= Retarded Fourier pole prescription
{title2=$e^{-i\omega t}:\quad\omega\mapsto\omega+i0$}
For inverse time <Fourier transform> $\int\widehat G(\omega)e^{-i\omega t}d\omega/(2\pi)$, 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 $\Theta(t)\sin(\omega_0t)/\omega_0$.
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