A new special function and its application in probability

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Zeraoulia function

Zeraoulia function is a new special function see , proposed by Zeraoulia Rafik in 17/02/2017 and has been studied by Zeraoulia Rafik, Alvaro Humberto Salas, Davide L.Ocampo and published in , it behaves like more than Error function and it is defined as :

$T(a)=\underset{0}{\overset{a}{\int }}{(e^{-x^2})}^{\mathrm{erf}(x)}dx$ mathematica gives for the first $100$ digits:

$T(\mathrm{\infty })=0.9721069927691785931510778754423911755542721833855699009722910408441888759958220033410678218401258734$ with $T(0)=0$ Here is the plot of $T(a)\text{for}a\in [0,10]$ which presents a numerical approximation of that function.

Approximation of Zeraoulia function by means of Chebyshev Polynomials

The function $f$ which is defined as: $f(x)=T(\frac{b+a}{2}+\frac{b-a}{2}x),-1\le x\le 1$ can be approximated by means of chebyshev polynomials, See , we may approximate the function $f$ on the interval $[-1,1]$ by using Chebyshev polynomials of the first kind. To this end, we choose some positive integer $n$ and we define the coefficients $c_n$ by the formulas:

$C_j=\frac{2}{\pi }\underset{-1}{\overset{1}{\int }}\frac{T_j(x)}{\sqrt{1-x^2}}f(x)dx$ for $j=0,...,n$ Then the polynomial

$P_n(x)=\frac{1}{2}{c}_0+\sum _{j=1}^n{c}_j{T}_j(x)$ approximates $f(x)$ in the best possible way. Since

$T(x)=f(\frac{a+b-2x}{a-b})\text{ for }a\le x\le b$ we see that the polynomial $Q_n(a,b,x)=P_n(\frac{a+b-2x}{a-b})$ is an approximant to $T(x)$ function on $[a,b]$ see , for the . Calculations give:

$\begin{array}{rlr}{Q}_{11}(0,3/2,x)& =& 0.0137936039435x^{11}-\\ & & 0.135129528505x^{10}+\\ & & 0.548169602543x^9-\\ & & 1.16161653976x^8+\\ & & 1.31691631085x^7-\\ & & 0.746480407376x^6+\\ & & 0.338453415662x^5-\\ & & 0.370071852413x^4+\\ & & 0.0133517048763x^3-\\ & & 0.00104123958376x^2+\\ & & 1.00003172454x\end{array}$

and

$\begin{array}{rlr}{Q}_{11}(3/2,3,x)& =& -0.0000675632422240x^{11}+\\ & & 0.00188305739843x^{10}-\\ & & 0.0239397852528x^9+\\ & & 0.183255163671x^8-\\ & & 0.937675010268x^7+\\ & & 3.35913844398x^6-\\ & & 8.55140470408x^5+\\ & & 15.3046428836x^4-\\ & & 18.4622672665x^3+\\ & & 13.5920479951x^2-\\ & & 4.69093970289x+\\ & & 1.04191571066\end{array}$

For both approximation the error is less than $10^{-6}$. Indeed, numerical integration gives:

$\begin{array}{rlr}||T(x)-Q_{11}(0,3/2,x)||& =& \\ \sqrt{\underset{0}{\overset{\frac{3}{2}}{\int }}(T(x)-Q_{11}(0,3/2,x){)}^{2}dx}& \approx & 2.26\times 10^{-7}\end{array}$

and

$\begin{array}{rlr}||T(x)-Q_{11}(3/2,3,x)||& =& \\ \sqrt{\underset{\frac{3}{2}}{\overset{3}{\int }}(T(x)-Q_{11}(3/2,3,x){)}^{2}dx}& \approx & 3.66\times 10^{-10}\end{array}$

Thus, we may evaluate the $T(x)$ function with high accuracy on the interval $[0,3]$. For $x>3$ we may use the following approximation formula in terms of the [[[Error function|error function]]:

$T(x)\approx \phi (x)=\underset{0}{\overset{3}{\int }}\exp (-t^2\text{erf}(t))dt+\frac{\sqrt{\pi }}{2}(\text{erf}(x)-\text{erf}(3)),x\ge 3.$

The mean squared error on $[3,100]$ is :

$||T(x)-\phi (x))||=\sqrt{\underset{3}{\overset{100}{\int }}(T(x)-\phi (x){)}^{2}dx}\approx 2.02\times 10^{-8}.$

Application of Zeraoulia function to probability

Let $F_{\lambda ,\mu }(x)=\underset{0}{\overset{x}{\int }}{e}^{-{\xi }^2(\lambda +\mu \text{erf}(\xi ))}d\xi ,(\lambda >0)$

Define

$c=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-{\xi }^2(\lambda +\mu \text{erf}(\xi ))}d\xi$

and let

$T_{\lambda ,\mu }(x):=c^{-1}{F}_{\lambda ,\mu }(x),(x\ge 0)$

The function $T(x)$ This function defines a cumulative distribution function (CDF) with probability distribution function (PDF)

$f_{\lambda ,\mu }(x)=e^{-x^2(\lambda +\mu \text{erf}(x))},(x\ge 0)$

Indeed, we have :

$T_{\lambda ,\mu }^{\prime }(x)=e^{-x^2(\lambda +\mu \text{erf}(x))}>0\text{ and }{T}_{\lambda ,\mu }(+\mathrm{\infty })=1.$

The ODE(ordinary differential equations) for this function not involving the error function erf may be obtained by differentiating the following equation twice and eliminating the expression containing that error function.

$\sqrt{\pi }x{y}^{\prime \prime }(x)=2e^{-x^2}{y}^{\prime }(x)(\sqrt{\pi }{e}^{x^2}\log (y^{\prime }(x))-\mu x^3)$

Letting $\mu =0$ gives the ODE(ordinary differential equations):

$x{y}^{\prime \prime }(x)=2y^{\prime }(x)\log (y^{\prime }(x))$

whose general solution is : $y(x)=\frac{1}{2}{e}^{-\frac{c_1}{2}}\sqrt{\pi }\text{erfi}\left(e^{\frac{c_1}{2}}x\right)+c_2$

One can show that in the case when $\mu =0$ our function $T_{\lambda ,0}(x)$ coincides with the error function $\text{erf}(\sqrt{\lambda }x)$ with the value $\lambda =\frac{\pi }{4}$.When $\mu \ne 0$ we cannot obtain the solution to the above ODE(ordinary differential equations) in closed form. We may try a numerical procedure or other method to solve it

Example of Application

We look for $\lambda$ and $\mu$ in order to adjust the error function by means of the function $y(x)=T_{\lambda ,\mu }(x).$ To this end, we impose the conditions :

$\text{erf}(1)=T_{\lambda ,\mu }(1)\text{ and }{\text{erf}}^{\prime }(1)=T_{\lambda ,\mu }^{\prime }(1)$

Solving this system gives :

$\lambda =0.1671645\text{ and }\mu =0.8449657.$

The function $T_{\lambda ,\mu }(x)$ converts into :

$T_{\lambda ,\mu }(x)=1.05021\underset{0}{\overset{x}{\int }}\exp (-{\xi }^2(0.167164+0.844966\text{erf}(\xi )))d\xi .$ Plotting the two functions gives the following picture as shown in Figure

Application of Zeraoulia function in Thermo-dynamic using Boltzmann distribution

The maxwell–Boltzmann distribution is the chi distribution with three degrees of freedom (the components of the velocity vector in euclidean space), with a scale parameter measuring speeds in units proportional to the square root of $\frac{T}{m}$ (the ratio of temperature and particle mass) see

The CDF (cumulative distribution function) for The maxwell–Boltzmann may be approximated by means of the new special function $T_{\kappa ,\mu }(x)$ as follows:

$\text{erf}(\frac{x}{\sqrt{2}a})-\sqrt{\frac{2}{\pi }}\frac{x}{a}\exp (-\frac{x^2}{2a^2})\approx T_{\lambda ,\mu }(x)-\sqrt{\frac{2}{\pi }}\frac{x}{a}\exp (-\frac{x^2}{2a^2}),$

where $T_{\lambda ,\mu }(x)$ is an approximation to $\text{erf}(\frac{x}{\sqrt{2}a})$ for some parameters: $\lambda$ and $\mu$ depending on $a$ This approximation may be obtained in a similar way we illustrated. See On the other hand, in the case when $0<a\le 1$ we may approximate the CDF for the maxwell–Boltzmann for the value $a=0.75$ , See

Representation of Golden Ratio using Zeraoulia Function

Golden ratio could be approximated or represented by one kind of Zeraoulia function class such that defined as:

$I(t)=\underset{0}{\overset{t}{\int }}(\exp (\sqrt{x}-x^2){)}^{\text{erf(x)}}\text{d}x$ and for $t=2.7495392638089838109896679573104092694836858310585$ , we have a Golden ratio with approximation of $\frac{1}{10^{50}}$

See Also

Related functions

• Gaussian integral, over the whole real line
• Gaussian function, derivative
• Dawson function, renormalized imaginary error function
• Goodwin–Staton integral
• Special function

In probability

• Normal distribution
• Normal cumulative distribution function, a scaled and shifted form of error function
• Maxwell-Boltzmann distribution

References

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