Retarded Fourier pole prescription ($e^{-i\omega t}:\omega \mapsto \omega +i0$)

For inverse time Fourier transform $\int \hat{G}(\omega )e^{-i\omega t}d\omega /(2\pi )$, a causal stable oscillator has its poles below the real frequency axis. Undamped real poles are therefore displaced by the prescription $\omega \mapsto \omega +i0$ in its frequency polynomial. For $t>0$, closing the contour below encloses poles with clockwise orientation; for $t<0$, closing above gives zero. Changing the transform's sign reverses the corresponding prescription. For a positive oscillator operator ${\partial }_t^2+{\omega }_0^2$, the inverse is $[{\omega }_0^2-(\omega +i0{)}^2{]}^{-1}$ and its time response is $\Theta (t)\sin ({\omega }_{0}t)/{\omega }_0$.