Some lecture notes of mine quote this apparently well known result:
$I(\omega)=\lim_{\eta\rightarrow0^+}\int_0^\infty e^{(i\omega-\eta)s}\text{d}s=\pi\delta(\omega)+i\mathcal{P}(\frac{1}{\omega})$,
where $\mathcal{P}(\frac{1}{\omega})$ behaves such that $\int_{-\infty}^\infty{P}(\frac{1}{\omega})\text{d}\omega=\lim_{\eta\rightarrow0^+}\left[\int_{-\infty}^\delta\frac{1}{\omega}\text{d}\omega+\int_{-\delta}^\infty\frac{1}{\omega}\text{d}\omega\right]$.
I tried to come to this result myself and I end up with:
$I(\omega)=\lim_{\eta\rightarrow0^+}\left[\frac{\eta}{\eta^2+\omega^2}+\frac{i\omega}{\eta^2+\omega^2}\right]$,
I see that the real term goes to the dirac delta as desired but I don't know the justification for wrapping the imaginary part in $\mathcal{P}()$.
Can someone shed some light on this please?
For some context, the lectures notes are for open quantum systems.
