= Initial-velocity memory in the overdamped slow mode
For $m\ddot x+\gamma\dot x+kx=0$, let $r_s$ and $r_f$ be the slow and fast <negative roots>. The coefficient of the slow <normal mode> is exactly
$$
A_s=\frac{v_0-r_fx_0}{r_s-r_f}.
$$
With $\epsilon=km/\gamma^2\ll1$ and $\tau_m=m/\gamma$, expansion gives $A_s=x_0+\tau_m v_0+O(\epsilon)(|x_0|+|\tau_m v_0|)$. Thus the initial <velocity> can leave a finite displacement after its own fast decay. In particular, $x_0=0$ and $v_0\ne0$ produce a nonzero slow tail. A first-order <overdamped approximation> needs an adjusted initial position after the <boundary layer>; setting it to $x_0$ requires the additional smallness of $\tau_m v_0$.
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