Principal-value Fourier transform of a real pole ($\mathcal{F}\left(\mathrm{PV}\frac{1}{x-a}\right)=-i\pi \mathrm{sgn}(k)e^{-ika}$)

A real simple pole is not locally integrable in the ordinary improper sense. Its symmetric Cauchy principal value defines a tempered distribution. For the $e^{-ikx}$ transform convention, contour indentation or sine integration gives the displayed nondecaying oscillatory contribution. Upper and lower boundary-value prescriptions instead change it by $\mp i\pi \delta (x-a)$ before transforming. A pole prescription must therefore be stated before claiming a Fourier asymptotic expansion.