Using the standard series expansion of exponential function gives

$\begin{array}{c}{I}_0={\displaystyle {\int }_0^1}{\text{e}}^{-t^p}\text{d}t={\displaystyle {\int }_0^1}\left(1-\frac{t^p}{1!}+\frac{t^{2p}}{2!}-\frac{t^{3p}}{3!}+\frac{t^{4p}}{4!}-\frac{t^{5p}}{5!}+\cdots \right)\text{d}t\\ ={t-\frac{t^{p+1}}{\left(p+1\right)\cdot 1!}+\frac{t^{2p+1}}{\left(2p+1\right)\cdot 2!}-\frac{t^{3p+1}}{\left(3p+1\right)\cdot 3!}+\cdots |}_0^1\\ =1-\frac{1}{\left(p+1\right)\cdot 1!}+\frac{1}{\left(2p+1\right)\cdot 2!}-\frac{1}{\left(3p+1\right)\cdot 3!}+\cdots \end{array}$ (10)

which is absolutely convergent as long as the upper limit of integration is equal to or less than one, as is the case here. A quite similar expression corresponding to the error function, $p=2$, for an arbitrary upper limit can be found in ( [2] , p. 297).

In order to evaluate $I_1$ some preliminary arrangements and splitting of it into separate integrals are necessary. First, introduce a change of variable $t^p=v$ hence $\text{d}t=\text{d}v/\left[p{v}^{1-1/p}\right]$ and define $\beta =1-1/p$ for ease of notation so that

$I_1={\displaystyle {\int }_1^x}{\text{e}}^{-t^p}\text{d}t=\frac{1}{p}{\displaystyle {\int }_1^{x^p}}\frac{{\text{e}}^{-v}}{v^{\beta }}\text{d}v$ (11)

Note that $0<\beta <1$ since $p>1$ and for the special case of the error function $p=2$ hence $\beta =1/2$. Let $v=u+1$,

$I_1=\frac{1}{p}{\displaystyle {\int }_0^{x^p-1}}\frac{{\text{e}}^{-\left(1+u\right)}\text{d}u}{{\left(1+u\right)}^{\beta }}=\frac{1}{p\cdot \text{e}}\left[{\displaystyle {\int }_0^1}\frac{{\text{e}}^{-u}\text{d}u}{{\left(1+u\right)}^{\beta }}+{\displaystyle {\int }_1^{x^p-1}}\frac{{\text{e}}^{-u}\text{d}u}{{\left(1+u\right)}^{\beta }}\right]$ (12)

Define the first integral within the brackets as

$I_{10}={\displaystyle {\int }_0^1}\frac{{\text{e}}^{-u}\text{d}u}{{\left(1+u\right)}^{\beta }}$ (13)

Since $0\le u\le 1$, $I_{10}$ is going to be treated by employing series expansion (8a). The second integral within the brackets is split into two parts as

${\displaystyle {\int }_1^{x^p-1}}\frac{{\text{e}}^{-u}\text{d}u}{{\left(1+u\right)}^{\beta }}={\displaystyle {\int }_1^{x^p}}\frac{{\text{e}}^{-u}\text{d}u}{{\left(1+u\right)}^{\beta }}-{\displaystyle {\int }_{x^p-1}^{x^p}}\frac{{\text{e}}^{-u}\text{d}u}{{\left(1+u\right)}^{\beta }}=I_{11}-I_{12}$ (14)

where, by definition,

$I_{11}={\displaystyle {\int }_1^{x^p}}\frac{{\text{e}}^{-u}\text{d}u}{{\left(1+u\right)}^{\beta }},I_{12}={\displaystyle {\int }_{x^p-1}^{x^p}}\frac{{\text{e}}^{-u}\text{d}u}{{\left(1+u\right)}^{\beta }}$ (15)

$I_{12}$ is elaborated further with two successive changes of variables; first $u=y+x^p-1$ then $y=x^p\bar{u}$, and finally $\bar{u}$ is renamed as u so that

$I_{12}={\displaystyle {\int }_{x^p-1}^{x^p}}\frac{{\text{e}}^{-u}\text{d}u}{{\left(1+u\right)}^{\beta }}=x\cdot {\text{e}}^{1-x^p}{\displaystyle {\int }_0^{1/x^p}}\frac{{\text{e}}^{-x^{p}u}\text{d}u}{{\left(1+u\right)}^{\beta }}$ (16)

Considering the entire $I_1$,

$\begin{array}{c}\left(p\cdot \text{e}\right)I_1={\displaystyle {\int }_0^1}\frac{{\text{e}}^{-u}\text{d}u}{{\left(1+u\right)}^{\beta }}+{\displaystyle {\int }_1^{x^p}}\frac{{\text{e}}^{-u}\text{d}u}{{\left(1+u\right)}^{\beta }}-x\cdot {\text{e}}^{1-x^p}{\displaystyle {\int }_0^{1/x^p}}\frac{{\text{e}}^{-x^{p}u}\text{d}u}{{\left(1+u\right)}^{\beta }}\\ =I_{10}+I_{11}-I_{12}\end{array}$ (17)

Integrals $I_{10}$, $I_{11}$, and $I_{12}$ are now evaluated in succession by employing the improved series expansions defined in (8) for $1/\left(1+u\right)$.

In order to evaluate $I_{10}$ it is necessary to express $1/{\left(1+u\right)}^{\beta }$ as a power series and to do so we first employ the improved series expansion of $1/\left(1+u\right)$ as given in Equation (8a) for the range $0\le u\le 1$. Let $g\left(u\right)=1/\left(1+u\right)$ so that

$g\left(u\right)=\frac{1}{1+u}=1+\frac{1}{2^N}{\displaystyle \underset{m=1}{\overset{N}{\sum }}}\left[{\displaystyle \underset{k=m}{\overset{N}{\sum }}}\left(\begin{array}{c}N\\ k\end{array}\right)\right]{\left(-1\right)}^m{u}^m=C_0+{\displaystyle \underset{m=1}{\overset{N}{\sum }}}{C}_m{u}^m$ (18)

where, by definition,

$C_0=1,C_m=\frac{{\left(-1\right)}^m}{2^N}{\displaystyle \underset{k=m}{\overset{N}{\sum }}}\left(\begin{array}{c}N\\ k\end{array}\right)\text{for}m=\mathrm{1,2,}\cdots \mathrm{,}N$ (19)

We now establish $f\left(u\right)=g^{\beta }\left(u\right)=1/{\left(1+u\right)}^{\beta }={\left[C_0+{\displaystyle {\sum }_{m=1}^N}{C}_m{u}^m\right]}^{\beta }$ as a Maclaurin series truncated at a definite order N:

$f\left(u\right)=f\left(0\right)+f^{\prime }\left(0\right)u+\frac{{f^{\prime }}^{\prime }\left(0\right)}{2!}{u}^2+\frac{{{f^{\prime }}^{\prime }}^{\prime }\left(0\right)}{3!}{u}^3+\cdots +\frac{f^{\left(N\right)}\left(0\right)}{N!}{u}^N$ (20)

As a demonstration, taking $N=3$ gives

$\begin{array}{l}f=g^{\beta }=1,f^{\prime }=\beta g^{\beta -1}{g}^{\prime },{f^{\prime }}^{\prime }=\beta \left(\beta -1\right)g^{\beta -2}{g^{\prime }}^2+\beta g^{\beta -1}{g^{\prime }}^{\prime },\\ {{f^{\prime }}^{\prime }}^{\prime }=\beta \left(\beta -1\right)\left(\beta -2\right)g^{\beta -3}{g^{\prime }}^3+3\beta \left(\beta -1\right)g^{\beta -2}{g}^{\prime }{g^{\prime }}^{\prime }+\beta g^{\beta -1}{{g^{\prime }}^{\prime }}^{\prime }\end{array}$ (21)

in which the functions are to be evaluated at $u=0$. Use of $g\left(u\right)=C_0+{\displaystyle {\sum }_{m=1}^N}{C}_m{u}^m$ results in

$\begin{array}{l}f\left(0\right)={\hat{C}}_0=\mathrm{0\; <mo>\; !\; </mo>}\cdot C_0=1,f^{\prime }\left(0\right)=\beta {\hat{C}}_0{\hat{C}}_1,{f^{\prime }}^{\prime }\left(0\right)=\beta \left(\beta -1\right){\hat{C}}_0{\hat{C}}_1^2+\beta {\hat{C}}_0{\hat{C}}_2,\\ {{f^{\prime }}^{\prime }}^{\prime }\left(0\right)=\beta \left(\beta -1\right)\left(\beta -2\right){\hat{C}}_0{\hat{C}}_1^3+3\beta \left(\beta -1\right){\hat{C}}_0{\hat{C}}_1{\hat{C}}_2+\beta {\hat{C}}_0{\hat{C}}_3\end{array}$ (22)

where ${\hat{C}}_m=m\mathrm{<mo>\; !\; </mo>}\cdot C_m$ and $C_m$’s are as defined in Equation (19). $m\mathrm{<mo>\; !\; </mo>}$ multiplying $C_m$ arises from successive differentiations of the polynomial function $g\left(u\right)$ given in (18). A practical method of carrying out higher derivatives of $f=g^{\beta }$ evaluated at $u=0$ is developed in Appendix A. For ease of notation we define a set of new coefficients:

$\begin{array}{l}{D}_0=f\left(0\right)/\mathrm{0\; <mo>\; !\; </mo>}={\hat{C}}_0=1\\ D_1=f^{\prime }\left(0\right)/\mathrm{1\; <mo>\; !\; </mo>}=\beta {\hat{C}}_0{\hat{C}}_1/\mathrm{1\; <mo>\; !\; </mo>}\\ D_2={f^{\prime }}^{\prime }\left(0\right)/\mathrm{2\; <mo>\; !\; </mo>}=\left[\beta \left(\beta -1\right){\hat{C}}_0{\hat{C}}_1^2+\beta {\hat{C}}_0{\hat{C}}_2\right]/\mathrm{2\; <mo>\; !\; </mo>}\mathrm{,}\\ D_3={{f^{\prime }}^{\prime }}^{\prime }\left(0\right)/\mathrm{3\; <mo>\; !\; </mo>}=\left[\beta \left(\beta -1\right)\left(\beta -2\right){\hat{C}}_0{\hat{C}}_1^3+3\beta \left(\beta -1\right){\hat{C}}_0{\hat{C}}_1{\hat{C}}_2+\beta {\hat{C}}_0{\hat{C}}_3\right]/\mathrm{3\; <mo>\; !\; </mo>}\\ ⋮\\ D_N=f^{\left(N\right)}\left(0\right)/N!=\cdots \end{array}$ (23)

With the use of above coefficients in Equation (20) a power series for $1/{\left(1+u\right)}^{\beta }$ for the range $0\le u\le 1$ is obtained. A simple manipulation, described for Equations (7b) and (8b), gives the corresponding expression for the range $1\le u\le +\infty$. First, changing the variable by setting $u=1/\bar{u}$, then dividing both sides by ${\bar{u}}^{\beta }$ and finally renaming $\bar{u}$ as u again, renders the series valid for $1\le u\le +\infty$. Thus,

$\frac{1}{{\left(1+u\right)}^{\beta }}=D_0+D_{1}u+D_2{u}^2+D_3{u}^3+\cdots +D_N{u}^N\text{for}0\le u\le 1$ (24a)

$\frac{1}{{\left(1+u\right)}^{\beta }}=\frac{D_0}{u^{\beta }}+\frac{D_1}{u^{\beta +1}}+\frac{D_2}{u^{\beta +2}}+\frac{D_3}{u^{\beta +3}}+\cdots +\frac{D_N}{u^{\beta +N}}\text{for}1\le u\le +\infty$ (24b)

The accuracy of equations given in (24) to represent $1/{\left(1+u\right)}^{\beta }$ for various $\beta =1-1/p$ values may be seen in Figure 2 , where the exact expression $1/{\left(1+u\right)}^{\beta }$ and Equation (24) are depicted for $p=3/2,2,3$, and 10 which correspond to $\beta =1/3,1/2,2/3$, and 9/10, respectively. Figure 2 is drawn by the use of (24a) for $0\le u\le 1$ and (24b) for $1\le u\le 5$ and both series are truncated at $N=4$.

Equation (24a) is now used for the evaluation of $I_{10}$ for the range $0\le u\le 1$ while Equation (24b) is employed for the evaluation of $I_{11}$ in §3.2.2 for the range $1\le u\le x^p$ where $x^p>1$ and may be set to $+\infty$.

$I_{10}={\displaystyle {\int }_0^1}\frac{{\text{e}}^{-u}\text{d}u}{{\left(1+u\right)}^{\beta }}={\displaystyle {\int }_0^1}\left(D_0+D_{1}u+D_2{u}^2+D_3{u}^3+\cdots +D_N{u}^N\right){\text{e}}^{-u}\text{d}u$ (25)

Successively integrating by parts yields

$\begin{array}{c}{I}_{10}=-{D_0{\text{e}}^{-u}|}_0^1-{D_1\left(1+u\right){\text{e}}^{-u}|}_0^1-{D_2\left(2+2u+u^2\right){\text{e}}^{-u}|}_0^1\\ -{D_3\left(6+6u+3u^2+u^3\right){\text{e}}^{-u}|}_0^1-{D_4\left(24+24u+12u^2+4u^3+u^4\right){\text{e}}^{-u}|}_0^1-\cdots \end{array}$(26)

which may be easily generalized as

$I_{10}={\displaystyle \underset{m=0}{\overset{N}{\sum }}}\left(m!D_m-{\text{e}}^{-1}{\displaystyle \underset{k=m}{\overset{N}{\sum }}}\frac{k!}{m!}{D}_k\right)$ (27)

We have thus completed the evaluation of $I_{10}$ in terms of the coefficients of the power series given in Equation (24).

For the evaluation of $I_{11}$ we make use of Equation (24b) as $1\le u\le x^p$ in this case.

$I_{11}={\displaystyle {\int }_1^{x^p}}\frac{{\text{e}}^{-u}\text{d}u}{{\left(1+u\right)}^{\beta }}={\displaystyle {\int }_1^{x^p}}\left(\frac{D_0}{u^{\beta }}+\frac{D_1}{u^{\beta +1}}+\frac{D_2}{u^{\beta +2}}+\cdots +\frac{D_N}{u^{\beta +N}}\right){\text{e}}^{-u}\text{d}u$ (28)

The first integral ${\displaystyle {\int }_1^{x^p}}\left({\text{e}}^{-u}/u^{\beta }\right)\text{d}u$ is equal to $p\cdot I_1$ as can bee seen from Equation (13) and cannot be evaluated directly; therefore, we keep it as $p\cdot I_1$. On the other hand, the remaining integrals are worked out by partial integration so that

${\displaystyle {\int }_1^{x^p}}\frac{{\text{e}}^{-u}}{u^{\beta +1}}={\text{e}}^{-1}\left[\frac{1}{\beta }\right]-x{\text{e}}^{-x^p}\left[\frac{x^{-p}}{\beta }\right]-\frac{p\cdot I_1}{\beta }$ (29a)

$\begin{array}{c}{\displaystyle {\int }_1^{x^p}}\frac{{\text{e}}^{-u}}{u^{\beta +2}}={\text{e}}^{-1}\left[\frac{1}{\beta +1}-\frac{1}{\beta \left(\beta +1\right)}\right]-x{\text{e}}^{-x^p}\left[\frac{x^{-2p}}{\beta +1}-\frac{x^{-p}}{\beta \left(\beta +1\right)}\right]\\ +\frac{p\cdot I_1}{\beta \left(\beta +1\right)}\end{array}$ (29b)

$\begin{array}{c}{\displaystyle {\int }_1^{x^p}}\frac{{\text{e}}^{-u}}{u^{\beta +3}}={\text{e}}^{-1}\left[\frac{1}{\beta +2}-\frac{1}{\left(\beta +1\right)\left(\beta +2\right)}+\frac{1}{\beta \left(\beta +1\right)\left(\beta +2\right)}\right]\\ -x{\text{e}}^{-x^p}\left[\frac{x^{-3p}}{\beta +2}-\frac{x^{-2p}}{\left(\beta +1\right)\left(\beta +2\right)}-\frac{x^{-p}}{\beta \left(\beta +1\right)\left(\beta +2\right)}\right]\\ -\frac{p\cdot I_1}{\beta \left(\beta +1\right)\left(\beta +2\right)}\end{array}$ (29c)

Continuing in this manner and adding all the results to establish $I_{11}$ as expressed in (28) give

$\begin{array}{c}{I}_{11}=p\cdot I_1\left[D_0-\frac{D_1}{\beta }+\frac{D_2}{\beta \left(\beta +1\right)}-\frac{D_3}{\beta \left(\beta +1\right)\left(\beta +2\right)}+\cdots \right]\\ +{\text{e}}^{-1}{D}_1\left[\frac{1}{\beta }\right]+{\text{e}}^{-1}{D}_2\left[\frac{1}{\beta +1}-\frac{1}{\beta \left(\beta +1\right)}\right]\\ +{\text{e}}^{-1}{D}_3\left[\frac{1}{\beta +2}-\frac{1}{\left(\beta +1\right)\left(\beta +2\right)}+\frac{1}{\beta \left(\beta +1\right)\left(\beta +2\right)}\right]+\cdots \\ -x{\text{e}}^{-x^p}\left[\frac{x^{-p}}{\beta }\right]-x{\text{e}}^{-x^p}\left[\frac{x^{-2p}}{\beta +1}-\frac{x^{-p}}{\beta \left(\beta +1\right)}\right]\\ -x{\text{e}}^{-x^p}\left[\frac{x^{-3p}}{\beta +2}-\frac{x^{-2p}}{\left(\beta +1\right)\left(\beta +2\right)}+\frac{x^{-p}}{\beta \left(\beta +1\right)\left(\beta +2\right)}\right]-\cdots \end{array}$ (30)

The general form of (30) may be expressed as

$\begin{array}{c}{I}_{11}=p\cdot I_1\left[D_0+{\displaystyle \underset{m=1}{\overset{N}{\sum }}}{\left(-1\right)}^m\left({\displaystyle \underset{k=1}{\overset{m}{\prod }}}{\left(\beta +m-k\right)}^{-1}\right)D_m\right]\\ +{\text{e}}^{-1}\left\{{\displaystyle \underset{m=1}{\overset{N}{\sum }}}\left[{\displaystyle \underset{k=1}{\overset{m}{\sum }}}{\left(-1\right)}^{k+1}\left({\displaystyle \underset{l=1}{\overset{k}{\prod }}}{\left(\beta +m-l\right)}^{-1}\right)\right]D_m\right\}\\ -x{\text{e}}^{-x^p}\left\{{\displaystyle \underset{m=1}{\overset{N}{\sum }}}\left[{\displaystyle \underset{k=1}{\overset{m}{\sum }}}{\left(-1\right)}^{k+1}{x}^{-p\left(m-k+1\right)}\left({\displaystyle \underset{l=1}{\overset{k}{\prod }}}{\left(\beta +m-l\right)}^{-1}\right)\right]D_m\right\}\end{array}$ (31)

To complete the evaluation of $I_1$ the only remaining integral is

$I_{12}=x\cdot {\text{e}}^{1-x^p}{\displaystyle {\int }_0^{1/x^p}}\left[{\text{e}}^{-x^{p}u}/{\left(1+u\right)}^{\beta }\right]\text{d}u$ as defined by (16). Considering only the integral itself and noting that $1/x^p<1$, the series expansion (24a) valid for $0\le u\le 1$ can be employed.

${\displaystyle {\int }_0^{1/x^p}}\frac{{\text{e}}^{-x^{p}u}\text{d}u}{{\left(1+u\right)}^{\beta }}={\displaystyle {\int }_0^{1/x^p}}\left(D_0+D_{1}u+D_2{u}^2+D_3{u}^3+\cdots +D_N{u}^N\right){\text{e}}^{-x^{p}u}\text{d}u$ (32)

Proceeding very similar to the evaluation of $I_{10}$, we get

$\begin{array}{c}{\displaystyle {\int }_0^{1/x^p}}\frac{{\text{e}}^{-x^{p}u}\text{d}u}{{\left(1+u\right)}^{\beta }}=-{\frac{D_0}{x^p}{\text{e}}^{-x^{p}u}|}_0^{x^{-p}}-{\frac{D_1}{x^p}\left(x^{-p}+u\right){\text{e}}^{-x^{p}u}|}_0^{x^{-p}}\\ -{\frac{D_2}{x^p}\left(2x^{-2p}+2x^{-p}u+u^2\right){\text{e}}^{-x^{p}u}|}_0^{x^{-p}}\\ -{\frac{D_3}{x^p}\left(6x^{-3p}+6x^{-2p}u+3x^{-p}{u}^2+u^3\right){\text{e}}^{-x^{p}u}|}_0^{x^{-p}}-\cdots \end{array}$ (33)

Generalizing the above expression and multiplying by $x\cdot {\text{e}}^{1-x^p}$ give $I_{12}$ as

$I_{12}=x\cdot {\text{e}}^{1-x^p}{\displaystyle \underset{m=0}{\overset{N}{\sum }}}\left(x^{-p\left(m+1\right)}m!D_m-{\text{e}}^{-1}{\displaystyle \underset{k=m}{\overset{N}{\sum }}}{x}^{-p\left(k+1\right)}\frac{k!}{m!}{D}_k\right)$ (34)

Finally, $I_1$ can be obtained by the use of Equation (17) $\left(p\cdot e\right)I_1=I_{10}+I_{11}-I_{12}$,

$\begin{array}{c}\left(p\cdot \text{e}\right)I_1={\displaystyle \underset{m=0}{\overset{N}{\sum }}}\left(m!D_m-{\text{e}}^{-1}{\displaystyle \underset{k=m}{\overset{N}{\sum }}}\frac{k!}{m!}{D}_k\right)\\ +p\cdot I_1\left[D_0+{\displaystyle \underset{m=1}{\overset{N}{\sum }}}{\left(-1\right)}^m\left({\displaystyle \underset{k=1}{\overset{m}{\prod }}}{\left(\beta +m-k\right)}^{-1}\right)D_m\right]\\ +{\text{e}}^{-1}\left\{{\displaystyle \underset{m=1}{\overset{N}{\sum }}}\left[{\displaystyle \underset{k=1}{\overset{m}{\sum }}}{\left(-1\right)}^{k+1}\left({\displaystyle \underset{l=1}{\overset{k}{\prod }}}{\left(\beta +m-l\right)}^{-1}\right)\right]D_m\right\}\\ -x{\text{e}}^{-x^p}\left\{{\displaystyle \underset{m=1}{\overset{N}{\sum }}}\left[{\displaystyle \underset{k=1}{\overset{m}{\sum }}}{\left(-1\right)}^{k+1}{x}^{-p\left(m-k+1\right)}\left({\displaystyle \underset{l=1}{\overset{k}{\prod }}}{\left(\beta +m-l\right)}^{-1}\right)\right]D_m\right\}\\ -x\cdot {\text{e}}^{1-x^p}{\displaystyle \underset{m=0}{\overset{N}{\sum }}}\left(x^{-p\left(m+1\right)}m!D_m-{\text{e}}^{-1}{\displaystyle \underset{k=m}{\overset{N}{\sum }}}{x}^{-p\left(k+1\right)}\frac{k!}{m!}{D}_k\right)\end{array}$ (35)

which can be re-arranged for $I_1$ as

$\begin{array}{l}p\cdot I_1\left[\text{e}-D_0-{\displaystyle \underset{m=1}{\overset{N}{\sum }}}{\left(-1\right)}^m\left({\displaystyle \underset{k=1}{\overset{m}{\prod }}}{\left(\beta +m-k\right)}^{-1}\right)D_m\right]\\ ={\displaystyle \underset{m=0}{\overset{N}{\sum }}}\left(m!D_m-{\text{e}}^{-1}{\displaystyle \underset{k=m}{\overset{N}{\sum }}}\frac{k!}{m!}{D}_k\right)\\ +{\text{e}}^{-1}\left\{{\displaystyle \underset{m=1}{\overset{N}{\sum }}}\left[{\displaystyle \underset{k=1}{\overset{m}{\sum }}}{\left(-1\right)}^{k+1}\left({\displaystyle \underset{l=1}{\overset{k}{\prod }}}{\left(\beta +m-l\right)}^{-1}\right)\right]D_m\right\}\\ -x{\text{e}}^{-x^p}\left\{{\displaystyle \underset{m=1}{\overset{N}{\sum }}}\left[{\displaystyle \underset{k=1}{\overset{m}{\sum }}}{\left(-1\right)}^{k+1}{x}^{-p\left(m-k+1\right)}\left({\displaystyle \underset{l=1}{\overset{k}{\prod }}}{\left(\beta +m-l\right)}^{-1}\right)\right]D_m\right\}\\ -x\cdot {\text{e}}^{1-x^p}{\displaystyle \underset{m=0}{\overset{N}{\sum }}}\left(x^{-p\left(m+1\right)}m!D_m-{\text{e}}^{-1}{\displaystyle \underset{k=m}{\overset{N}{\sum }}}{x}^{-p\left(k+1\right)}\frac{k!}{m!}{D}_k\right)\end{array}$ (36)

Computing $I_0$ from (10) and $I_1$ from (36) enable us to compute the integral sought $G_p\left(+\infty \right)\cdot G_p\left(x\right)={\displaystyle {\int }_0^x{\text{e}}^{-t^p}\text{d}t}=I_0+I_1$ as stated in (9).

The well-known error function corresponds to $p=2$ case hence it is considered first. Table 1 compares the tabulated values of ( [2] , pp. 310-11), labeled “exact”, with computations from present formulation for the truncation order $N=4$. Integral value at infinity can be obtained either by simply setting the upper limit contribution of $I_{11}$ to zero or taking a relatively large x value such as $x=100$; but even $x=5$ is sufficient to obtain a perfectly acceptable result for $x=+\infty$. For the present case of $p=2$ this value corresponds to $\Gamma \left(1/2\right)/2=\sqrt{\pi }/2$ and the values in Table 1 are normalized by multiplying with $2/\Gamma \left(1/2\right)=2/\sqrt{\pi }$. Finally, the relative error percentages are computed from 100∙(1 – Approximate/Exact).

For $0.00\le x\le 1.00$ the computed results agree perfectly well with the tabulated values because both are obtained from the series expansion in (10) with upper limit of integration set to the given x value. In computations the number of terms was restricted to 150 due to the limited capability of the machine to compute large factorials. Tables of Abramowitz and Stegun [2] terminate at $x=2$; therefore, the comparisons accord with this upper limit.

For $x\ge 1$ the present formulation uses (36) with $N=4$ as no improvement in accuracy is observed for higher truncation orders $N=\mathrm{5,6,7,8}$. However, as Table 3 in §4 reveals, higher truncation orders give improved results for higher p values, especially for $x=+\infty$. The present comparisons have the same tendency concerning x; namely, increasing the upper limit of integration decreases the relative error percentage appreciably. For instance, the error is approximately 0.07% for $x=1.25$; but it reduces tenfold to −0.007% for $x=2.00$. Despite these moderately satisfactory results, even for a truncation order of as small as $N=4$, the lack of improvement in accuracy for higher truncation orders indicates an inherently difficult problem, which is also observed for numerical integrations performed by Simpson’s rule of integration for purposes of extra checking. High precision computations in the range $0<x\le 1$ did not yield results as good as the series solutions despite taking an extremely small step of $\Delta x=1/1000000$.

Another group of tables given in ( [2] , p. 320] is for the integral

$G_3\left(x\right)=\frac{3}{\Gamma \left(1/3\right)}{\displaystyle {\int }_0^x{\text{e}}^{-t^3}\text{d}t}$, which corresponds to $p=3$ in the present notation. Table 2 shows the comparisons up to $x=2.30$, which is the highest value included in [2] . Error percentages show similar characteristics with those of $p=2$ case; yet, overall the errors are smaller. The listed highest error is 0.056% for $x=1.20$ and it reduces to 0.004% for $x=2.30$. Again, values for $0.00\le x\le 1.00$ agree perfectly well because they are computed from the series expansion in complete agreement with the one used in [2] .

For different truncation orders $N=\mathrm{4,}\cdots \mathrm{,8}$, Table 3 presents computational results for the normalized integral $G_p\left(+\infty \right)$, which is exactly equal to unity for any p. Deviation from the unity corresponds to the absolute error in computational result. It is clear from Table 3 that, regardless of the truncation number,

computational accuracy increases with increasing p values. In particular, $N=8$ case with four/five decimal places accuracy is remarkably good for $p\ge 5.75$. On the other hand, lower truncation orders $N=\mathrm{4,5}$ appear to perform better for relatively smaller $p\le 3$ values. Nevertheless, there appears no definite trend of truncation orders. Figure 3 depicts $G_p\left(x\right)$ normalized integral function within the range $0\le x\le 5$ for five different p values. Computations for $p=1$ may be performed by setting p to a slightly higher value such as $p=1+{10}^{-6}$; otherwise, $\beta =1-1/p$ becomes exactly zero and the scheme fails. As expected, all the curves tend to unity with increasing x, even as low as x = 5.

A perspective view of $G_p\left(x\right)$ function for x and p pairs is presented in Figure 4 . The most notable characteristic of both Figure 3 and Figure 4 is the sharp increase observed in $G_p\left(x\right)$ in the neighborhood of $x=1$ as p values increase.