Fig. \[fig1\] (left panel). The region marked as [1]{} corresponds to the frequencies smaller than $\omega _{\rm c}$. The region [2]{} defining the Drude parameters extends from the cutoff frequency to the edge of the interband absorption $\omega _{0}$. The high energy domain $\omega>\omega _{0}$ is denoted by [3]{}.

We may now deduce the dielectric function at imaginary frequencies (\[K-K\]) using the Kramers-Kronig relation

$$\varepsilon \left( i\zeta \right) =1+\varepsilon _{1}\left( i\zeta \right)
+\varepsilon _{2}\left( i\zeta \right) +\varepsilon _{3}\left( i\zeta
\right) ,  \label{split}$$

where the indices 1, 2, and 3 indicate respectively the integration ranges $0\leq \omega <\omega _{\rm c}$, $\omega _{\rm
c}\leq\omega <\omega _{0}$, and $\omega _{0}\leq \omega <\infty $. $\varepsilon _{1}$ can be derived using the Drude model (\[Drude\]) leading to

$$\varepsilon _{1}\left( i\zeta \right) =\frac{2}{\pi }\frac{\omega
_{p}^{2}}{\zeta ^{2}-\omega _{\tau }^{2}}\left[ \tan ^{-1}\left( \frac{\omega
_{c}}{\omega _{\tau }}\right) -\frac{\omega _{\tau }}{\zeta }\tan ^{-1}\left(
\frac{\omega _{c}}{\zeta }\right) \right] .  \label{eps1}$$

The two other functions $\varepsilon _{2}$ and $\varepsilon _{3}$ have to be calculated numerically. The results for all three functions as well as for $ \varepsilon \left( i\zeta
\right) $ are shown in Fig. \[fig1\] (right). One can clearly see that $\varepsilon _{1}\left( i\zeta \right) $ dominates the dielectric function at 