$${\varepsilon = 1+ \frac{Ne^2}{\varepsilon_0 m_e} \frac{1}{[(\omega_0^2 - \omega^2) + i \omega \gamma]} = 1+ \frac{Ne^2}{\varepsilon_0 m_e} \frac{1}{[(\omega_0^2 - \omega^2 ) + i\omega \gamma]} \times \frac{(\omega_0^2-\omega^2 ) - i \omega \gamma}{(\omega_0^2 - \omega^2 ) - i\omega \gamma} = \\ =\bigg\{1+ \frac{Ne^2}{\varepsilon_0 m_e} \frac{\omega_0^2 - \omega^2}{(\omega_0^2 - \omega^2 )^2 + (\omega \gamma)^2 )}\bigg\} - i \bigg\{\frac{Ne^2}{\varepsilon_0 m_e} \frac{\omega \gamma}{(\omega_0^2-\omega^2 )^2+(\omega \gamma)^2 )} \bigg\}}$$