## 5 Effect of Temperature on Theoretical Spectra of Complex Dielectric Permittivity
In the case of \(\mathrm{T}<<\mathrm{T}_{\mathrm{cr,relax}}\), \(T_{\mathrm{D}}\)\(<<\)\(<\)\(\mathrm{T}_{\mathrm{M}}\), by rewriting (46) while taking into account (42), we have  
\[\left[\hat{\varepsilon}_{1}^{(\omega)}\right]_{\mathrm{D}}=\varepsilon_{\infty} \!\left(1+\frac{4T_{\mathrm{D}}}{\pi^{2}T_{\mathrm{M}}}\sum_{\mathrm{n=1}}^{ \infty}\left[\frac{(1-(-1)^{\mathrm{n}})}{\mathrm{n}^{4}+\omega^{2}T_{ \mathrm{D}}^{2}}\right]\right)\!,\left[\hat{\varepsilon}_{1}^{(\omega)}\right]_ {\mathrm{D}}^{//}=\frac{4T_{\mathrm{D}}\varepsilon_{\infty}}{\pi^{2}T_{ \mathrm{M}}}\sum_{\mathrm{n=1}}^{\infty}\left[\frac{(1-(-1)^{\mathrm{n}})\cdot \omega T_{\mathrm{D}}}{\mathrm{n}^{2}\big{(}\mathrm{n}^{4}+\omega^{2}T_{ \mathrm{D}}^{2}}\right]\right]\!.\] (46d) From here, by virtue of \[\left[\hat{\varepsilon}_{1}^{(\omega=0)}\right]_{\mathrm{D}}=\varepsilon_{ \mathrm{S,D,1}}(\mathrm{T})=\varepsilon_{\infty}\!\left(1+\Gamma_{1,\mathrm{D}}^ {(\omega=0)}\right)=\varepsilon_{\infty}\!\left(1+\frac{\pi^{2}T_{\mathrm{D}}}{ \mathrm{i}2T_{\mathrm{M}}}\right)\!,\] \[\Gamma_{1}^{(\omega)}(\rm T)=\frac{8}{\pi^{2}}\sum_{\rm n=1}^{ \infty}\left[\frac{\frac{T_{\rm M}}{2}\sin^{2}\left(\frac{\pi\eta}{2}\right) }{n^{2}(1+\omega^{2}\Gamma_{\rm M}^{2})}\right]\approx\frac{1}{1+\omega^{2} \Gamma_{\rm M}^{2}},\]0 entering the designation \[\Gamma_{1}^{(\omega)}(\rm T)=\frac{8}{\pi^{2}}\sum_{\rm n=1}^{ \infty}\left[\frac{\frac{T_{\rm M}}{2}\sin^{2}\left(\frac{\pi\eta}{2}\right) }{n^{2}(1+\omega^{2}\Gamma_{\rm M}^{2})}\right]\approx\frac{1}{1+\omega^{2} \Gamma_{\rm M}^{2}},\]1, we have \[\Gamma_{1}^{(\omega)}(\rm T)=\frac{8}{\pi^{2}}\sum_{\rm n=1}^{ \infty}\left[\frac{\frac{T_{\rm M}}{2}\sin^{2}\left(\frac{\pi\eta}{2}\right) }{n^{2}(1+\omega^{2}\Gamma_{\rm M}^{2})}\right]\approx\frac{1}{1+\omega^{2} \Gamma_{\rm M}^{2}},\]2 \[\Gamma_{1}^{(\omega)}(\rm T)=\frac{8}{\pi^{2}}\sum_{\rm n=1}^{ \infty}\left[\frac{\frac{T_{\rm M}}{2}\sin^{2}\left(\frac{\pi\eta}{2}\right) }{n^{2}(1+\omega^{2}\Gamma_{\rm M}^{2})}\right]\approx\frac{1}{1+\omega^{2} \Gamma_{\rm M}^{2}},\]3 (46e) \[\Gamma_{1}^{(\omega)}(\rm T)=\frac{8}{\pi^{2}}\sum_{\rm n=1}^{ \infty}\left[\frac{\frac{T_{\rm M}}{2}\sin^{2}\left(\frac{\pi\eta}{2}\right) }{n^{2}(1+\omega^{2}\Gamma_{\rm M}^{2})}\right]\approx\frac{1}{1+\omega^{2} \Gamma_{\rm M}^{2}},\]4 (46f) which is also consistent [2] in the temperature region \(\mathrm{T}<<\mathrm{T}_{\mathrm{cr,relax}}\). In the theoretical studies of the solution schemes of the nonlinear kinetic Equation (2) and its annees to the issues of studying the mechanisms of the relaxation polarization described by formulas (8), (25)generalized calculation of the relaxation parameters (33a) for the wide range of field and temperature parameters. Introducing dimensionless variables \(\alpha_{1}=\frac{T_{\rm D}}{T_{\rm M}}\), \(\alpha_{2}=\omega T_{\rm D}\), we present the generalized expressions (33a) in the form of  
\[\Gamma_{1}^{(\omega)}(\rm T)=\frac{8}{\pi^{2}}\sum_{\rm n=1}^{ \infty}\left[\frac{\frac{T_{\rm M}}{2}\sin^{2}\left(\frac{\pi\eta}{2}\right) }{n^{2}(1+\omega^{2}\Gamma_{\rm M}^{2})}\right]\approx\frac{1}{1+\omega^{2} \Gamma_{\rm M}^{2}},\]5  
It is convenient for further analysis of the effects of various kinds of the polarization nonlinearities (interactions of the multiple relaxation modes of both the same frequency and different field frequencies; quantum microscopic effects associated with proton tunneling in HBC; the phenomena of nonlinear volume-charge polarization, etc.) and temperature per dispersion expressions (38) with numerical calculation of frequency-temperature spectra of the complex dielectric permittivity (CDP).