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Find the inverse Fourier transform of $\widehat{f}(\xi) = \frac{1}{1+\xi^2}$.

I wanted to use the inverse Fourier formula to solve the problem.

$$ f(x) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} e^{ix\xi} \frac{1}{1+\xi^2} d\xi $$

Integration by part is not helpful at the first stage, and I am not sure about differentiating both side w.r.t $x$ and then taking the derivative inside and then using integration by parts. (I have issue with taking the derivative as well, in the sense how do we know that function is differentiable and even ignoring that issue, I am not able to find a closed form).

LearningMath
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  • Try checking https://math.stackexchange.com/questions/348067/inverse-fourier-transform-of-1-1s2 – Yalikesifulei Jan 07 '21 at 21:26
  • I had looked at both the problem. @Yalikesifulei one of the comment says that a simpler way to do it it using Leibniz rule inside integral rule. That's what I was trying. – LearningMath Jan 07 '21 at 21:30
  • @LearningMath not exactly yours question, but may be helpful https://math.stackexchange.com/questions/2236261/find-the-fourier-cosine-transform-of-the-function-defined-by-fx-frac11x2 – Yalikesifulei Jan 07 '21 at 22:12

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