A question about Fourier transform

Question

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

Answer 1

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

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

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

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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:

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