= Solution
Choose the $e^{-i\omega t}$ convention. The steady displacement in the <Lorentz oscillator model> has <complex amplitude>
$$
u_\omega=\frac{qE_\omega/m}{\omega_T^2-\omega^2-i\gamma\omega}.
$$
The induced <electric dipole moment> is $p_\omega=qu_\omega$, so the microscopic <electric polarizability> is $\alpha(\omega)=q^2/[m(\omega_T^2-\omega^2-i\gamma\omega)]$. For $N$ identical oscillators per unit volume, $P_\omega=Np_\omega=\epsilon_0\chi_\omega E_\omega$. Consequently
$$
\boxed{\chi_\omega=\frac{\omega_p^2}{\omega_T^2-\omega^2-i\gamma\omega},\qquad \epsilon_r(\omega)=1+\chi_\omega,\qquad \omega_p^2=\frac{Nq^2}{m\epsilon_0}.}
$$
Here the dimensionless quantity called polarizability in the paper is the <electric susceptibility>; the single-electron polarizability $\alpha$ has different units. The sign of the <Electron> charge disappears because both the driving force and dipole moment contain $q$.
Multiplying by the <complex conjugate> of the denominator gives
$$
\epsilon'_r=1+\frac{\omega_p^2(\omega_T^2-\omega^2)}{(\omega_T^2-\omega^2)^2+\gamma^2\omega^2},\qquad \epsilon''_r=\frac{\omega_p^2\gamma\omega}{(\omega_T^2-\omega^2)^2+\gamma^2\omega^2}.
$$
For positive <damping> and positive <frequency>, $\epsilon''_r\geq0$, as required for passive <wave absorption> in this Fourier convention. At zero <frequency>, $\epsilon'_r=1+\omega_p^2/\omega_T^2$ and $\epsilon''_r=0$. Near a weakly damped <resonance> the imaginary part has a narrow positive peak, while the real part rises on the low-frequency side and then falls steeply through $\epsilon'_r=1$ at $\omega_T$. It can become negative above <resonance> for sufficiently strong oscillator strength and weak <damping>. At large <frequency>, $\epsilon'_r=1-\omega_p^2/\omega^2+O(\omega^{-4})$ and $\epsilon''_r\sim\omega_p^2\gamma/\omega^3$: both approach the vacuum values. The absorption maximum is close to, but need not be exactly at, $\omega_T$ for finite <damping>.
\Image[/past-exam-of-the-quantum-condensed-matter-course-of-the-university-of-cambridge/2017-lorentz-permittivity.png]
{title=Real and imaginary permittivity for an illustrative damped bound-charge resonance}
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