Rational approximations of the Arrhenius and general temperature integrals, expansion of the incomplete gamma function

Abstract When analyzing materials non-isothermally using the Arrhenius equation under linear heating, a temperature integration is necessary. While the frequency factor in this equation is typically assumed to be constant, it can actually vary with temperature for certain solid-state reactions. The resulting temperature integral, known as the Arrhenius or general temperature integrals, usually have no analytical solutions. Therefore, special functions and approximation functions are often used to estimate them. In this particular study, new rational approximations for the Arrhenius and general temperature integrals were derived through the expansion of the incomplete gamma function. Two sets of these rational approximations, which exhibit excellent accuracy, are presented. One set of approximations for the Arrhenius integral matches the widely used Senum and Yang’s approximations, while the other set, which offers even greater accuracy, has not been previously reported. An obtained rational approximation has been utilized to simulate the thermal degradation of a commercially available PMMA, illustrating a practical application example.


Introduction
Thermal analysis is a valuable technique used to examine and predict the thermal properties of materials.This analysis can be conducted under either isothermal or non-isothermal conditions.Numerous reactions take place under non-isothermal conditions, making non-isothermal analysis crucial for comprehending the impact of temperature variations on a system's behavior.This provides essential insights into a broad spectrum of thermal and chemical processes (Xin et al. 2022;Zhang, Wang, et al. 2023;Zhang, Zheng, et al. 2023;Sun et al. 2023;Sheng et al. 2023).The rate of reaction is commonly expressed using the Arrhenius equation.The pre-exponential factor in the Arrhenius equation, which measures the frequency of reactions, is generally assumed to remain constant and unaffected by temperature.This assumption leads to the simplified form of the Arrhenius equation.However, for certain solid-state reactions, the pre-exponential factor may vary with temperature due to factors like changes in diffusion rates within bulk phases or confinement of reactants in porous materials.Understanding the temperature dependence of this factor is crucial for accurately modeling thermally activated reactions, enabling the optimization of industrial processes and the design of effective materials.
When the heating rate is linear in non-isothermal conditions, the Arrhenius integral is derived assuming a constant pre-exponential factor.However, if the pre-exponential factor of reactions varies with temperature according to a power law relationship (Criado et al. 2005), a generalized temperature integral is obtained instead (Cai and Liu 2007).Both integrals do not have analytical solutions and necessitate approximation functions to estimate the integration.Implicitly assuming the constancy of activation energy, the approximation functions are typically utilized in model-fitting methods.These methods find application in various processes, such as crystallization and melting of polymers, solid-solid transitions, crystallization from solution, glass transitions, gelation and gel melting, denaturation of proteins, polymerization, and crosslinking of polymers, as well as thermal and thermo-oxidative degradation (Vyazovkin 2017(Vyazovkin , 2018)).Different approximations with varying accuracies have been proposed for estimating the general temperature integral: Gorbachev (1976) suggested a first-degree rational function, Wanjun et al. (2005) and Wanjun and Donghua (2009) obtained multiple approximations using an integration by parts approach, Cai andLiu (2007, 2008) and Cai et al. (2007) proposed several rational or semi-empirical functions, Chen and Liu (2007, 2009a, 2009b) derived a series of rational functions with different degrees or semi-empirical functions, Capela et al. (2009) recommended an approximation based on the Gaussian quadrature procedure, Xia and Liu (2018) obtained several Pad� e type functions using the Stieltjes integral, Casal and Marb� an (2020) suggested a first-degree rational function, Lei et al. (2020) proposed a function utilizing the asymptotic expansion of the exponential integral function, and Aghili et al. (2021) obtained rational functions based on the minimization of the maximal deviation of bivariate functions.The previous study (Aghili 2021) showcased the precise representation and evaluation of the general temperature integral using various special functions such as the exponential integral function, incomplete gamma function, confluent hypergeometric function, Whittaker function, and generalized hypergeometric function.It is worth noting that employing dedicated software tools like Matlab, Mathematica, Maple, and Gnu Octave is necessary for implementing the special function models.However, it is essential to recognize that using these methods might come with certain restrictions, rendering approximation functions a preferable and more practical option, especially when applying iterative techniques for computing the temperature integral.This is due to the fact that approximate calculations can be executed faster than those obtained through numerical integration or special function models.In the mentioned study (Aghili 2021), only a few examples of rational approximations with a degree of 4 were provided.
In the latest study (Aghili, forthcoming) rational approximations of the Arrhenius and general temperature integrals with any arbitrary degree were systematically obtained based on expansion of the confluent hypergeometric function.
This study systematically derives rational approximations of the Arrhenius and general temperature integrals of any arbitrary degree by expanding the incomplete gamma function.

Theory
The rate of reaction, da=dt, depends on the temperature and the type of reaction, as follows: where kðTÞ and f ðaÞ are the rate constant and a function of conversion, respectively.The rate constant can be expressed by the modified Arrhenius equation (Laidler 1996) as: where the term A T m ð Þ is temperature dependent pre-exponential factor, A and m are constants, E is activation energy, R is universal gas constant, and T is absolute temperature.The simple form of the Arrhenius equation can be obtained by setting m ¼ 0, but when it comes to solid-state reactions, the value of m typically falls between −1:5 and 2:5 (Criado et al. 2005); m ¼ 1 for thermal decomposition of a single reactive solid, m ¼ 0:5 for reactions between gases and the surface of a solid, values between 0 and 2:5 for desorption of gases from the surface of solids, and values between −1:5 and 0 for shrinkage processes depending on the sintering mechanism.It has also been demonstrated theoretically that m for solid decomposition reactions or congruent dissociative vaporization is −0:5 and 0:75, respectively, in vacuum and under an inert gas (L'vov 2007).If b is the constant heating rate, then dT ¼ b dt, and then: which, upon integration, becomes: where T 0 is the initial reaction temperature and I m T ð Þ is: The temperature integral in Equation ( 4) is known as the general temperature integral.When we set m ¼ 0, it transforms into the popular Arrhenius integral.The Arrhenius integral can be represented by the following function: If we let x ¼ E=RT, then: and and for the case of m ¼ 0, These integrals have no analytical solutions and their series expansions comprise an infinite number of terms, rendering them as non-elementary functions.Typically, the integrals are estimated using various elementary approximation functions that contain finite terms.Numerical methods hold paramount importance as they provide approximate solutions to integrals and differential equations that otherwise lack exact analytical solutions (Song et al. 2023;Zhou et al. 2023;Guo et al. 2023;Yin et al. 2023;Qi et al. 2023).
In the literature, g 0 ðxÞ has been approximated by several series, asymptotic, rational, or semiempirical functions.Excellent reviews of these functions have been provided by Flynn and Wall (1966), Lyon (1997), P� erez-Maqueda andCriado (2000), � Orfão (2007), and Deng et al. (2009).The function g m x ð Þ has also been estimated with different series, rational or semi-empirical functions with different accuracies.The introduction provides a listing of reviews for these approximations.The earlier work, Aghili (2021) showcased the precise representation and evaluation of the general temperature integral using various special functions.The special functions play a critical role in scientific research, providing advanced analytical tools essential for solving complex differential equations and modeling phenomena in multiple domains of physics and engineering (Zhou et al. 2014(Zhou et al. , 2018;;Rashid et al. 2023;Mianowski et al. 2023;Vanitha et al. 2023).
The function g m ðxÞ can be written in terms of the incomplete gamma function (Aghili 2021), as follows: where C �, x ð Þ is the incomplete gamma function of x, defined as (Abramowitz and Stegun 1965): Evidently, � ¼ −m − 1 in Equation ( 10).The Pad� e approximation of the following function has been presented in Luke (1975), as follows: where n is the degree of the approximation function.E n �, x ð Þ and F n �, x ð Þ are rational functions that each has two forms.Therefore, two sets of rational approximations, denoted here by RA1 and RA2 series, will be obtained.These functions can be evaluated by the following recurrence formulae: for RA1 series: and for RA2 series: The zeroth and first degrees of the functions E n �, x ð Þ and F n �, x ð Þ have the following forms:for RA1 series: and for RA2 series: Consequently, it is possible to assess the higher degrees of these functions for both RA1 and RA2 series effortlessly, using Equations ( 13)-( 20).Additionally, there exists an inequality relationship (Luke 1975) for the approximations stated as follows: Consequently, the exact values for the Arrhenius or general temperature integrals are expected to lie within the range bounded by the RA1 and RA2 series.

Results and discussion
The rational approximations of x 1−� e x C �, x ð Þ were calculated using Equations ( 13)-(20).Then � was substituted by ð−m − 1Þ and consequently, the rational approximations of g m x ð Þ were obtained for both RA1 and RA2 series and several degrees of n, denoted here by p m, n x ð Þ: The results are listed in Tables 1-3.Table 1 shows the RA1 series of the rational approximations of the general temperature integral, p m, n x ð Þ, for n ¼ 1-6, while the same results for the RA2 series are listed in Table 2.The first degree rational approximate in RA1 series is equivalent to the approximation proposed by Gorbachev (1976).
As mentioned earlier, the Arrhenius integral which has been used extensively in the literature, is a specific case of the general temperature integral, with m ¼ 0: Thus, the rational approximations of the Arrhenius integral, p 0, n x ð Þ, are also listed in Table 3.
It can be observed that the RA1 series of the Arrhenius integral (Table 3) are exactly the same as P� erez-Maqueda and Criado (2000) or the popular Senum and Yang's approximations (1977).However, with the exception of the fourth degree in the previous study (Aghili 2021), the RA2 series (Table 3) has not been reported elsewhere.

Accuracy of new approximation functions
Comparing the new rational approximations with numerical integration will be done to assess their accuracy.The equation that defines the relative deviation is as follows: To evaluate the accuracy of the approximates, it is necessary to compute g m x ð Þ with high precision using numerical methods.In our past research (Aghili 2021), we detailed the evaluation of the function g m x ð Þ to a high degree of accuracy by employing Variable-Precision Arithmetic (VPA) within the Symbolic Math Toolbox TM of MATLAB, referred to as the NumVPA2 method.This approach was designed to achieve precision beyond the machine epsilon on a 64-bit computer system.We have applied the same methodology in this study to calculate g m x ð Þ with high precision in the aforementioned equation.The value of m in solid state reactions typically ranges from −1:5 to 2:5 (Criado et al. 2005).Starink (2003) demonstrated that the majority of reactions occur within the range of 15 < x < 60, while only the range of 9 < x < 100 is practically significant.However, there are some reactions that occur at x values as low as 5 (Starink 2018).Therefore, it would be sufficient to use the ranges of −1:5 < m < 2:5 and 5 < x < 100 for evaluating the accuracy of the approximations.
Figures 1 and 2 demonstrate the results of assessing the precision of the rational approximations for both the Arrhenius and general temperature integrals.These figures display logarithmic graphs of the absolute values of relative error, � j j, for the RA1 and RA2 series across various combinations of m and n values.The study investigates approximations up to the 6th degree and includes four representative values of m: It is worth noting that the figures also depict the approximations of the Arrhenius integral The findings indicate that higherdegree approximations exhibit less deviation, thus demonstrating higher accuracy.
The comparison between the series with the same degrees indicates that the RA2 functions exhibit less deviation compared to RA1 across a Accuracy of RA1 series of rational approximations in logarithmic scale.For comparison, � j j ¼ 0:1% is also shown with a dashed line.
wide range of x values.Figure 3 illustrates a contrast between p m, 4 x ð Þ for both the RA1 and RA2 series, considering different values of m: It is evident that the RA2 series demonstrates smaller deviations than the RA1 series.However, when it comes to x values near zero, which lack practical significance, the RA2 functions exhibit higher deviations than the RA1 series.Notably, the RA1 series consistently display negative deviations throughout all x values, whereas the RA2 series consistently exhibit positive deviations.This outcome supports Equation ( 21), as the function g m x ð Þ always surpasses the RA1 series and remains lower than the RA2 series.The function 12), has an asymptotic behavior, implying that as x approaches to infinity, the function approaches an asymptote that can be demonstrated through rational approximation functions.Consequently, the rational approximations presented in this work offer higher precision for large x values, while their accuracy diminishes for smaller values of x, particularly as x approaches zero.The relative error close to x ¼ 0 will only approach zero if n, the degree of the rational functions, tends toward infinity.
To have a perception of the accuracy of the rational approximations, the ranges of x associated with relative deviations lower than 0:1% and 0:01%, for typical values of m between −1:5 and 2:5, are listed in Tables S1-S4 in the Supplementary information.Except for the first and second degrees, the rational approximations demonstrate high levels of precision for a wide range of x: The versatility of the rational approximations obtained lies in their ability to be applied to any value of m, as both the RA1 and RA2 series can be used.It is worth noting, however, that higherdegree approximations (n) provide greater accuracy.For example, when m is an integer number less than −1, the general temperature integral has an analytical solution, and fortunately, the rational approximations can yield these analytical solutions.For example, if m ¼ −2 : and the functions p −2, n x ð Þ are the analytical solutions for n > 0 in RA1 and RA2 series.If m ¼ −3 : and the functions p −3, n x ð Þ are the analytical solutions for n > 1 in RA1 series as well as for n > 0 in RA2 series.If m ¼ −4 : and the functions p −4, n x ð Þ are the analytical solutions for n > 2 in RA1 series as well as for n > 1 in RA2 series and so on.In general, if m is an integer number less than −1, the functions p m, n x ð Þ are the analytical solutions for n > −ðm þ 2Þ in RA1 series as well as for n > −ðm þ 3Þ in RA2 series.
Not all values of m may be practically useful from a thermal analysis perspective.However, in mathematical terms, the rational approximations obtained in this study demonstrate exceptional accuracy for almost all values of m:

Application and comparison with other approximation functions
In thermal analysis, the Arrhenius integral is commonly employed to simulate material  kinetics.When seeking to ascertain the most suitable thermal behavior model for a substance with a one-step reaction and consistent activation energy (E), the equations below are utilized to compute the functions y a ð Þ and z a ð Þ: These calculated values are then compared against the reference graphs found in relevant literature.
Approximation functions are commonly employed to estimate the Arrhenius integral g 0 x ð Þ in this equation.Furthermore, to simulate reactions under linear heating conditions, it is necessary to calculate the function g 0 x ð Þ, which is commonly achieved through the use of rational approximation functions (Li, Gan, et al. 2023;Li, Li, et al. 2023;Mianowski et al. 2023;Ochieng et al. 2023).
Table 4 presents the maximum relative error, � j j max , and sum of squared error (SSE) for assessing the accuracy of recently suggested approximations of g 0 x ð Þ within the range of 5 < x < 100 with a step size of Dx ¼ 1: It is expected that higher-degree approximations would show better accuracy.Additionally, newly obtained approximations of g 0 x ð Þ were compared to highly accurate approximations proposed in the literature.The Arrhenius integral has been extensively reviewed in several studies, including Flynn and Wall (1966), Lyon (1997), P� erez-Maqueda andCriado (2000), � Orfão (2007), and Deng et al. (2009).Zhang, Wang, et al. (2023), Zhang, Zheng, et al. (2023) has discovered a numerical method that leverages multi-step processes and the 20th Taylor expansion to substantially refine the accuracy of the Arrhenius integral across different temperature intervals, minimizes error related to activation energy sensitivity, and elucidates the reasons for divergence at extreme values of x, as illustrated by 3D and contour visualizations.Among these studies, � Orfão ( 2007) considered the O model to be the most accurate, while Deng et al. (2009) found both the O and J models to be the most precise approximations to the Arrhenius integral.However, the previous study (Aghili 2021) revealed that the rational approximations obtained by expanding the exponential integral function, specifically EI4 and EI8, exhibited the highest level of accuracy among the approximations.Therefore, it is sufficient to compare the new approximations with only the O, J, EI4, and EI8 models.These approximations can be found in Table 5, and the coefficients of the EI4 and EI8 models are listed in Tables S5 and S6 in the Supplementary information.
The maximum relative error, � j j max , and SSE, obtained from evaluation of the approximates found in the literature, have been shown in Table 5 within the range of 5 < x < 100 and Dx ¼ 1: According to these results, while some of the newly suggested approximations are more accurate than J and O models, the EI4 and EI8 models still maintain their position as the most precise approximations.Figure S1 in the Supplementary information file illustrates a comparison between these approximations and the newly proposed approximations with 4th degree.
The literature has also employed the general temperature integral for thermal analysis of materials.For instance, Casal and Marb� an (2020) proposed a method called the Integral Method of Combined Kinetic Analysis of reactions (ICKA), which estimates f ðaÞ using a general relationship in the following manner: where Z, a, b, and c are parameters that depend on the kinetic model.Then the conversion (a) will be calculated by the following relationship: In this equation, g m ðxÞ is the general temperature integral which is estimated by the approximation functions.
The newly proposed approximations of g m ðxÞ have been evaluated for accuracy within the range of 5 < x < 100 and Dx ¼ 1: The evaluations were conducted for two distinct intervals: −1:5 < m < 2:5 and −5 < m < 5, with Dm ¼ 0:05: The results are presented in Table 6.It is anticipated that higher degree approximations would demonstrate greater accuracy.According to the results, the RA2 series is more accurate than the RA1 series.For further details, the Supplementary information includes visual representations in the form of 3D plots.These plots depict the relative deviations obtained from both RA1 and RA2 series.The plots encompass the intervals 5 < x < 95 and −5 < m < 5, and are labeled as Figures S2-S13.
Table 7 contains documented approximations of the function g m ðxÞ found in literature.It is important to note that the G model is essentially the same as the RA1 series with n ¼ 1: Furthermore, for the Ch1 model (Chen and Liu 2007) and the X model (Xia and Liu 2018), fourth-degree rational functions are utilized.In the X model, the coefficients of the numerator are explicitly reported only for certain values of m, which are indicated in Table S7 in the Supplementary information.The evaluation of the X model is based on these specific values.On the other hand, the CHG and GHG models (Aghili 2021) are rational approximations obtained by expanding the confluent hypergeometric function and generalized hypergeometric function, respectively.Lastly, the A model (Aghili et al. 2021) is presented in its expanded form and exhibits the following relationship: The coefficients of this model are listed in Table S8 in the Supplementary information.
Table 8 presents the maximum relative error, denoted as � j j max , and the sum of squared error, referred to as SSE, for the approximations of the function g m ðxÞ: The evaluation was conducted within the range of 5 < x < 100, with a step size of Dx ¼ 1: Two distinct intervals were considered for the parameter m : −1:5 < m < 2:5 and −5 < m < 5, with a step size of Dm ¼ 0:05: The results indicate that the A model is the most accurate approximation cited in the literature.Furthermore, certain recently proposed approximations, namely the RA2 series with n ¼ 5-6, exhibit comparable or even higher levels of precision than the A model Table 6.Accuracy of new approximates to the general temperature integral.
in specific cases.Supplementary information contains Figures S14-S29, which depict 3D plots illustrating the relative deviations obtained from these approximation functions.These plots cover the intervals 5 < x < 95 and −5 < m < 5: As a result, we have found that it is possible to expand the incomplete gamma function in order to obtain rational approximations with high precision for the Arrhenius and general temperature integrals of any desired degree.ffi ffi Xia and Liu (2018) X e −x x mþ2 � � x 4 þa 3 x 3 þa 2 x 2 þa 1 x x 4 þ16x 3 þ72x 2 þ96xþ24 � � � � ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi

Simulation of thermal degradation of PMMA
Utilizing the rational approximations detailed in this research, we explored the practical aspect by examining thermal degradation in a commercially available polymethylmethacrylate, PMMA (IH830 grade supplied by LG Chem Company), and simulated its kinetic behavior with the ICKA model.Thermal analytical experiments were conducted using a METTLER TOLEDO TGA/DSC 1 STARe System with a nitrogen atmosphere.The experiments employed four distinct heating rates, 2, 5, 10, and 20 � C=min, across a temperature range from 25 � C to 600 � C: The degree of conversion, a, is defined as the ratio of the actual mass loss to the total mass loss, where w is the actual mass at time t (or at temperature T), w 0 is the initial mass, and w 1 is the mass at the end of experiment.Figure 4(a) presents the thermogravimetric (TG) profiles acquired from experimental data, and Figure 4(b) displays the variation of conversion with reaction temperature across various heating rates.The experimental conversion values, ranging from 0:01 to 0:99 in increments of 0:01, were analyzed using the ICKA method as outlined in Equation (29).In the original reference (Casal and Marb� an 2020), the suggestion is made to employ the kinetic parameters derived from model-free approaches as preliminary estimations for the ICKA model.Nevertheless, we implemented an adapted technique that is based on the W2 model, which was originally proposed by Wanjun and Donghua (2009), as: where Þ and the activation energy can be obtained from the slope.The intercept of this straight line is , and from the intercept, one can evaluate ln ZA ð Þ : The thermal degradation of PMMA results in the formation of small gas molecules, including the MMA monomer, which then evaporate, causing a reduction in sample mass.Assuming that the degradation of PMMA adheres to the model of congruent dissociative vaporization, it is appropriate to use m ¼ 0:75 as the starting point, as this was theoretically derived for this particular mechanism operating under an inert atmosphere (L'vov 2007).With m set to 0:75, a trial-anderror method was utilized to fine-tune the values of a, b, and c such that the plotting of ln bF a ð Þ=T B m � � against 1=T ð Þ, as per Equation (31), yields a straight line with the least sum of the squares of errors in linear regression.The results are: ln ðZAÞ ¼ 18:5646, where AT m ð Þ is in ð1=sÞ, E ¼ 241:30 ðkJ=molÞ, a ¼ 0:6456, b ¼ 3:7563 � 10 −11 , and c ¼ 1:7376: Figure S30 provides in the Supplementary information illustrates the linear graph corresponding to Equation (31).Subsequently, these determined values, along with the preset m value of 0:75, were employed as the initial estimations for Equation (29).In this equation, the function g m ðxÞ was calculated by applying the RA2 with n ¼ 6: A further trialand-error process was undertaken to identify the optimal kinetic parameter values that would minimize the SSEs in the nonlinear regression analysis based on Equation (29).The results, including their 95% confidence intervals, were derived as follows: ln ðZAÞ ¼ 18:552560:2606, m ¼ 0:748860:01, E ¼ 241:2861:12 ðkJ=molÞ,   31), a fixed value for m is necessary; and if the minimization of the SSEs in linear regression is applied to derive the optimal value for m, in the same way as for parameters a, b, and c, the result would be a value of approximately 17 for m: However, this result is not physically interpretable.Indeed, the determinant of the least-squares matrix, which this calculation is based on, is likely influenced by random experimental data errors, making such a value unrealistic.Thus, the most accurate value for m was derived from ICKA model, Equation (29).

Conclusion
The non-isothermal analysis with linear heating with the application of the Arrhenius equation involves calculating a temperature integral.When the frequency factor in the Arrhenius equation is constant, the resulting temperature integral is commonly referred to as the Arrhenius integral.However, if the frequency factor varies with temperature, it is called the general temperature integral.Unfortunately, both the Arrhenius and general temperature integrals do not have analytical solutions in their general forms and must be represented by special functions.As a result, researchers often rely on approximation functions to estimate these integrals.In this article, new rational approximations for the Arrhenius and general temperature integrals were derived using the expansion of the incomplete gamma function.The article presents two series of approximations: RA1 and RA2 series.The RA1 series for the Arrhenius integral matches the well-known Senum and Yang's approximations.On the other hand, the RA2 series offers even higher accuracy and has not been previously published.Within these specified ranges of 5 < x < 100 and −1:5 < m < 2:5, the RA1 and RA2 series with n > 3 stand out as the preferred approximations if an accuracy within 0:01% is desired for the general temperature integral.Nevertheless, the A model surpasses them in accuracy across the entire ranges of x and m values.The RA1 and RA2 series with n > 3 become more precise than the A model specifically for x values above approximately 25, which encompasses the ranges where most chemical reactions occur.Moreover, when applying the simplified Arrhenius equation, the EI8 model is the most accurate approximation cited in the literature.An example of practical application involves the simulation of the thermal degradation of a commercial grade PMMA, which was conducted by utilizing a derived rational approximation.

Figure 2 .
Figure 2. Accuracy of RA2 series of rational approximations in logarithmic scale.For comparison, � j j ¼ 0:1% is also shown with a dashed line.

Figure 3 .
Figure 3.Comparison of p m, 4 x ð Þ for RA1 and RA2 series with different values of m:

a
¼ 0:642060:0108, b ¼ ð3:555960:0808Þ � 10 −11 , and c ¼ 1:737460:0153: The simulation results were compared with the experimental data in Figure 4(b), showing a strong concordance between the model and the empirical findings.It should be noted that for Equation (

Figure 4 .
Figure 4. a) TG Curves for thermal degradation of PMMA under different heating rates; b) comparison of conversions obtained from experiments and simulation.

Table 1 .
The rational approximations of the general temperature integral, RA1 series.

Table 2 .
The rational approximations of the general temperature integral, RA2 series.

Table 3 .
The rational approximations of the Arrhenius integral, p 0, n x ð Þ:

Table 4 .
Accuracy of new approximates to the Arrhenius integral ðm ¼ 0Þ:

Table 5 .
The most accurate approximates to the Arrhenius integral in the literature and their accuracy.

Table 7 .
Approximation functions for the general temperature integral in the literature.

Table 8 .
Accuracy of approximates to the general temperature integral in the literature.