Advanced Green function ($G_A(t,t^{\prime })=0(t>t^{\prime })$)

An advanced Green function has support only before its source time. It is selected by a future boundary condition, rather than an initial causal response. For $L={\partial }_t^2+2\lambda {\partial }_t+{\omega }_0^2$ with $\lambda <0$ and ${\Omega }^2={\omega }_0^2-{\lambda }^2>0$, the real-axis inverse Fourier transform of $(-{\omega }^2-2i\lambda \omega +{\omega }_0^2{)}^{-1}$ is $-H(-t)e^{-\lambda t}\sin (\Omega t)/\Omega$. Its first derivative jumps by one at zero. A growing retarded solution exists with a different inverse contour; this illustrates causality versus stability of a Green function.