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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

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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 MUCH –  dbzhu Nov 8 '12 at 8:12
    
If my answer was helpful, then please upvote/accept it. Thanks. –  Charles Boyd Nov 8 '12 at 9:13
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