Retarded unit-frequency oscillator response ($G_R(t)=\Theta (t)\sin t$)

The retarded Green function for $d^2/d{t}^2+1$ is $G_R(t)=\Theta (t)\sin t$, with Fourier transform $[1-(\omega +i{0}^{+}{)}^2{]}^{-1}$ in the $e^{-i\omega t}$ convention. Both poles lie below the real axis. Convolution with a force gives the solution with no incoming homogeneous oscillation. A source that is nonzero at negative times can have a nonzero response before a chosen zero of time without violating causality.

For $g(t)=\alpha e^{-\alpha |t|}/2$ with $\alpha >0$, convolution givesThe coordinate and velocity are continuous at zero. At late times the undamped oscillation has amplitude ${\alpha }^2/(1+{\alpha }^2)$; causality fixes the homogeneous solution only after the no-incoming-oscillation condition is specified.