## 4 The Complex Dielectric Permittivity  
The dielectric polarization, in the general case, calculated as  
\[\mathrm{P}^{(\Omega)}(\mathrm{t})=\sum_{\lambda=0}^{\infty}\hat{\alpha}_{2 \lambda}^{(\Omega_{2\lambda+1})}\mathrm{E}_{\mathrm{pol}}^{2\lambda+1}(\mathrm{t }). \tag{26}\]  
In (26) \(\hat{\alpha}_{2\lambda}^{(\Omega_{2\lambda+1})}\) is a complex function defined on set \(\Omega_{2\lambda+1}=\{\omega;2\omega;3\omega,\dots;2\lambda\omega;(2\lambda+1)\omega\}\) of parameters of relaxers and temperature \(\hat{\alpha}_{2\lambda}^{(\Omega_{2\lambda+1})}(\mathrm{U}_{0},\delta_{0}, \
u_{0},n_{0},a;\mathrm{T})=\frac{\mathrm{P}^{((2\lambda+1)\omega)}(\mathrm{t})} {\mathrm{E}_{\mathrm{pol}}^{2\lambda+1}(\mathrm{t})}\), which has meaning of \(2\lambda\) order component from decomposition of the polarization \(\mathrm{P}^{(\Omega)}(\mathrm{t})\) into a series according to odd degrees of intensity of the polarizing field \(\mathrm{E}_{\mathrm{pol}}(\mathrm{t})\). Here \(\lambda=\{0,1,2,3,\dots\}\). The frequency harmonic of the polarization, of order \(r=2\lambda+1\), according to (7), \(\mathrm{P}^{((2\lambda+1)\omega)}(\mathrm{t})=\frac{\mathrm{q}}{\mathrm{d}} \sum_{\lambda=0}^{\infty}\int_{0}^{\mathrm{d}}\mathrm{x}\wp^{((2\lambda+1)\omega )}(\xi,\tau)\mathrm{dx}\), taking into account (20), becomes  
\[\mathrm{P}^{((2\lambda+1)\omega)}(\mathrm{t})=-\frac{2\mathrm{q}\mathrm{d}}{\pi^ {2}}\sum_{n=1}^{\infty}\sum_{k=2\lambda+1}^{\infty}\frac{\gamma^{k}\cdot\Re_{k}^{( (2\lambda+1)\omega)}(\mathrm{n},\tau)}{\mathrm{n}^{2}}\times\sin^{2}\Bigl{(} \frac{\pi n}{2}\Bigr{)}. \tag{27}\]  
Comparing the formula \(\mathrm{P}^{(\Omega)}(\mathrm{t})=\hat{\alpha}^{(\Omega)}\cdot\mathrm{E}_{ \mathrm{pol}}(\mathrm{t})\) ([4], p. 151) with the expression (26), let us write the complex dielectric susceptibility (CDS) in the form of  
\[\hat{\alpha}^{(\Omega)}=\sum_{\lambda=0}^{\infty}\hat{\alpha}_{2\lambda}^{(\Omega_ {2\lambda+1})}\mathrm{E}_{\mathrm{pol}}^{2\lambda}(\mathrm{t}),\hat{\alpha}^{( \Omega)}=\sum_{\lambda=0}^{\infty}\hat{\alpha}^{(\Omega_{2\lambda+1})}. \tag{28}\]  
In (28) \(\hat{\alpha}_{2\lambda}^{(\Omega_{2\lambda+1})}\cdot\mathrm{E}_{\mathrm{pol}}^{2 \lambda}(\mathrm{t})=\hat{\alpha}^{(\Omega_{2\lambda+1})}=\frac{\mathrm{P}^{((2 \lambda+1)\omega)}(\mathrm{t})}{\mathrm{E}_{\mathrm{pol}}(\mathrm{t})}\). Taking the relaxation complex polarization as \(\mathrm{P}^{(\Omega)}(\mathrm{t})=\varepsilon_{0}\Big{(}\xi^{(\Omega)}- \varepsilon_{\infty}\Big{)}\mathrm{E}_{\mathrm{pol}}(\mathrm{t})\) ([4], p. 151), we find the complex dielectric permittivity (CDP)  
\[\varepsilon^{(\Omega)}=\varepsilon_{\infty}+\frac{\hat{\alpha}^{(\Omega)}}{ \varepsilon_{0}}. \tag{29}\]  
Substitution (28) into (29) gives  
\[\varepsilon^{(\Omega)}=\varepsilon_{\infty}+\sum_{\lambda=0}^{\infty} \varepsilon_{2\lambda}^{(\Omega_{2\lambda+1})}\mathrm{E}_{\mathrm{pol}}^{2 \lambda}(\mathrm{t}),\ \varepsilon^{(\Omega)}=\varepsilon_{\infty}+\sum_{\lambda=0}^{\infty} \varepsilon^{(\Omega_{2\lambda+1})}. \tag{30}\]  
In (30) \(\hat{\varepsilon}^{(\Omega_{2\lambda+1})}=\hat{\varepsilon}_{2\lambda}^{( \Omega_{2\lambda+1})}\mathrm{E}_{\mathrm{pol}}^{2\lambda}(\mathrm{t})=\frac{ \hat{\alpha}^{(\Omega_{2\lambda+1})}}{\varepsilon_{0}}\); limiting the approximation of \(\lambda=1\),  
\[\varepsilon^{(\Omega_{1},\Omega_{3})}=\varepsilon_{\infty}+\frac{1}{ \varepsilon_{0}}\Big{(}\hat{\alpha}^{(\Omega_{1}=\{\omega\})}+\hat{\alpha}^{( \Omega_{3}=\{\omega 2\omega 3\omega\})}\Big{)} \tag{31}\]  
we find the CDP with accuracy to the quadratic term along the field  
\[\varepsilon^{(\Omega_{1},\Omega_{3})}=\varepsilon_{\infty}+\varepsilon_{0}^{( \Omega_{1}-\{\omega\})}+\hat{\varepsilon}_{2}^{(\Omega_{3}=\{\omega 2\omega 3\omega\})} \mathrm{E}_{\mathrm{pol}}^{2}(\mathrm{t}). \tag{32}\]  
Substituting (8) and (25) in expressions of the form  
\[