## 4 The Complex Dielectric Permittivity
\begin{split}\hat{\varepsilon}^{(\Omega_{1}-\{\omega\})}=\hat{ \varepsilon}_{0}^{(\Omega_{1}-\{\omega\})}=\frac{\hat{\kappa}^{(\Omega_{1}- \{\omega\})}}{\varepsilon_{0}}=\frac{\hat{\
u}^{(\omega)}(\mathrm{t})}{ \varepsilon_{0}\mathrm{E}_{\mathrm{pol}}(\mathrm{t})},\\ \hat{\varepsilon}^{(\Omega_{3}=\{\omega 2\omega 3\omega\})}=\hat{ \varepsilon}_{2}^{(\Omega_{3}-\{\omega 2\omega 3\omega\})}=\hat{ \varepsilon}_{\mathrm{pol}}^{(\Omega_{3}-\{\omega 2\omega 3\omega\})}=\frac{\hat{\
u}^{( \Omega_{3}-\{\omega 2\omega 3\omega\})}}{\varepsilon_{0}\mathrm{E}_{\mathrm{pol}}(\mathrm{t})},\end{split} \]  
we obtain  
\[\hat{\varepsilon}^{(\Omega_{1}-\{\omega\})}=\hat{\varepsilon}_{0}^{(\Omega_{1}- \{\omega\})}=\frac{\varepsilon_{\infty}\Big{(}\Gamma_{1}^{(\omega)}-\mathrm{ i}\Gamma_{2}^{(\omega)}\Big{)}}{1-\Gamma_{1}^{(\omega)}+\mathrm{i}\Gamma_{2}^{( \omega)}}, \tag{33}\]  
where relaxation parameters are introduced that characterize the first frequency harmonic of the polarization (a multiple of the first power of the alternating field (8))  
\[\Gamma_{1}^{(\omega)}(\mathrm{T})=\frac{8}{\pi^{2}}\sum_{n=1}^{\infty}\Bigg{[} \frac{\frac{T_{n}}{T_{M}}\sin^{2}\!\left(\frac{\pi n}{2}\right)}{\mathrm{n}^{2 }(1+\omega^{2}T_{n^{2}})}\Bigg{]},\ \Gamma_{2}^{(\omega)}(\mathrm{T})=\frac{8}{\pi^{2}}\sum_{n=1}^{\infty}\Bigg{[} \frac{\frac{\omega T_{n}^{2}}{T_{M}}\sin^{2}\!\left(\frac{\pi n}{2}\right)}{ \mathrm{n}^{2}(1+\omega^{2}T_{n^{2}})}\Bigg{]}. \tag{33a}\]  
The expression \(\hat{\varepsilon}^{(\Omega_{1})}=\varepsilon_{\infty}+\hat{\varepsilon}_{0}^{( \Omega_{1}-\{\omega\})}\) formally coincides with the formulae (2) of ([1], p. 44), which indicates the validity of the \"zero\" field of approximation and consistent with the expressions obtained in [3] for CDP ([3], p. 17). Also, based on (25), we obtain a multiple of the second power of the alternating field component of the complex permittivity  
\[ \begin{split}\hat{\varepsilon}^{(\Omega_{3}=\{\omega 2\omega 3\omega\})}=\frac{ \varepsilon_{0}^{2}\varepsilon_{\infty}^{3}\pi^{4}\mathrm{E}_{0}^{2}}{4\mathrm{ n}^{2}\mathrm{n}_{0}^{2}\mathrm{d}^{2}}\Bigg{(}\frac{\Gamma_{1,1}^{(\omega 2\omega 3\omega)}-\mathrm{i}\Gamma_{1,2}^{(\omega 2\omega 3\omega)}}{ \Phi_{1}^{(\omega)}+\mathrm{i}\Phi_{2}^{(\omega)}}+\frac{\Gamma_{2,1}^{(\omega 2\omega 3\omega)}-\mathrm{i}\Gamma_{2,2}^{(\omega 2 \omega 3\omega)}}{\Phi_{1}^{(\omega)}+\mathrm{i}\Phi_{2}^{(\omega 3\omega)}}\Bigg{)}\cdot\exp(2\mathrm{i}\omega t).\end{split}  \tag{34}\]  
In (34) the following notations are adopted:  
\[ \begin{split}\Gamma_{1,1}^{(\omega 2\omega 3\omega)}=\left(\frac{8}{\pi^{2}} \right)^{3}\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\sum_{s=1}^{\infty}\Bigg{\{} \frac{\mathrm{m}^{2}\cos^{2}\!\left(\frac{\pi n}{2}\right)\sin^{2}\!\left( \frac{\pi n}{2}\right)\sin^{2}\!\left(\frac{\pi n}{2}\right)}{\left(\mathrm{m}^ {2}-\mathrm{m}^{2}\right)\cdot\left(\mathrm{m}^{2}-\mathrm{s}^{2}\right)\cdot \left(1+\omega^{2}T_{n}^{2}\right)\cdot\left(1+4\omega^{2}T_{n}^{2}\right)\cdot \left(1+9\omega^{2}T_{n}^{2}\right)}\times\\ \times\frac{T_{4}T_{n}T_{n}}{T_{M}^{3}}\times\left(1-\omega^{2} (2T_{s}T_{m}+6T_{m}T_{n}+3T_{n}T_{s})\right)\Bigg{\}};\end{split}