 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.