= Solution
Seek the steady forced displacement as $x=x_0e^{-i\omega t}$. The <oscillator> equation gives
$$
x_0=\frac{qE_0/\mu}{\omega_{\rm TO}^2-\omega^2-i\gamma\omega}.
$$
For $n$ ion pairs per volume, the ionic polarization is $P_{\rm ion}=nqx$. Add the electronic polarization $\epsilon_0(\epsilon_\infty-1)E$, assumed frequency-independent over this infrared range. The complex <relative permittivity> is therefore
$$
\epsilon_r(\omega)=\epsilon_\infty+
\frac{nq^2/(\epsilon_0\mu)}{\omega_{\rm TO}^2-\omega^2-i\gamma\omega}.
$$
For a longitudinal free oscillation without external free charge, $\mathbf k\cdot\mathbf D=0$ and the longitudinal <electric field> requires $\epsilon_r(\omega_{\rm LO})=0$ in the lossless limit. Hence
$$
\omega_{\rm LO}^2-\omega_{\rm TO}^2=\frac{nq^2}{\epsilon_0\mu\epsilon_\infty}.
$$
Eliminating the microscopic <oscillator> strength gives
$$
\boxed{\epsilon_r(\omega)=\epsilon_\infty
\frac{\omega_{\rm LO}^2-\omega^2-i\gamma\omega}
{\omega_{\rm TO}^2-\omega^2-i\gamma\omega}.}
$$
Here $\omega_{\rm LO}$ denotes the undamped longitudinal frequency; with finite <damping> the zero is at a complex frequency. The sign of the imaginary term follows the stated $e^{-i\omega t}$ convention and gives positive absorptive $\operatorname{Im}\epsilon_r$ for $\omega>0$, $\gamma>0$. At zero frequency the same model yields the <Lyddane-Sachs-Teller relation> $\epsilon_r(0)/\epsilon_\infty=\omega_{\rm LO}^2/\omega_{\rm TO}^2$. The high-frequency limit here is above the <phonon> resonance but below electronic resonances, not arbitrarily high <photon> <energy>.
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