## 4 The Complex Dielectric Permittivity
 \tag{34a}\]\[\Gamma_{1,2}^{(\omega 2\omega 3\omega)}=\left(\frac{8}{\pi^{2}}\right)^{3} \sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\sum_{s=1}^{\infty}\Bigg{\{}\frac{m^{2} \cos^{2}\left(\frac{m}{2\omega}\right)\sin^{2}\left(\frac{m}{2}\right)\sin^{2} \left(\frac{m}{2}\right)}{(m^{2}-m^{2})\cdot(m^{2}-s^{2})\cdot(1+\omega^{2}T_{ s}^{2})\cdot(1+4\omega^{2}T_{R}^{2})\cdot(1+9\omega^{2}T_{R}^{2})}\times \tag{34b}\] \[\frac{T_{s}T_{m}T_{n}}{T_{M}}\times\left(\omega(T_{s}+2T_{m}+3T_{n})-6\omega^{3}T _{s}T_{m}T_{n}\right)\Bigg{\}};\] \[\Gamma_{2,1}^{(\omega 2\omega 3\omega)}=\left(\frac{8}{\pi^{2}}\right)^{4} \sum_{n=1}^{\infty}\sum_{p=1}^{\infty}\sum_{m=1}^{\infty}\sum_{s=1}^{\infty} \frac{(1-\omega^{2}(2T_{s}T_{m}+6T_{m}T_{p}+3T_{p}T_{s})-3\omega T_{n}(a(T_{s} +2T_{m}+3T_{p})-6\omega^{3}T_{s}T_{m}T_{p}))}{(1+\omega^{2}T_{s}^{2})(1+4 \omega^{2}T_{m}^{2})\left(1+9\omega^{2}T_{p}^{2}\right)(1+9\omega^{2}T_{n}^{2} )};\] (34c) \[\Gamma_{2,2}^{(\omega 2\omega 3\omega)}=\left(\frac{8}{\pi^{2}}\right)^{4} \sum_{n=1}^{\infty}\sum_{p=1}^{\infty}\sum_{m=1}^{\infty}\frac{(3\omega T_{n}(1 -\omega^{2}(2T_{s}T_{m}+6T_{m}T_{p}+3T_{p}T_{s})+a(T_{s}+2T_{m}+3T_{p})-6 \omega^{3}T_{s}T_{m}T_{p}))}{(1+\omega^{2}T_{s}^{2})(1+4\omega^{2}T_{m}^{2}) \left(1+9\omega^{2}T_{p}^{2}\right)(1+9\omega^{2}T_{n}^{2})};\] (34d) \[\[PERSON]_{1}^{(\omega\omega)}=\[PERSON]_{1}^{(\omega)}\cdot\[PERSON]_{1}^{(3 \omega)}-\[PERSON]_{2}^{(\omega)}\cdot\[PERSON]_{2}^{(3\omega)},\;\[PERSON]_{2}^{(\omega \omega)\omega}=\[PERSON]_{2}^{(3\omega)}\cdot\[PERSON]_{1}^{(\omega)}+\[PERSON]_{2}^{(\omega) }\cdot\[PERSON]_{1}^{(3\omega)};\] (34e) \[\[PERSON]_{1}^{(\omega)}=\left(1-\Gamma_{1}^{(\omega)}\right)\cdot \left(\left(1-\Gamma_{1}^{(\omega)}\right)^{2}-3\Big{(}\Gamma_{2}^{(\omega)} \Big{)}^{2}\right)\Bigg{\}},\;\[PERSON]_{2}^{(\omega)}=\Gamma_{2}^{(\omega)}\bigg{(} 3\Big{(}1-\Gamma_{1}^{(\omega)}\Big{)}^{2}-\Big{(}\Gamma_{2}^{(\omega)}\Big{)} ^{2}\bigg{)}\] (34f) \[\Phi_{1}^{(3\omega)}=1-\Gamma_{1}^{(3\omega)},\;\Phi_{2}^{(3 \omega)}=\Gamma_{2}^{(3\omega)};\] (34g) \[\Gamma_{1}^{(3\omega)}(T)=\frac{8}{\pi^{2}}\sum_{n=1}^{\infty} \Bigg{[}\frac{\frac{3T_{m}}{T_{M}}\sin^{2}\left(\frac{m}{2}\right)}{n^{2}(1+9 \omega^{2}T_{n}^{2})}\Bigg{]},\Gamma_{2}^{(3\omega)}(T)=\frac{8}{\pi^{2}} \sum_{n=1}^{\infty}\Bigg{[}\frac{\frac{3\omega T_{n}^{2}}{T_{M}}\sin^{2}\left( \frac{m}{2}\right)}{n^{2}(1+9\omega^{2}T_{n}^{2})}\Bigg{]}. \tag{34h}\]  
Based on (32), taking (33) and (34) into account, we obtain  
\[\varepsilon^{(\Omega_{1},\Omega_{3})}=\varepsilon_{\infty}\Bigg{(}\frac{1- \Gamma_{1}^{(\omega)}-\mathrm{i}\Gamma_{2}^{(\omega)}}{\left(1-\Gamma_{1}^{( \omega)}\right)^{2}+\left(\Gamma_{2}^{(\omega)}\right)^{2}}+\frac{\varepsilon_{ 0}^{2}\varepsilon_{\infty}^{2}\pi^{4}\mathrm{E}_{0}^{2}}{4\mathrm{q}^{2}n_{0}^{2 }\mathrm{d}^{2}}\Big{(}\chi_{1}^{(\omega;2\omega;3\omega)}-\mathrm{i}\chi_{2}^ {(\omega;2\omega;3\omega)}\Big{)}\cdot\exp(2\mathrm{i}\omega t)\Bigg{)}. \tag{35}\]  
The abridged notations are adopted in (35):  
\[\chi_{1}^{(\omega 2\omega;3\omega)}=\frac{\kappa_{1}^{2}-\kappa_{2}^{2 }}{\kappa_{1}^{2}+\kappa_{2}^{2}}\chi_{2}^{(\omega 2\omega;3\omega)}=\frac{2\kappa_{1}\kappa_{2}}{ \kappa_{1}^{2}+\kappa_{2}^{2}};\] \[\frac{T_{s}T_{m}T_{n}}{T_{M}}\times\left(\omega(T_{s}+2T_{m}+3T_{n})-6\omega^{3}T _{s}T_{m}T_{n}\right)\Bigg{\}};\]0 \[\frac{T_{s}T_{m}T_{n}}{T_{M}}\times\left(\omega(T_{s}+2T_{m}+3T_{n})-6\omega^{3}T _{s}T_{m}T_{n}\right)\Bigg{\}};\]1 \[\frac{T_{s}T_{m}T_{n}}{T_{M}}\times\left(\omega(T_{s}+2T_{m}+3T_{n})-6\omega^{3}T _{s}T_{m}T_{n}\right)\Bigg{\}};\]2 \[\frac{T_{s}T_{m}T_{n}}{T_{M}}\times\left(\omega(T_{s}+2T_{m}+3T_{n})-6\omega^{3}T _{s}T_{m}T_{n}\right)\Bigg{\}};\]3  
Separating the real and imaginary parts in (35), we get  
\[\frac{T_{s}T_{m}T_{n}}{T_{M}}\times\left(\omega(T_{s}+2T_{m}+3T_{n})-6\omega^{3}T _{s}T_{m}T_{n}\right)\Bigg{\}};\]4 \[\frac{T_{s}T_{m}T_{n}}{T_{M}}\times\left(\omega(T_{s}+2T_{m}+3T_{n})-6\omega^{3}T _{s}T_{m}T_{n}\right)\Bigg{\}};\]5  
It is easy to see that the first terms in the right parts of (36), (37) also formally coincide with the CDP components calculated in the \"zero\" field approximation ([1], p. 44; [3], p. 17).