PACS: 78.20.Bh; 78.60.Kn

1. Introduction

The ^{1}^{,}^{2}, and thermoluminescence dosimetry (TLD)^{3}^{,}^{4}. The TI integral has been approximated by several methods, mainly by asymptotic series^{3}^{,}^{5} and continued fractions as the most sought-after^{6}^{,}^{7}. Likewise, a number of smart algebraic equations have also been proposed in order to approximate, analyze and compare the TI^{8}^{,}^{9}^{,}^{10}^{,}^{11}^{,}^{12} from the fact that *z*), it is possible to obtain algebraic equations that accurately estimate the ^{11}^{,}^{13}^{,}^{14}^{,}^{15}^{,}^{16}^{,}^{17}. In this paper, some approximations by continued fractions to function *a =* -1are presented. These function approximations involve one to six fitting parameters, and show to have relative errors from 0.15% to 0.00042% within the interval

2. Basic equations

The upper incomplete gamma function

for all *a* with Re(*z*) > 0, and Re(*a*) < 0 if Re(*z*) = 0. This integral is of great importance in several areas of mathematical-physics surging along a diversity of contexts and applications, such as the application of ^{1}.

Equation (1) can be written as

Where *h*(*a*, *z*) is defined as

The coupling of Eqs. (2) and (3) is carried out in a similar way by Chen and Liu^{16}. Here, the multiplier *z*
^{
R
} is introduced (crucial in our methodology), and *R* is a real number. Equation (3) can be transformed to

and by differentiating Eq. (4) once in *z*, we obtain

Repeating several times this procedure, we obtain the *n* + 1order derivative in the form

where *h*(*a*, *z*) is always lower than that for the function itself (for increasing *n*), we can approximate the left hand side of Eqs. (5) and (6) to zero. Thus, we obtain the solution of the *n* + 1-th equation which contains all the other lower order solutions, *i.e*., we obtain different approximations for *h*(*a*, *z*). From equation (5), and taking into account the first two equations from (6) we have

where the sub-index denotes the n-th order approximant. In Eqs. (7-9) as for all the system of equations (6), no restrictions are imposed to the value of *R* despite that in our case these values are just represented by continued fractions of the function *h*(*a*, *z*) (with different degree of approximation and complexity).

3. Numerical approximations to *z*

In this section we show a family of continued fractions that approximate the *h*(*a*, *z*) there are four subsets characterized by their simplicity of a continued fraction. Three of them are obtained for *R* = -1, 0, 1 (Tables I to III).

One representation by continued fractions of *R* = -1 in the form (we adopt the notation introduced in Ref. 18)

Similarly, the rational fractions for R = 0,1 and

The other one arises from the combination of *n* and *R* values, respectively. Substituting the continued fractions for *h*(*a*, *z*), obtained from the first five rational fractions with a set of pairs ^{18}

Expression (13) is the well-known representation by continued fractions of ^{5}. Equation (13) was applied to thermoluminescence dosimetry in Refs. ^{19} and ^{20}. All these arrangements of rational fractions by continued ones, Eqs. (10-13), were obtained from the “confrac" Maple function^{21}.

The relative truncation errors of Eq. (13) assuming a real *z*, is resumed in Table (V), where the relative truncation error is defined as

here *n* corresponds to the *n*-th approximant and *F* (*a*, *z*) is the full approximation to

The evaluation of the special function for the selected arguments is rounded to exact seven decimal digits and the truncation errors are upward rounded to 2 decimal digits. Equations (11-12) are the standard approximations to the gamma function and Eq. (10) becomes the least accurate approximation to the *e.g.* in the calculation of the thermoluminescence line shape^{22} where only the real values of *z* > 5 with accuracy of up to two or three significant figures of merit are needed. Therefore, continued fractions Eqs. (10-13) can be employed with up to five approximants. Another criterion used to compare several approximations for

This criterion will be used in the following, as well as the assumption of a real *z*.

**4. Two simple approximations for a real z
**

In order to simplify the first three approximants of continued fraction, Eq. (13), we account for the fact that the polynomial order in the numerator is less that one as compared to the order in the denominator. In this sense, we look for a ratio of polynomials of order (*n*-1)/*n* in the form

or

where *A*, *B* and *C* are the parameters to be fitted. The rational approximations on the right hand side of Eqs. (14-15) have been taken as test functions in the least-squares fitting, looking for the minimal number of coefficients to be fitted. The least-squares fitting using the NLREG code^{23}, gives the parameters *A*, *B* and *C* accounting for the set of values

Now, we obtain better approximations as compared to Eqs. (14) and (15). Equation (9) with *n* = 2, as a continued fraction takes the form,

Introducing the quantity

Expanding the denominator in Eqn. (17) into a binomial series and conserving only the first two terms, we have

Equation (18) is valid for

Finally, turning back the value of

In Eq. (19), the constant number

where *A*, *B*, *C* and *a* and *z* (summarized in Table VI). In case of the Eq. (25), we obtain A = 1.7256, B = 1.3219, C = 0.29606, D = -3.0279, E = 3.6887, F = 0.51297 and

The fitting is carried out, as previously, comparing the left hand side of Eqs. (21-25) for several values of *a* and *z* with test functions as on the right hand side. From the fitted results, we observe that the more complicated the test functions the approximations result better.

**5. Comparison with other published approximations with a real z
**

The generalized temperature integral is defined as^{24},

There are several algebraic expressions that approximate Eq. (26) using the *m* = -*a* - 1 in (1), and obtain the relation

Six reported approximations for the ^{13}^{,}^{14}^{,}^{16}^{,}^{17}^{,}^{25}

Where *A, B, C, D, E* and *F* in Eqs. (28-32) are the fitted parameters, summarized in Table VII.

From the comparison of the values for maximal relative errors from the proposed (Table VI) and reported approximations (Table VII), we note that for those proposed ones the error becomes smaller.

6. Case of the temperature integral for Arrhenius equation (*a* = -1, *m* = 0)

An important particular case, for the *a* = -1 in the fourth approximant (*n* = 4) of continued fraction Eq. (13),

This equation represents the Senum and Yang approximation^{8} used as a test function by several authors^{9}^{,}^{12}. The maximal relative errors for the cases of Eqs. (24), (25), (34) (*n* = 4) and (13) (*n* = 10) are *a* = -1 from Eq. (15), written in the form

where *A* = 1.132, *B* = -0.620 and *C* = 3.124. Considering these values as the initial ones in a least-squares fitting to synthetic generated data from the left hand side of Eq. (35) on a set of points *A*, *B* and *C* summarized in Table VIII. In order to reduce the maximal relative error of Eq. (24) and Eq. (25), in a one-dimensional least-squares fitting, we propose the next test expression

to determine the parameters *A*, *B*, *C*, *D* and *E* (Table VIII). This test equation is compared to that equation proposed in Ref. 12, in the form

where the values for their corresponding parameters are summarized also in Table VIII. Comparing the maximal relative errors of Eq. (36) and Eq. (37), we observe that approximation Eq. (36) containing only five fitting parameters, not six as in Eq. (37), reduces the error in two orders of magnitude.

We can observe that it is a compromise between the minimal error reached and the simplicity of the approximation. Thus, expressions systematically obtained from the continued fractions method, show that these two properties could be rather simple or become complicated as needed for numerical calculations.

7. Conclusions

New modifications that approximate the incomplete gamma function *z*, a required condition in experimental data analysis. The expressions obtained for the TI could be incorporated in glow deconvolution line treated by different methods such as in the project GLOCANIN^{26} and the TGCD package^{27}. Thus, expressions systematically obtained from the continued fractions method, show an equilibrium between the minimal error reached and the simplicity of approximations, and could be simple or complicated as needed for numerical calculations.