Help with an Inverse Fourier transform

Question

Can anybody please guide me how to compute the inverse Fourier Transform of: $$ f(k) = \frac{1}{1+k^2} \frac{\pi}{4}(\rm{sgn}(1-k) + \rm{sgn}(1+k)) $$

Answer 1

$$ q\cdot(sgn(1-k) + sgn(1+k)) =\begin{cases} 0 \ \ k > 1 \\ 2q \ \ -1<k<1 \\ 0 \ \ k < - 1 \end{cases} $$ Which yields

$$ f(t) = \frac{1}{2\pi} \int_{-1}^{1} \frac{2e^{itw}}{1+w^2} \frac{\pi}{4} dw $$

Answer 2

If we define $$\widehat{f}(s)=\mathcal{F}(f)(s) = \frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}f(x)e^{isx}\,dx,\qquad \mathcal{F}^{-1}(f)(s) = \frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}f(x)e^{-isx}\,dx$$ we have: $$ \mathcal{F}^{-1}\left(\frac{1}{1+x^2}\right) = \sqrt{\frac{\pi}{2}}\,e^{-|s|} \tag{1} $$ $$ \mathcal{F}^{-1}\left(\mathbb{1}_{(-1,1)}(x)\right)=\sqrt{\frac{2}{\pi}}\,\text{sinc}(s)\tag{2}$$ hence the inverse Fourier transform of $\frac{\mathbb{1}_{(-1,1)}(x)}{1+x^2}$ can be computed through the convolution of the RHSs of $(1)$ and $(2)$, but it is not an elementary function: it involves $\text{Si}(z)$ and $\text{Ci}(z)$ (the sine and cosine integrals) evaluated at complex arguments.