Past exam of the theoretical physics I course of the University of Cambridge / 2017 / 4 / b / Perturbation of the oscillatory poles

At $\epsilon =0$ the algebraic poles are $\boxed{{\omega }_{\pm }^{(0)}=\pm 1}$; a retarded limiting prescription displaces them infinitesimally below the real axis. Set ${\omega }_s=s+{\delta }_s$, with $s=\pm 1$ and ${\delta }_s=O(\epsilon )$. The denominator becomesConsequentlyFor positive $\epsilon$ these two poles move below the real axis, giving slowly damped oscillations. For negative $\epsilon$ they move above it and the slow oscillations grow. This is only the slow part of the radiation-reaction oscillator pole structure; the third pole must still be examined.