complex polarization: $\mathbf {P}(t)=\tilde{\chi}\epsilon_0\tilde{\mathbf{E}}_0e^{-\mathrm{i}\omega t}$

I have not studied the Fourier transform (FT) in great detail, but came across a problem in electrodynamics in which I assume it is needed. The problem goes as follows:

Evaluate $\chi (t)$ for the model function

$\tilde{\chi}(\omega)=\frac{Nq^2}{Vm\epsilon_0}\frac{1}{\omega_0^2-\omega^2-\mathrm{i}\omega\gamma}=\frac{Nq^2}{Vm\epsilon_0}\frac{1}{2\sqrt{\omega_0^2-\frac{\gamma^2}{4}}}\left(\frac{1}{\omega+\mathrm{i}\frac{\gamma}{2}+\sqrt{\omega_0^2-\frac{\gamma^2}{4}}}-\frac{1}{\omega+\mathrm{i}\frac{\gamma}{2}-\sqrt{\omega_0^2-\frac{\gamma^2}{4}}}\right) \ ,$

and interpret the result.​

To find $\chi (t)$, one needs to evaluate the integral $\chi (t) = \frac{1}{2\pi}\int_{-\infty}^{\infty}\tilde{\chi}(\omega) e^{-i\omega t} \mathrm{d}\omega$, right? Are there any tricks to simplify the integral?

 

[hutchphd]

There are a multitude of tricks in the realm of complex analysis, but absent that it is probably easiest to make a simple substitution of variable for each of the two terms and the result is not too difficult for each.

 

[schniefen]

There are a multitude of tricks in the realm of complex analysis, but absent that it is probably easiest to make a simple substitution of variable for each of the two terms and the result is not too difficult for each.

Thanks for the reply. I am not too familiar with complex analysis either, but I assume making a substitution in a complex integral follows the same procedure as in a real integral?

 

[jasonRF]

To find $\chi (t)$, one needs to evaluate the integral $\chi (t) = \int_{-\infty}^{\infty}\tilde{\chi}(\omega) e^{i\omega t} \mathrm{d}\omega$, right?

How did your course/book define the Fourier transform? The details of the inverse Fourier tranform depend on how the Fourier transform is defined. For example, if
$\tilde{\chi}(\omega) = \int_{-\infty}^\infty \chi(t) \, e^{-i \omega t} \, dt$
then the inverse transform is,
$\chi (t) = \frac{1}{2\pi}\int_{-\infty}^{\infty}\tilde{\chi}(\omega) e^{i\omega t} \mathrm{d}\omega$
But not everyone uses the same conventions. In fact, based on your equation
$\mathbf {P}(t)=\tilde{\chi}\epsilon_0\tilde{\mathbf{E}}_0e^{-\mathrm{i}\omega t}$
I suspect that the signs in the exponents are the opposite of what I posted above. But the location of the factor of $2\pi$ can vary (again, no standard...)

If your course allows you to use a table of Fourier transforms, then that is one approach that can help. Equivalently, your professor may have computed the forward Fourier transform of a function that looks like your $\tilde{\chi}(\omega)$.Note that if you use a table, you need to make sure they are defining their Fourier transform the same way you are (it really is a pain that there are multiple conventions...). Otherwise you will need to perform the integral yourself.

jason

 

[schniefen]

I have edited my original question. I meant something else, what you wrote (but with minus signs in the exponents reversed, see here).

There are some tables on Wikipedia and I’ve been referred to equations 103, 205 and 309 there. However, I don’t see yet how to apply all of them.

 

[jasonRF]

The table in the wikipedia article isn't that great. Here is one that has more useful functions
Table of Fourier Transform Pairs (purdue.edu)
The table uses standard electrical engineering notation, so it uses $j$ instead of $i$ for $\sqrt{-1}$, $u(t)$ is $0$ for $t<0$and $1$ for $t>0$, etc.

jason

 

[rude man]

If you're familiar with the laplace transform you can substitute $s=i\omega~$ in your $\tilde X(\omega)$ then go to a table of laplace transforms. There are oodles of laplace tables but few fourier.
BTW this looks like some kind of damped oscillation.