= Time support from poles of a Fourier Green function
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.
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