= Perturbation of the oscillatory poles
At $\varepsilon=0$ the algebraic poles are $\boxed{\omega_\pm^{(0)}=\pm1}$; a retarded limiting prescription displaces them infinitesimally below the real axis. Set $\omega_s=s+\delta_s$, with $s=\pm1$ and $\delta_s=O(\varepsilon)$. The denominator becomes
$$
1-(s+\delta_s)^2-i\varepsilon(s+\delta_s)^3
=-2s\delta_s-i\varepsilon s^3+O(\varepsilon^2).
$$
Consequently
$$
\boxed{\omega_\pm=\pm1-\frac{i\varepsilon}{2}+O(\varepsilon^2)}.
$$
For positive $\varepsilon$ these two poles move below the real axis, giving slowly damped oscillations. For negative $\varepsilon$ 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.
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