I was given the problem of calculating the Fourier transform of $$g(x) = \frac{1}{1 + x^2}$$ however I didn't know where to start. More specifically, I know that we should be calculating this. $$\hat{g}(\xi): = \int_{-\infty}^{\infty} \frac{1}{1 + x^2}e^{-2\pi ix\xi}dx$$
I think I should start with For $\xi \leq 0$, try integrating $$f(z)= \frac{1}{1+z^2}e^{−2\pi iz \xi}$$over an appropriate keyhole contour (the residue theorem). I think the hole here should be $i$. But I really didn't know how to start with the calculation! Appreciated your help in advance.