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I just don't know how to calculate the the fourier transform of $1/(1+x^2)$.Can you help me guys? Thx

1 Answers 1

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You need to use the duality property of the Fourier transform:


Theorem: Let $\hat{f}(w) = \mathcal{F}[f(t)]$, then the following duality property holds:

\mathcal{F} \, \big [\hat{f}(t) \big ] = 2 \pi f(-w).


So, look at your table and see this convenient identity:

\mathcal{F} \, \Big [\frac{1}{2a}e^{-a|t|} \Big ] = \frac{1}{a^2 + w^2}.

Now apply the duality property:

\mathcal{F} \, \Big [ \frac{1}{a^2 + t^2} \Big ] = \frac{\pi}{a} e^{-a|w|}

Then, the answer to your question is:

\mathcal{F} \, \Big [ \frac{1}{1 + t^2} \Big ] = \pi e^{-|w|}.$$

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    I got , THANK YOU VERY MUCH2012-11-08