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

I've been given the following function:

[tex]g(x) = \frac{\gamma}{2}e^{-\gamma \left|x\right|}[/tex] with [tex]\gamma>0[/tex]

First thing I needed to do was to prove [tex]\int^{-\infty}_{\infty}g(x)dx=1[/tex] which was simple enough.

I hit a problem when trying to find the Fourier transform [tex]\widetilde{g}(k)[/tex] of [tex]g(x)[/tex]

I'm asked to show the transform is of the form

[tex]\widetilde{g}(k)=a\frac{1}{1+\frac{k^{2}}{s^{2}}}[/tex]

and find a and s

2. Relevant equations

[tex]\widetilde{g}(k) = \frac{1}{\sqrt{2\pi}}\int^{-\infty}_{\infty}e^{-ikx}g(x)dx[/tex]

3. The attempt at a solution

I try to perform integration by parts on

[tex]\widetilde{g}(k) = \frac{1}{\sqrt{2\pi}}\int^{-\infty}_{\infty}e^{-ikx}\frac{\gamma}{2}e^{-\gamma \left|x\right|}dx[/tex]

but whenever I try this I end up with uv equalling vdu so the whole expression turns out as zero. I think this is probably because I don't really know what I'm doing with the integration with those limits to infinity. I can split up g(x) because its an even function but not the [tex]e^{-ikx}[/tex] part.

I've plugged this into wolfram alpha and it churned out [tex]a = \frac{1}{2\pi}[/tex] and [tex]s = \gamma[/tex], I'm just not sure how to get there.

Any help appreciated!

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# Fourier Transfrom problem

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