# 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  
Traditional colorimetric temperature measurement selects two narrow-band wavelength ranges at two specific wavelengths and calculates the measured temperature using the ratio of radiative energies within these bands. However, when calculating the radiative energy within each band, the conventional approach approximates the definite integral as the area of a rectangle, as shown in Equation (6). This article avoids such approximations and calculates the radiative energies within the two narrow bands using Equation (5). It proves the strictly monotonic relationship between the ratio of radiative energies \(Q(T)\) and temperature \(T\) through derivation. The following is the proof of the strict monotonicity of \(Q(T)\) with respect to \(T\):  
According to the monotonicity principle, for any function \(y=f(x)\), if it is continuous on \([a,b]\) and differentiable on \((a,b)\), then if \(f^{\prime}(x)>0\) within \((a,b)\), then the function \(y=f(x)\) is strictly increasing on \([a,b]\). Similarly, if \(f^{\prime}(x)<0\) within \((a,b)\), the function \(y=f(x)\) is strictly decreasing on \([a,b]\). Therefore, demonstrating that the sign of \(Q^{\prime}(T)\) is unique establishes the strict monotonicity of \(Q(T)\) with respect to \(T\). Before thermopiles receive the radiative energy from the measured object, they are affected by the spectral absorption characteristics of absorbing media, such as air impurities, during the transmission process, resulting in a certain degree of energy attenuation. The attenuation function of the radiance emitted during transmission through a medium is denoted as \(K(\lambda,T,H)\), Where \(\lambda\) represents the wavelength, \(T\) represents the ambient temperature, and \(H\) represents the ambient humidity. In this study, the primary variables under discussion are wavelength and temperature, while humidity is considered constant. Considering the spectral absorption characteristics of the absorbing medium, and in conjunction with Equation (4) (assuming the object being measured is a grey body), the radiance emitted by the object is:  
\[M(\lambda,T)=\varepsilon C_{1}\lambda^{-5}e^{-C_{2}/\lambda T}K(\lambda,T)= \varepsilon C_{1}M_{0}(\lambda,T), \tag{10}\]  
in the equation, \(M_{0}(\lambda,T)=\lambda^{-5}e^{-C_{2}/\lambda T}K(\lambda,T)\). Substituting Equation (10) into Equation (5), we obtain the radiative energies \(M_{1}(T)\) and \(M_{2}(T)\) within the two narrow  
in the equation, \(\lambda_{a}=\lambda_{1}\), \(\lambda_{b}=\lambda_{1}+\Delta\lambda_{1}\), \(\lambda_{c}=\lambda_{2}\), \(\lambda_{d}=\lambda_{2}+\Delta\lambda_{2}\). Therefore, the ratio of radiative energies \(Q(T)\) is given by:  
\[Q(T)=\frac{M_{1}(T)}{M_{2}(T)}=\frac{\int_{\lambda_{a}}^{\lambda_{b}}M(\lambda, T)d\lambda}{\int_{\lambda_{c}}^{\lambda_{d}}M(\lambda,T)d\lambda}=\frac{\int_{ \lambda_{a}}^{\lambda_{b}}M_{0}(\lambda,T)d\lambda}{\int_{\lambda_{c}}^{ \lambda_{d}}M_{0}(\lambda,T)d\lambda}. \tag{12}\]  
Taking the logarithm of both sides of Equation (12), we obtain:  
\[\ln Q(T)=\ln\int_{\lambda_{a}}^{\lambda_{b}}M_{0}(\lambda,T)d\lambda-\ln\int_{ \lambda_{c}}^{\lambda_{d}}M_{0}(\lambda,T)d\lambda. \tag{13}\]  
Taking the derivative of both sides of Equation (13) with respect to \(T\), we obtain:  
\[