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F. transform problem

  • #1

Homework Statement



Find Fourier transform of function

[tex]f(x)=\frac{1}{x^2+a^2}[/tex], [tex]a>0[/tex]



Homework Equations



[tex]\mathcal{F}[\frac{1}{x^2+a^2}]=\frac{1}{\sqrt{2\pi}}\int^{\infty}_{-\infty}\frac{e^{-ikx}dx}{x^2+a^2}[/tex]



The Attempt at a Solution



Two different case

[tex]k>0[/tex]

and

[tex]k<0[/tex]

How to solve integral

[tex]\mathcal{F}[\frac{1}{x^2+a^2}]=\frac{1}{\sqrt{2\pi}}\int^{\infty}_{-\infty}\frac{e^{-ikx}dx}{x^2+a^2}[/tex]

Probably using complex analysis?! I forget this. I have two poles [tex]ia[/tex] and [tex]-ia[/tex]. How to integrate this? Is there some other method without using complex analysis?
 

Answers and Replies

  • #2
121
0
"Is there some other method without using complex analysis?"

Nope. But complex analysis isn't that bad -- it just seems like black magic until you get used to it. You might want to look at the example at http://en.wikipedia.org/wiki/Residue_theorem .
 

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