The field of astrophysics routinely applies the standard distribution of probabilities in different environments such as the distributions in mass and luminosity of stars and galaxies. The normal or Gaussian distribution, introduced by Pearson in 1902 [1] is in principle the first distribution to be considered, but due to the fact that the mass and luminosity are always positive, it presents an inconvenience. The half Gaussian and the truncated half Gaussian [2] are defined for positive values of the variate and therefore can be used to model the stars’ heights above the Galactic plane. Another example is the case of the gamma family for the luminosity function for galaxies, which started with Schechter in 1976 [3] . An improvement in the fit of the luminosity function for galaxies has been made by considering the generalized gamma distribution [4] and the truncated generalized gamma distribution [5] . Another distribution useful in astrophysics is the lognormal, which dates back to Galton and McAlister in 1879 [6] [7] . Other names are also used for the lognormal, such as Cobb-Douglas in 1928 [8] , Gibrat in 1931 [9] and logarithmic-normal in 1919 [10] . We now review some recent applications of the lognormal distribution: to model the distribution function of the flux and the dependence of the standard deviation of the flux on the mean flux [11] , the Fermi data of the blazar 3FGL J0730.2-1141, showing that its $\gamma$-ray flux is consistent with a lognormal distribution [12] , data analysis of the results of the three-dimensional hydro-dynamical simulations of shocks [13] , analysis of the distribution properties of the areas of sunspots [14] , and the distribution of $\gamma$-ray flux and the statistical characteristics of a large sample of 1414 variable blazars from the Fermi-LAT LCR catalog [15] . We now pose some questions not yet solved.

1) Is it possible to introduce the effect of truncation in the modified lognormal with a power-law (MLP) distribution?

2) Can the truncated MLP distribution explain the initial mass function for the stars which usually is reported between ${10}^{-2}M/{ℳ}_{\odot }<{10}^2$?

In order to answer these questions, we review the following distributions in Section 2: the lognormal, the truncated lognormal, and the double Pareto-lognormal. Section 3 reviews the statistics connected with the MLP distribution and Section 4 introduces the truncated MLP. Section 5 applies the new and old distributions to five clusters of stars and one of galaxies and Section 6 derives the parameters of the truncated MLP for the mass function of stars.

In the following, PDF means probability density function and DF distribution function. The function $\text{erf}\left(x\right)$, the error function, is defined in Appendix A.

We now review the lognormal distribution, the truncated lognormal distribution and the double Pareto-lognormal distribution.

The modified lognormal with a power-law (MLP) defined in the interval $\left[0,\infty \right]$ has the following PDF

$f\left(x:\alpha ,\sigma ,\mu \right)=\frac{x^{-1-\alpha }\text{erfc}\left(\frac{\sqrt{2}\left(\alpha \sigma -\frac{\text{ln}\left(x\right)-\mu }{\sigma }\right)}{2}\right)\alpha }{2{\text{e}}^{-\frac{1}{2}{\alpha }^2{\sigma }^2-\alpha \mu }},$(15)

see formula (14) in [20] . Its DF is

$\begin{array}{c}F\left(x:\alpha ,\sigma ,\mu \right)=-\frac{\text{erfc}\left(\frac{\sqrt{2}\left(\alpha {\sigma }^2-\text{ln}\left(x\right)+\mu \right)}{2\sigma }\right)x^{-\alpha }{\text{e}}^{\frac{1}{2}{\alpha }^2{\sigma }^2+\alpha \mu }}{2}\\ +1-\frac{\text{erfc}\left(\frac{\sqrt{2}\left(\text{ln}\left(x\right)-\mu \right)}{2\sigma }\right)}{2},\end{array}$(16)

see formula (16) in [20] . The first moment or mean, $E\left(\alpha ,\sigma ,\mu \right)$, is defined for $\alpha >1$:

$E\left(\alpha ,\sigma ,\mu \right)=\frac{\alpha {\text{e}}^{\frac{{\sigma }^2}{2}+\mu }}{\alpha -1},$(17)

see formula (19) in [20] .

The variance, $Var\left(\alpha ,\sigma ,\mu \right)$, is defined for $\alpha >2$:

$Var\left(\alpha ,\sigma ,\mu \right)=-\frac{{\alpha }^2{\left({\text{e}}^{\frac{{\sigma }^2}{2}+\mu }\right)}^2}{{\left(\alpha -1\right)}^2}+\frac{\alpha {\text{e}}^{2{\sigma }^2+2\mu }}{\alpha -2},$(18)

see formula (21) in [20] . The mode should be evaluated numerically. The random generation of the variate $X$ of the truncated MLP is obtained by solving the following nonlinear equation in $x$:

$F\left(x:\alpha ,\sigma ,\mu \right)=R,$(19)

where $R$ is the unit rectangular variate. We give an approximation for the error function among eleven others; see Table 1 in [21] for more details,

$\text{erf}\left(x\right)\approx \tanh \left(\frac{39x}{2\sqrt{\pi }}-\frac{111\arctan \left(\frac{35x}{111\sqrt{\pi }}\right)}{2}\right),$(20)

see formula (3.1) in [22] . With the above approximation, the PDF is

$\begin{array}{c}f\left(x:\alpha ,\sigma ,\mu \right)\approx \frac{1}{2{\text{e}}^{-\frac{1}{2}{\alpha }^2{\sigma }^2-\alpha \mu }}\times (-x^{-1-\alpha }(\tanh (\frac{39\sqrt{2}\left(\alpha {\sigma }^2-\ln \left(x\right)+\mu \right)}{4\sigma \sqrt{\pi }}\begin{array}{c}\\ \\ \\ \end{array}\\ -\frac{111\arctan \left(\frac{35\sqrt{2}\left(\alpha {\sigma }^2-\ln \left(x\right)+\mu \right)}{222\sigma \sqrt{\pi }}\right)}{2})-1)\alpha ).\end{array}$(21)

An example of such an approximation is presented in Figure 1 , which has a maximum percentage error of 7%. The approximated DF is

$\begin{array}{l}f\left(x:\alpha ,\sigma ,\mu \right)\\ \approx \frac{1}{2}\times (x^{-\alpha }{\text{e}}^{\frac{1}{2}{\alpha }^2{\sigma }^2+\alpha \mu }\tanh \left(\frac{39\sqrt{2}\left(\alpha {\sigma }^2-\ln \left(x\right)+\mu \right)}{4\sigma \sqrt{\pi }}-\frac{111\arctan \left(\frac{35\sqrt{2}\left(\alpha {\sigma }^2-\ln \left(x\right)+\mu \right)}{222\sigma \sqrt{\pi }}\right)}{2}\right)\\ -x^{-\alpha }{\text{e}}^{\frac{1}{2}{\alpha }^2{\sigma }^2+\alpha \mu }-\tanh \left(-\frac{39\sqrt{2}\left(\ln \left(x\right)-\mu \right)}{4\sigma \sqrt{\pi }}+\frac{111\arctan \left(\frac{35\sqrt{2}\left(\ln \left(x\right)-\mu \right)}{222\sigma \sqrt{\pi }}\right)}{2}\right)+1),\end{array}$(22)

see Figure 2 , which has a maximum percentage error of 0.05%.

The three parameters $\alpha$ , $\sigma$ and $\mu$ can be obtained by two methods. The first method is via maximum likelihood (MLE), which maximizes

$\begin{array}{c}\Lambda =\frac{n{\alpha }^2{\sigma }^2}{2}+n\alpha \mu -n\ln \left(2\right)+n\ln \left(\alpha \right)\\ +{\displaystyle \underset{j=1}{\overset{n}{\sum }}\left(\text{ln}\left(\text{erfc}\left(\frac{\sqrt{2}\left(\alpha {\sigma }^2-\text{ln}\left(x_j\right)+\mu \right)}{2\sigma }\right)\right)-\text{ln}\left(x_j\right)\alpha -\text{ln}\left(x_j\right)\right)},\end{array}$(23)

where $n$ is the number of elements in the sample $x_j$. This method was introduced by Fisher in 1921 and at the moment of writing is widely used. We extract the words used by Fisher “The solution of the problems of calculating from a sample the parameters of the hypothetical population, which we have put forward in the method of maximum likelihood, consists, then, simply of choosing such values of these parameters as have the maximum likelihood…” The derivatives of $\text{Λ}$ with respect to $\alpha$, $\sigma$ and $\mu$ form a system of three nonlinear equations:

$\frac{\partial \text{Λ}}{\partial \alpha }=n\alpha {\sigma }^2+n\mu +\frac{n}{\alpha }+{\displaystyle \underset{j=1}{\overset{n}{\sum }}\left(-\frac{{\text{e}}^{-\frac{{\left(\alpha {\sigma }^2-\text{ln}\left(x_j\right)+\mu \right)}^2}{2{\sigma }^2}}\sqrt{2}\sigma }{\sqrt{\pi }\text{erfc}\left(\frac{\sqrt{2}\left(\alpha {\sigma }^2-\text{ln}\left(x_j\right)+\mu \right)}{2\sigma }\right)}-\text{ln}\left(x_j\right)\right)}=0$(24a)

$\frac{\partial \text{Λ}}{\partial \mu }=n\alpha +{\displaystyle \underset{j=1}{\overset{n}{\sum }}\left(-\frac{{\text{e}}^{-\frac{{\left(\alpha {\sigma }^2-\text{ln}\left(x_j\right)+\mu \right)}^2}{2{\sigma }^2}}\sqrt{2}}{\sqrt{\pi }\sigma \text{erfc}\left(\frac{\sqrt{2}\left(\alpha {\sigma }^2-\text{ln}\left(x_j\right)+\mu \right)}{2\sigma }\right)}\right)}=0$(24b)

$\frac{\partial \text{Λ}}{\partial \sigma }=n{\alpha }^2\sigma +{\displaystyle \underset{j=1}{\overset{n}{\sum }}\left(-\frac{2{\text{e}}^{-\frac{{\left(\alpha {\sigma }^2-\text{ln}\left(x_j\right)+\mu \right)}^2}{2{\sigma }^2}}\left(\sqrt{2}\alpha -\frac{\sqrt{2}\left(\alpha {\sigma }^2-\text{ln}\left(x_j\right)+\mu \right)}{2{\sigma }^2}\right)}{\sqrt{\pi }\text{erfc}\left(\frac{\sqrt{2}\left(\alpha {\sigma }^2-\text{ln}\left(x_j\right)+\mu \right)}{2\sigma }\right)}\right)}=0$(24c)

which can be solved numerically. The second method minimizes the following ${\chi }^2$

${\chi }^2={\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left(datad{f}_i-DF\left(x_i\right)\right)}^2},$(25)

where the $x_i$ are the elements of the sample sorted into ascending order, $datad{f}_i$ are the values of the empirical DF corresponding to $x_i$, see subroutine KSONE in [23] , and $DF\text{}\left(x_i\right)$ are the theoretical values of the DF. This second method is classified as nonlinear least squares (NLLS). The derivatives of this ${\chi }^2$ with respect to $\alpha$, $\sigma$ and $\mu$ form a system of three nonlinear equations:

$\frac{\partial {\chi }^2}{\partial \alpha }=0$(26a)

$\frac{\partial {\chi }^2}{\partial \mu }=0$(26b)

$\frac{\partial \text{Λ}}{\partial \sigma }=0$(26c)

which can be solved as non-linear least squares (NLLSQ). At the moment of writing, the above method is occasionally used in current research.

The truncated version of the MLP defined in the interval $\left[x_l,x_u\right]$ has the following PDF:

$f_T\left(x:\alpha ,\sigma ,\mu ,x_l,x_u\right)=\frac{x^{-1-\alpha }\text{erfc}\left(\frac{\sqrt{2}\left(\alpha \sigma -\frac{\text{ln}\left(x\right)-\mu }{\sigma }\right)}{2}\right)}{A},$(27)

where

$\begin{array}{c}A=\frac{1}{\alpha }\times [{\text{e}}^{-\frac{1}{2}{\alpha }^2{\sigma }^2-\alpha \mu }\text{erf}\left(\frac{\sqrt{2}\left(\text{ln}\left(x_u\right)-\mu \right)}{2\sigma }\right)\\ -{\text{e}}^{-\frac{1}{2}{\alpha }^2{\sigma }^2-\alpha \mu }\text{erf}\left(\frac{\sqrt{2}\left(\text{ln}\left(x_l\right)-\mu \right)}{2\sigma }\right)\\ -x_l^{-\alpha }\text{erf}\left(\frac{\sqrt{2}\left(\alpha {\sigma }^2-\text{ln}\left(x_l\right)+\mu \right)}{2\sigma }\right)\\ +x_u^{-\alpha }\text{erf}\left(\frac{\sqrt{2}\left(\alpha {\sigma }^2-\text{ln}\left(x_u\right)+\mu \right)}{2\sigma }\right)+x_l^{-\alpha }-x_u^{-\alpha }].\end{array}$(28)

The DF is

$\begin{array}{c}{F}_T\left(x:\alpha ,\sigma ,\mu ,x_l,x_u\right)=\frac{1}{\alpha A}\times [-x_l^{-\alpha }\text{erf}\left(\frac{\sqrt{2}\left(\alpha {\sigma }^2-\text{ln}\left(x_l\right)+\mu \right)}{2\sigma }\right)\\ +{\text{e}}^{-\frac{1}{2}{\alpha }^2{\sigma }^2-\alpha \mu }\text{erf}\left(\frac{\sqrt{2}\left(\text{ln}\left(x\right)-\mu \right)}{2\sigma }\right)\\ -{\text{e}}^{-\frac{1}{2}{\alpha }^2{\sigma }^2-\alpha \mu }\text{erf}\left(\frac{\sqrt{2}\left(\text{ln}\left(x_l\right)-\mu \right)}{2\sigma }\right)\\ +x^{-\alpha }\text{erf}\left(\frac{\sqrt{2}\left(\alpha {\sigma }^2-\text{ln}\left(x\right)+\mu \right)}{2\sigma }\right)-x^{-\alpha }+x_l^{-\alpha }].\end{array}$(29)

The first moment or average value is

$\begin{array}{l}E\left(x:\alpha ,\sigma ,\mu ,x_l,x_u\right)\\ =\frac{1}{A\left(-1+\alpha \right)}\times [{\text{e}}^{-\frac{1}{2}{\alpha }^2{\sigma }^2-\alpha \mu +\frac{1}{2}{\sigma }^2+\mu }\text{erf}\left(\frac{\sqrt{2}\left({\sigma }^2-\text{ln}\left(x_l\right)+\mu \right)}{2\sigma }\right)\\ -{\text{e}}^{-\frac{1}{2}{\alpha }^2{\sigma }^2-\alpha \mu +\frac{1}{2}{\sigma }^2+\mu }\text{erf}\left(\frac{\sqrt{2}\left({\sigma }^2-\text{ln}\left(x_u\right)+\mu \right)}{2\sigma }\right)\\ -x_l^{1-\alpha }\text{erf}\left(\frac{\sqrt{2}\left(\alpha {\sigma }^2-\text{ln}\left(x_l\right)+\mu \right)}{2\sigma }\right)\\ +x_u^{1-\alpha }\text{erf}\left(\frac{\sqrt{2}\left(\alpha {\sigma }^2-\text{ln}\left(x_u\right)+\mu \right)}{2\sigma }\right)+x_l^{1-\alpha }-x_u^{1-\alpha }],\end{array}$(30)

and the second moment is

$\begin{array}{l}E\left(x^2:\alpha ,\sigma ,\mu ,x_l,x_u\right)\\ =\frac{1}{A\left(-2+\alpha \right)}\times [{\text{e}}^{-\left(-2+\alpha \right)\left(\frac{1}{2}\alpha {\sigma }^2+{\sigma }^2+\mu \right)}\text{erf}\left(\frac{\sqrt{2}\left(-2{\sigma }^2+\text{ln}\left(x_u\right)-\mu \right)}{2\sigma }\right)\\ -{\text{e}}^{-\left(-2+\alpha \right)\left(\frac{1}{2}\alpha {\sigma }^2+{\sigma }^2+\mu \right)}\text{erf}\left(\frac{\sqrt{2}\left(-2{\sigma }^2+\text{ln}\left(x_l\right)-\mu \right)}{2\sigma }\right)\end{array}$

$\begin{array}{l}-x_l^{2-\alpha }\text{erf}\left(\frac{\sqrt{2}\left(\alpha {\sigma }^2-\text{ln}\left(x_l\right)+\mu \right)}{2\sigma }\right)\\ +x_u^{2-\alpha }\text{erf}\left(\frac{\sqrt{2}\left(\alpha {\sigma }^2-\text{ln}\left(x_u\right)+\mu \right)}{2\sigma }\right)+x_l^{2-\alpha }-x_u^{2-\alpha }].\end{array}$ (31)

The variance, in an implicit form, is

$Var\left(\alpha ,\sigma ,\mu ,x_l,x_u\right)=E\left(x^2:\alpha ,\sigma ,\mu ,x_l,x_u\right)-{\left(E\left(x:\alpha ,\sigma ,\mu ,x_l,x_u\right)\right)}^2.$(32)

The first method to derive the three basic parameters is the maximum likelihood (MLE), which maximizes

$\begin{array}{c}\text{Λ}=n\ln \left(-\frac{1}{\left(B-C\right){\text{e}}^{-\frac{1}{2}{\alpha }^2{\sigma }^2-\alpha \mu }+\left(D-1\right)x_l^{-\alpha }-x_u^{-\alpha }\left(E-1\right)}\right)+n\ln \left(\alpha \right)\\ +{\displaystyle \underset{j=1}{\overset{n}{\sum }}\left(-\ln \left(x_j\right)\alpha -\text{ln}\left(x_j\right)+\text{ln}\left(\text{erfc}\left(\frac{\sqrt{2}\left(\alpha {\sigma }^2-\text{ln}\left(x_j\right)+\mu \right)}{2\sigma }\right)\right)\right)},\end{array}$(33)

where

$B=\text{erf}\left(\frac{\sqrt{2}\left(\text{ln}\left(x_l\right)-\mu \right)}{2\sigma }\right),$(34a)

$C=\text{erf}\left(\frac{\sqrt{2}\left(\text{ln}\left(x_u\right)-\mu \right)}{2\sigma }\right),$(34b)

$D=\text{erf}\left(\frac{\sqrt{2}\left(\alpha {\sigma }^2-\text{ln}\left(x_l\right)+\mu \right)}{2\sigma }\right),$(34c)

$E=\text{erf}\left(\frac{\sqrt{2}\left(\alpha {\sigma }^2-\text{ln}\left(x_u\right)+\mu \right)}{2\sigma }\right).$(34d)

where $n$ is the number of elements of the sample $x_j$ and $x_l$ and $x_u$ are respectively the minimum and maximum of the sample $x_j$. The derivatives of $\text{Λ}$ with respect to $\alpha$, $\sigma$ and $\mu$ form a system of three nonlinear equations here given in implicit form:

$\frac{\partial \text{Λ}\left(\alpha ,\sigma ,\mu ,x_l,x_u\right)}{\partial \alpha }=0,$(35a)

$\frac{\partial \text{Λ}\left(\alpha ,\sigma ,\mu ,x_l,x_u\right)}{\partial \mu }=0,$(35b)

$\frac{\partial \text{Λ}\left(\alpha ,\sigma ,\mu ,x_l,x_u\right)}{\partial \sigma }=0,$(35c)

which can be solved numerically. The second method implements the NLLSQ method, see Equation (25), for the truncated MLP distribution.

We now introduce these distributions and analyse five clusters of stars and one cluster of galaxies.

In the following, $m$ is the stellar mass in units of ${ℳ}_{\odot }$. The first model for the initial mass function (IMF) is given by three power laws of the type

$p_{stars}\left(m\right)\propto m^{-{\alpha }_i},$(36)

each zone being characterized by a different exponent ${\alpha }_i$. In order to have a PDF normalized to unity, one must have

$\underset{i=1,3}{{\displaystyle \sum }}{\displaystyle {\int }_{m_i}^{m_{i+1}}{c}_i{m}^{-{\alpha }_i}\text{d}m}=1.$(37)

For example, we start with $c_1=\text{1}$: $c_2$ will be determined by the following equation

$c_1{\left(0.5-ϵ\right)}^{-{\alpha }_1}=c_2{\left(0.5+ϵ\right)}^{-{\alpha }_2},$(38)

where $ϵ$ is a small number, e.g. $ϵ={10}^{-4}$. In the previous equation we insert ${\alpha }_1=1.3$ and ${\alpha }_2=2.3$ and therefore $c_2=0.\text{5}0\text{3}$. The same procedure applied for $c_3$ gives $c_3=0.\text{5}0\text{6}$. The integral of $p_{stars}\left(m\right)$ over the field of existence now gives 4.14, but according to the requirement of normalization as given by Equation (37), it should be 1. As a consequence, the three constants are now $c_1=0.24$, $c_2=0.1205$, and $c_3=0.1206$, which is the same as Equation (59) in [33]

$p\left(m\right)=\{\begin{array}{ll}\frac{0.1206}{m^{2.7}}\hfill & 1.0\le m\hfill \\ \frac{0.1205}{m^{2.3}}\hfill & 0.50\le m\hfill \\ \frac{0.2409}{m^{1.3}}\hfill & 0.07\le m\hfill \\ 0\hfill & 10.0<m\hfill \end{array}.$(39)

The mean of the galactic IMF is given by a numerical integration over the three zones

$\bar{m}=\underset{i=1,3}{{\displaystyle \sum }}{\displaystyle {\int }_{m_i}^{m_{i+1}}{c}_{i}m{m}^{-{\alpha }_i}\text{d}m}=\mathrm{0.389.}$(40)

The presence of the brown dwarfs means the use of four power laws instead of three power laws:

$p\left(m\right)=\{\begin{array}{ll}\frac{0.07675}{m^{2.7}}\hfill & 1.0\le m\hfill \\ \frac{0.0767}{m^{2.3}}\hfill & 0.50\le m\hfill \\ \frac{0.1535}{m^{1.3}}\hfill & 0.07\le m\hfill \\ \frac{2.1928}{m^{0.3}}\hfill & 0.01\le m\hfill \\ 0\hfill & 100.0\le m\hfill \end{array},$(41)

where in order to have a continuous PDF, the BDs have the range $0.01<m\le 0.07$ rather than $0.01<m\le 0.15$, see Equation (59) in [33] . The mean of the galactic IMF with four power laws is 0.2788. More details on this first model can be found in [34] . The second model for the IMF is due to Chabrier 2005, more precisely formula (1) in [35] , which is here presented as a PDF in a normalized form

$\xi \left(m\right)=\frac{3.3799\left(\{\begin{array}{ll}0.093{\text{e}}^{-1.6528{\left(\frac{\text{ln}\left(m\right)}{\text{ln}\left(10\right)}+0.6989700043\right)}^2}\hfill & m<1\hfill \\ \frac{0.0414}{m^{1.35}}\hfill & m<\infty \hfill \end{array}\right)}{m}.$(42)

The average value for Chabrier’s PDF is 0.623. Astronomers usually report the data on the IMF in logarithmic bins and therefore the IMF is

$\frac{\text{d}N}{\text{d}\log \left(m\right)}=mf\left(m\right),$(43)

where $f\left(m\right)$ is the considered normalized PDF and the two IMFs are given in Figure 10 .

The comparison between the IMF of Chabrier 2005 and the MLP IMF has already been done by [20] , obtaining the set of parameters $\mu =-2.04$, $\sigma =1.044$ and $\alpha =1.396$. We therefore evaluated the parameters of the best fit between Chabrier 2005 and truncated MLP, obtaining $\mu =-2.34$, $\sigma =1.086$ and $\alpha =1.406$ when the lower boundary for $\log \left(m\right)$ is −2 and the upper 2, see Figure 11 .

The truncated MLP

We derived the PDF, the DF, the average value, the second moment, using the MLE and NLLSQ methods to extract the parameters for the truncated MLP distribution.

Fitting single clusters of stars

Table 7 presents the best distribution for the five catalogs used here. The truncated MLP yields the best result in three clusters over the five analysed.

The results for the mass distribution of the $\gamma$ Velorum cluster compared with other distributions are shown in Table 8 , in which the truncated MLP distribution occupies the first position together with the Benini distribution.

A careful reading of the above table allows us to conclude that when the range in mass analysed covers ≈ one decade, other distributions, such as the Benini, are competitive with the lognormal family.

Mass function

We have fitted the data of the IMF for stars with the truncated MLP function and the results are presented in Figure 11 . Now, the insertion of a lower and an upper logarithm of the mass has a theoretical explanation. In other words, the basic question (2) in the introduction has now been answered. The analogy between the scaling of the mass of the stars and that of the galaxies, as shown in Figure 12 , is interesting. The above figure points toward a similar mechanism of formation for stars and galaxies.

The error function is defined by

$\text{erf}\left(x\right)=\frac{2}{\sqrt{\pi }}{\displaystyle {\int }_0^x{\text{e}}^{-t^2}\text{d}t}.$(A.1)

The complementary error function is

$\text{erfc}\left(x\right)x=\frac{2}{\sqrt{\pi }}{\displaystyle {\int }_x^{\infty }{\text{e}}^{-t^2}\text{d}t}=1-\text{erf}\left(x\right),$(A.2)

see the handbook [44] .