I would like to compute the Fourier transform for the function:
\begin{equation} f(x)=\begin{cases} 1/x&, x\in [a,b] \\ 0,& x \notin [a,b]\end{cases} \end{equation} but I cannot do the integral:
\begin{equation} \hat{f}(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(x)e^{-ikx}dx=\frac{1}{\sqrt{2\pi}}\int_{a}^{b}\frac{e^{-ikx}}{x}dx \end{equation} I do know that the improper integral of this Fourier transform can be computed as a principal value (PV) integral but I cannot find the result in a closed form.