= Principal-value Fourier transform of a real pole
{title2=$\mathcal F\!\left(\operatorname{PV}\frac1{x-a}\right)=-i\pi\operatorname{sgn}(k)e^{-ika}$}
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>.
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