# A Thermopile-Based Colorimetric Temperature Measurement Method for Arbitrary Bandwidth
## 3 Mathematical Proof of Temperature Measurement Methods
### Relationship Between Ratio of Radiative Energies Within Two Bands and Temperature
\begin{split}\frac{d\ln Q(T)}{dT}&=\frac{d\Big{(} \int_{\lambda_{a}}^{\lambda_{b}}M_{0}(\lambda,T)d\lambda\Big{)}}{\frac{dT}{K_{1 }(T)}}-\frac{d\Big{(}\int_{\lambda_{c}}^{\lambda_{d}}M_{0}(\lambda,T)d\lambda \Big{)}}{K_{2}(T)}\\ &=\frac{\int_{\lambda_{a}}^{\lambda_{b}}M_{0}(\lambda,T)C_{2} \lambda^{-1}T^{-2}d\lambda}{K_{1}(T)}-\frac{\int_{\lambda_{c}}^{\lambda_{d}}M _{0}(\lambda,T)C_{2}\lambda^{-1}T^{-2}d\lambda}{K_{2}(T)}\\ &=\frac{K_{2}(T)\int_{\lambda_{a}}^{\lambda_{b}}M_{0}(\lambda,T)C _{2}\lambda^{-1}T^{-2}d\lambda-K_{1}(T)\int_{\lambda_{c}}^{\lambda_{d}}M_{0}( \lambda,T)C_{2}\lambda^{-1}T^{-2}d\lambda}{K_{1}(T)K_{2}(T)}\\ &=\frac{D(T)}{K_{1}(T)K_{2}(T)},\end{split}  \tag{14}\]  
in the equation,  
\[ \begin{cases}K_{1}(T)=\int_{\lambda_{a}}^{\lambda_{b}}M_{0}(\lambda,T)d\lambda \\ K_{2}(T)=\int_{\lambda_{c}}^{\lambda_{d}}M_{0}(\lambda,T)d\lambda^{\prime} \end{cases}  \tag{15}\]  
clearly \(K_{1}(T)>0\), \(K_{2}(T)>0\), \(K_{1}(T)K_{2}(T)>0\), therefore, according to Equation (14), the sign of \(\frac{d\ln Q(T)}{dT}\) is determined solely by \(D(T)\). Then discuss \(D(T)\) from Equation (14) individually:  
\[ \begin{array}{ll}D(T)&=\int_{\lambda_{c}}^{\lambda_{d}}M_{0}(\lambda_{1},T) d\lambda_{1}\int_{\lambda_{a}}^{\lambda_{b}}M_{0}(\lambda_{2},T)C_{2}\lambda_{2} ^{-1}T^{-2}d\lambda_{2}\\ &-\int_{\lambda_{a}}^{\lambda_{b}}M_{0}(\lambda_{2},T)d\lambda_{2}\int_{ \lambda_{c}}^{\lambda_{d}}M_{0}(\lambda_{1},T)C_{2}\lambda_{1}^{-1}T^{-2}d \lambda_{1}\\ &=\iint\limits_{D}M_{0}(\lambda_{1},T)M_{0}(\lambda_{2},T)C_{2}\lambda_{2} ^{-1}T^{-2}d\lambda_{1}d\lambda_{2}\\ &-\iint\limits_{D}M_{0}(\lambda_{1},T)M_{0}(\lambda_{2},T)C_{2}\lambda_{1}^{- 1}T^{-2}d\lambda_{1}d\lambda_{2}\\ &=\iint\limits_{D}C_{2}T^{-2}\Big{(}\lambda_{2}^{-1}-\lambda_{1}^{-1}\Big{)}M _{0}(\lambda_{1},T)M_{0}(\lambda_{2},T)d\lambda_{1}d\lambda_{2},\end{array}  \tag{16}\]  
in the equation, the integration range \(D=\{(\lambda_{1},\lambda_{2})|\lambda_{c}\leq\lambda_{1}\leq\lambda_{d}, \lambda_{a}\leq\lambda_{2}\leq\lambda_{b}\}\), because \(C_{2}T^{-2}M_{0}(\lambda_{1},T)M_{0}(\lambda_{2},T)>0\), the sign of \(D(T)\) is determined solely by \(\lambda_{2}{}^{-1}-\lambda_{1}{}^{-1}\). When \(\lambda_{a}\leq\lambda_{2}\leq\lambda_{b}<\lambda_{c}\leq\lambda_{1}\leq\lambda _{d}\) is satisfied, \(\lambda_{2}{}^{-1}-\lambda_{1}{}^{-1}>0\). When \(\lambda_{c}\leq\lambda_{1}\leq\lambda_{d}<\lambda_{a}\leq\lambda_{2}\leq\lambda _{b}\) is satisfied, \(\lambda_{2}{}^{-1}-\lambda_{1}{}^{-1}<0\). Therefore, when the wavelength ranges \([\lambda_{a},\lambda_{b}]\) and \([\lambda_{c},\lambda_{d}]\) do not overlap, the sign of \(\lambda_{2}{}^{-1}-\lambda_{1}{}^{-1}\) is uniquely determined. Consequently, the sign of \(D(T)\) is uniquely determined, implying that the sign of \(\frac{d\ln Q(T)}{dT}\) is also uniquely determined. \[Q^{\prime}(T)=\frac{dQ(T)}{dT}=Q(T)\frac{d\ln Q(T)}{dT}, \tag{17}\]  
since \(Q(T)>0\), combined with Equation (17), it is known that when the wavelength ranges of the narrow bands \([\lambda_{a},\ \lambda_{b}]\) and \([\lambda_{c},\ \lambda_{d}]\) do not overlap, specifically when \([\lambda_{1},\ \lambda_{1}+\Delta\lambda_{1}]\) and \([\lambda_{2},\ \lambda_{2}+\Delta\lambda_{2}]\) do not overlap, the sign of \(Q^{\prime}(T)\) is uniquely determined, indicating that \(Q(T)\) is strictly monotonic with respect to \(T\). Based on this conclusion, we can further extend it to two arbitrarily wide bands. Regardless of their specific widths, as long as their wavelength ranges do not overlap, the ratio of radiation energies within these bands and the measured temperature exhibit a strictly monotonic relationship.