(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the Fourier transform of

[tex]f(x)=\frac{1}{(x^2+a^2)^2},\ a>0[/tex], and show by direct calculation that with inverse Fourier transform you'll get the original function [tex]f(x)[/tex]!

2. Relevant equations

Fourier transform and it's inverse:

[tex]F(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty f(x)e^{i k x}dx[/tex]

[tex]f(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty F(k)e^{-ikx}dk[/tex]

3. The attempt at a solution

I have put my function into the transform and I got:

[tex]F(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty \frac{e^{ikx}}{(x^2+a^2)^2}dx[/tex]

I have transformed it to complex integral and since k is arbitrary I have two possible paths of integration, for k>0 in the upper plane and for k<0 in the lower plane. My resudues are:

[tex]Res(f,ia)=-i\frac{e^{-ka}}{4a^3}(1+ka)[/tex] and

[tex]Res(f,-ia)=i\frac{e^ka}{4a^3}(1-ka)[/tex].

So for k>0 my integral (without the [tex](2\pi)^{-1/2}[/tex]) is:

[tex]I=\pi \frac{e^{-ka}}{2a^3}(1+ka)[/tex]

And for k<0:

[tex]I=\pi\frac{e^{ka}}{2a^3}(1-ka)[/tex]

Now the problem is: which one do I use? Mathematica gives me this:

[tex]\frac{1}{2a^3}\cdot\ \sqrt{\frac{2}{{\pi}}}e^{-ak}\left((ak+1)\theta(k)-e^{2ak}(ak-1)\theta(-k)\right)[/tex]

Where [tex]\theta(x)[/tex] is Heaviside Step function.

I have noticed that the solutions of those integral are similar but how do I implement the step function? Plus how would I integrate it? I know that the derivative of step function is Delta function, but I don't know what the integral of it is?

Thanks!

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# Homework Help: Finding Fourier transform

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