Principal-value residue rule
= Principal-value residue rule
For real $a$ and nonzero real $\omega$,
$$
\operatorname{PV}\int_{-\infty}^{\infty}\frac{e^{i\omega x}}{x-a}\,dx
=i\pi\operatorname{sgn}(\omega)e^{i\omega a}.
$$
= Principal-value residue rule
For real $a$ and nonzero real $\omega$,
$$
\operatorname{PV}\int_{-\infty}^{\infty}\frac{e^{i\omega x}}{x-a}\,dx
=i\pi\operatorname{sgn}(\omega)e^{i\omega a}.
$$