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Fourier transform.

1. Homework Statement
I need to take the inverse Fourier transform of

[tex]\frac{b}{\pi(x^2+b^2)}[/tex]


2. Homework Equations

f(t)=[tex]\int_{-\infty}^{\infty}e^{itx}\frac{b}{\pi(x^2+b^2)}dx[/tex]

It might be useful that [tex]\frac{2b}{\pi(x^2+b^2)}=\frac{1}{b+ix}+\frac{1}{b-ix}[/tex]


3. The Attempt at a Solution
I know the result is [tex]e^{(-b|t|)}[/tex], and I can get from [tex]e^{(-b|t|)}[/tex] to
[tex]\frac{b}{\pi(x^2+b^2)}[/tex], but how do I do it in reverse if I didn't already know the pair existed? This doesn't require complex integration does it?
 
Last edited:

Answers and Replies

979
1
I have to admit my first thought was a contour integral...

In my experience, things involving |x| tend to require them.
 
pam
455
1
It is a standard contour integral. Close the contour with a semicircle above the real axis.
 
Okay guys, thanks, that is what I was thinking, but the book I'm in doesn't have anything else involving complex integration, so I assumed that I was just missing a trick.
 

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