## 1 Introduction
### Quasi-Classical Dielectric Relaxation Functions
With respect to the study of the laws of dielectric relaxation at arbitrary temperatures (including close to a critical \(T_{\mathrm{cr,relax}}\)), when \(\alpha_{1}=\frac{T_{\mathrm{D}}}{T_{M}}>0\), which is within the low-frequency range of the variable field \(\alpha_{2}=\omega T_{\mathrm{D}}<<\)1, and when expressions (121) and (122) are at the frequency of the field tending toward zero, whether the conditions \(\alpha_{2}\to 0\), \(\alpha_{1}=\frac{T_{\mathrm{D}}}{T_{\mathrm{M}}}>0\), \(\Delta_{1}=0\), and \(\Delta_{2}=\frac{\pi}{2}\sqrt{\alpha_{1}}\)can be assumed is an important question for theory. From there, entering the variable \(\zeta=\frac{\pi}{2}\sqrt{\alpha_{1}}\) and going from (121) and (122) to limits \(\Gamma_{11}^{(\omega=0)}(\alpha_{1};0)=\lim\limits_{\alpha_{2}\to 0}\left( \Gamma_{11}^{(\omega)}(\alpha_{1};\alpha_{2})\right)\), \(\Gamma_{12}^{(\omega=0)}(\alpha_{1};0)=\lim\limits_{\alpha_{2}\to 0}\left( \Gamma_{12}^{(\omega)}(\alpha_{1};\alpha_{2})\right)\), we write the following approximate expressions:  
\[\Gamma_{11}^{(\omega=0)}(\alpha_{1};0)=\sum_{n=1}^{\infty}\left[\frac{(1-(-1)^ {n})}{\left(n^{2}+\alpha_{1}\right)^{2}}\right]=\frac{\pi^{4}}{32}\times\frac{ \text{th}\zeta-\zeta\times\left(1-\text{th}^{2}\zeta\right)}{\zeta^{3}}, \tag{123}\]  
\[\Gamma_{12}^{(\omega=0)}(\alpha_{1};0)=\sum_{n=1}^{\infty}\left[\frac{(1-(-1)^ {n})}{n^{2}(n^{2}+\alpha_{1})^{2}}\right]=\frac{\pi^{6}}{128}\times\left(\frac{ 3(\zeta-\text{th}\zeta)-\zeta\text{th}^{2}\zeta}{\zeta^{5}}\right). \tag{124}\]  
Substituting (123) and (124) into (120) and assuming \(\alpha_{2}\to 0\) yields the following:  
\[ \begin{split}&\Gamma_{1}^{(\omega=0)}(\alpha_{1};0)=\frac{4 \alpha_{1}}{\pi^{2}}\left(\Gamma_{11}^{(\omega=0)}(\alpha_{1};0)+\alpha_{1} \Gamma_{11}^{(\omega=0)}(\alpha_{1};0)\right)=\frac{1}{2\zeta}\times\\ &\times\left(\text{th}\zeta-\zeta\times\left(1-\text{th}^{2} \zeta\right)+3(\zeta-\text{th}\zeta)-\zeta\text{th}^{2}\zeta\right)=1-\frac{ \text{th}\zeta}{\zeta},\end{split}  \tag{125}\]  
\[\Gamma_{2}^{(\omega=0)}(\alpha_{1};0)=\frac{4\alpha_{1}\alpha_{2}}{\pi^{2}} \Gamma_{12}^{(\omega=0)}(\alpha_{1};0)=\frac{\pi^{4}\alpha_{2}}{8}\times \left(\frac{3(\zeta-\text{th}\zeta)-\zeta\text{th}^{2}\zeta}{\zeta^{3}}\right) =0. \tag{126}\]  
A similar result is obtained when \(\alpha_{2}\to 0\) in (119a) and (119b):  
\[ \begin{split}&\Gamma_{1}^{(\omega=0)}(\alpha_{1};0)=\frac{4 \alpha_{1}}{\pi^{2}}\sum\limits_{n=1}^{\infty}\left[\frac{(1-(-1)^{n})}{n^{2}( n^{2}+\alpha_{1})}\right]=\\ &=\frac{4\alpha_{1}}{\pi^{2}}\times\frac{1}{\alpha_{1}}\sum \limits_{n=1}^{\infty}\left((1-(-1)^{n})\right)\left[\frac{1}{n^{2}}-\frac{1}{ n^{2}+\alpha_{1}}\right]=\\ &=1-\frac{\text{th}\left(\frac{3}{2}\sqrt{\alpha_{1}}\right)}{ \frac{\pi}{2}\sqrt{\alpha_{1}}}.\end{split} \]  
The number series  
\[\sum\limits_{n=1}^{\infty}\left[\frac{(1-(-1)^{n})}{n^{2}(n^{2}+\alpha_{1})} \right]=\frac{\pi^{2}}{4}\times\frac{1}{\alpha_{1}}\left(1-\frac{\text{th} \left(\frac{3}{2}\sqrt{\alpha_{1}}\right)}{\frac{\pi}{2}\sqrt{\alpha_{1}}}\right)\]  
represents the expression \(\Gamma_{11}^{(\omega=0)}(T)+\alpha_{1}\Gamma_{12}^{(\omega=0)}(T)\), and it is calculated by means of some special cases of \(S^{(0)}(\alpha_{1},0)=\sum_{n=1}^{\infty}\frac{1-(-1)^{n}}{n^{2}+\alpha_{1}}= \frac{\pi^{2}}{4}\times\frac{\text{th}(\Delta_{2})}{\Delta_{2}}\) from the more general equality expression  
\[S^{(\pm\omega)}(\alpha_{1},\alpha_{2})=\sum\limits_{n=1}^{\infty}\frac{1-(-1)^ {n}}{n^{2}+\alpha_{1}\pm\text{i}\alpha_{2}}=\frac{\pi^{2}}{4}\times\frac{ \text{tg}(\Delta_{1}\mp\text{i}\Delta_{2})}{\Delta_{1}\mp\text{i}\Delta_{2}}\]  
used previously in calculating functions (121) and (122):  
\[