How to integrate this fourier transform?

Question

I want to integrate $$\int_{-\infty}^{\infty} \frac{e^{itx}}{{1+x^2}} dx.$$ I don't see how substitution or integration by parts could help here. Does anybody know how to do this?

Answer 1

Consider the complex function $f(z,t)=\frac{e^{itz}}{1+z^2}$ and consider the complex upper semicircle, from $+\infty$ to $-\infty$, the only pole in that region is a simple pole, $z=i$, so we can summarise our integral $\int_{-\infty}^\infty\frac{e^{itx}}{1+x^2}=2\pi iRes(f(z,t),z=i))$

Here $Res(f(z,t),z=i))=\lim_{z\to i}(z-i)f(z,t)=\lim_{z\to i}\frac{e^{itz}}{(z+i)}=\frac{e^{-t}}{2i}$

Thus:

$\int_{-\infty}^\infty\frac{e^{itx}}{1+x^2}=2\pi i\frac{e^{-t}}{2i}=\pi e^{-t}$

Answer 2

Do you know how to do a contour integral? This specific problem is actually solved on Wikipedia's page on this subject.