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For a bosonic operator $A$, the retarded Green function is $G_R(t)=-i\theta (t)⟨[A(t),A(0)]⟩$. Causality makes its Fourier transform analytic in the upper half frequency plane.

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 the inverse transform $G(\tau )=\int d\omega e^{-i\omega \tau }g(\omega )/(2\pi )$, positive $\tau$ requires closing a decaying contour below the real axis, clockwise; negative $\tau$ requires closing above, counterclockwise. Poles strictly below give a retarded Green function, while poles strictly above give an advanced Green function. Real-axis poles need a prescription. The sign convention in the Fourier exponential is essential.