## 5 Effect of Temperature on Theoretical Spectra of Complex Dielectric Permittivity
The curves \(\text{tg}\delta\Big{(}\frac{T_{\text{D}}}{T_{\text{M}}};\omega T_{\text{M}} \Big{)}\) in the area of large values \(\alpha_{2}\) are nearly congruent. In conclusion, it should be pointed out that the anomalously high dielectric permittivities \(\varepsilon=5\) million found experimentally in [5], in samples of corundum-zirconium ceramics (CZC), with an alternating field frequency of 1 kHz, at the point T = 1250 K, can be theoretically explained and further investigated at a higher analytical level using expressions (64a), (64b), (65) that indicate the correspondence of the formulated theoretical methodology to the experimental regularities that manifest themselves in high-temperature ionic superconductors near the second-order phase transition temperature (quasi-ferroelectric effect), which is topically for the design of the theoretical methods for forecasting the nonlinear electrophysical properties of hydrogen-bond ferroelectrics (KDP, DKDP) used in laser technology as regulators of electromagnetic radiation parameters and electric gates [1; 2; 3; 4]. In this regard, promising are the designs of high-speed nonvolatile memory devices based on thin films of ferroelectrics with a rectangular hysteresis loop (RHL), characterized by anomalously long times remanent polarization relaxation, increased mechanical performance and thermal stability. For the practical application of the calculation formulas (64a), (64b), (65), it is convenient to express them in terms of the static permittivity \(\varepsilon_{S}\) and permeability at high frequencies \(\varepsilon_{\infty}\). In order to obtain such phenomenological relations, we will use the expression obtained from (61) in the Maxwellian relaxation region, when \(\alpha_{1}=\frac{T_{\rm D}}{T_{\rm M}}>>1\) and, at \(\zeta=\frac{\pi}{2}\sqrt{\alpha_{1}}>>1\), by virtue of \(\zeta ch\zeta\approx\zeta\), we have \(\varepsilon_{S}\approx\varepsilon_{\infty}\frac{\pi}{2}\sqrt{\alpha_{1}}\), where \(\frac{2\,\varepsilon_{S}}{\varepsilon_{\infty}}=\pi\sqrt{\frac{T_{\rm D}}{T_{ \rm M}}}\). Then, based on (64a), (64b), (65) we have  
\[ \begin{array}{c}\left(\hat{\varepsilon}^{(\omega)}\left(\frac{T_{\rm D}}{T_ {\rm M}};\omega T_{\rm M}\right)\right)^{/}\approx\varepsilon_{s}\frac{1+ \frac{\varepsilon_{\infty}}{\varepsilon_{\infty}}(\omega T_{\rm M})^{2}}{1+ \frac{\left(\varepsilon_{\infty}\right)}{\left(\varepsilon_{\infty}\right)} ^{2}(\omega T_{\rm M})^{2}},\\ \left(\hat{\varepsilon}^{(\omega)}\left(\frac{T_{\rm D}}{T_{\rm M}};\omega T_{ \rm M}\right)\right)^{/}\approx\varepsilon_{s}\frac{\frac{\varepsilon_{s}}{ \varepsilon_{\infty}}(\omega T_{\rm M})}{1+\frac{\left(\varepsilon_{\infty} \right)}{\left(\varepsilon_{\infty}\right)}^{2}(\omega T_{\rm M})^{2}},\\ \text{tgg}^{(\omega)}\left(\frac{T_{\rm D}}{T_{\rm M}};\omega T_{\rm M}\right) \approx\frac{\varepsilon_{\infty}\omega T_{\rm M}}{1+\frac{\left(\varepsilon _{\infty}\right)}{\varepsilon_{\infty}}(\omega T_{\rm M})^{2}}.\end{array}  \tag{66}\]  
Additional terms  
\[\left(\omega T_{M}\right)_{|\max(\hat{\varepsilon}^{(\omega)})^{//}|}\approx \frac{2}{\pi}\sqrt{\frac{T_{\rm M}}{T_{D}}}=\frac{\varepsilon_{\infty}}{ \varepsilon_{s}},(\omega T_{M})_{|\max\{\text{tg}^{(\omega)}|\xi|\}}\approx \sqrt{\frac{2}{\pi}\sqrt{\frac{T_{\rm M}}{T_{\rm D}}}}=\sqrt{\frac{ \varepsilon_{\infty}}{\varepsilon_{s}}}. \tag{67}\]  
In formal form, the dispersion relations (66) resemble the classical expressions for the Debye dispersion [3], but differ from them in coefficients, which leads to other expressions for determining the maxima of the functions \(\left(\hat{\varepsilon}^{(\omega)}\left(\frac{T_{\rm D}}{T_{\rm M}};\omega T_ {\rm M}\right)\right)^{//}\) n \(\text{tg}^{(\omega)}\left[\delta\left(\frac{T_{\rm D}}{T_{\rm M}};\omega T_ {\rm M}\right)\right]\), as well as their values.