Zeraoulia function

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History for Zeraoulia function

File:Rafik Zeraoulia3.jpg

Zeraoulia Rafik is a Researcher in mathematics interesting to special functions and Number theory, "He has got a new Special function with it's application in probability and thermo-Dynamics"

Zeraoulia Function is a new special function derived from error function such that during the course of the 19th century the function from the theory of errors appeared in several contexts unrelated to probability, e.g. refraction and heat conduction. In 1871 J. W. Glaisher wrote that "Erf(x) may fairly claim at present to rank in importance next to the trigonometrical and logarithmic functions." Glaisher introduced the symbol Erf and the name error function for a particular form of the law as follows: As it is necessary that the function should have a name, and as I do not know that any has been suggested,, I propose to call it the Error-function. In the same time Zeraoulia function is expressed as a power of Exponential function and Error function and it is called Zeraoulia function referring to Zeraoulia Rafik since it's not refer to anyone. This function was discovered by Zeraoulia Rafik in 2018, when he asked in Stackexchange math about integral of a function power it's derivative defining by this way a new Probability distribution and a new Approach. Both of researchers Zeraoulia Rafik and Alvaro H Salas and Davide L Ocampo provided a huge efforts for studying that new special function such they found it's interesting with the incrementations of $\pi$ and They gave it's Series expansion to be defined as well and to be applied in Modern science as probability theory

Name

The term special function. has a historical connection with solutions of ordinary differential equations, often second order differential equations that arise from a separation of variables treatment of second order linear partial differential equations with constant coefficients. There is no general formal definition, but the list of mathematical functions contains functions which are commonly accepted as special. In particular, elementary functions are also considered as special functions.

"A vague bad definition could be "A function is a special function if it has some resemblance to some Hypergeometric function" or "A function is a special function if it fits into the Bateman manuscript project.

The Sage documentation of special functions , for example, lists the ODE(Ordinary differential equation) satisfied by the function together with remarks about boundary conditions. Stephen Wolfram gave a nice historical review in his talk The History and Future of Special Functions , in which he says, "Most continuous special functions are in effect defined implicitly, usually from differential equations." Naturally he is not shy about what he perceives as Mathematica's contribution to this history, developing automated searches for accurate numerical methods that proceed directly from the underlying definition by differential equations.

He is not alone in exploring the automatic generation of algorithms for special functions using symbolic computation. Focusing on special functions that are the solutions of linear ordinary differential equations with polynomial coefficients, Meunier and Salvy (2003) describe the design of their website The Encyclopedia of Special Functions (ESF)

The elementary transcendental functions fit easily into this scheme as special special functions, since they solve differential equations too. For example, the exponential function solves $y^{\prime }=y$ and sine and cosine, with different boundary conditions, solve the constant coefficient ODE(Ordinary differential equation) $y^{″}=-y$

Zeraoulia Function is a new special function such that it is non elementary Hypergeometric function satisfied by this Ordinary differential equation see, That function considered to be a new special function because there is no bibliography includes it and it doesn't studied before however it is one of class of error function

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

Zeraoulia function

File:The-plot-of-Ia-for-a-0-1 W640.jpg

Plot of zeraoulia function with some approximations. using Taylor Expansion

Zeraoulia function is a new special function see , proposed by Zeraoulia Rafik in 17/02/2017 and have been studied by Zeraoulia rafik, Alvaro Humberto Salas , davide L.Ocampo and published in , it behave 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 Plote of $T(a)\text{for}a\in [0,10]$ which present a numerical approximation of that function .

Approximation of Zeraoulia function by means of Chebychev Polynomials

The function $f$ which is defined as: $f(x)=T(\frac{b+a}{2}+\frac{b-a}{2}x),-1\le x\le 1$ Could 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 the terms of the error:

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

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 < math >\mu =0</math > 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

Application of Zeraoulia function in Thermo-dynamic using Boltzman 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 in 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 ration with approximation of $\frac{1}{10^{50}}$

Seee 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
• Chi distribution

External links

• Higher Transcendental Functions Volumes 1, 2, 3 by Arthur Erdélyi ISBN 0-486-44614-X Search this book on . ISBN 0-486-44615-8 Search this book on . ISBN 0-486-44616-6 Search this book on . . By permission of the copyright owner, scanned copies are publicly available: Volumes I-III or Vol.1, Vol. 2, Vol. 3.
• Tables of Integral Transforms Volumes 1, 2 by Arthur Erdélyi ISBN 0-07-019549-8 Search this book on .. Scanned copies are publicly available: Volumes I&II
• Cherry, G.W. (1985). "Integration in Finite Terms with Special Functions: The Error Function". Journal of Symbolic Computation. 1 (3): 283–302. doi:10.1016/S0747-7171(85)80037-7.

• A. Berk, Voigt equivalent widths and spectral-bin single-line transmittances:

Exact expansions and the MODTRANr5 implementation, J. Quantit. Spectrosc. Radiat. Transfer, 118 (2013) 102-120.

• B. M. Quine and S. M. Abrarov, Application of the spectrally integrated Voigt

function to line-by-line radiative transfer modelling, J. Quantit. Spectrosc. Radiat. Transfer, 127 (2013) 37-48.

• S. J. McKenna, A method of computing the complex probability function and

other related functions over the whole complex plane, Astrophys. Space Sci.,

107 (1984) 71-83.

• B. H. Armstrong and B. W. Nicholls, Emission, absorption and transfer of

radiation in heated atmospheres, Pergamon Press, New York, 1972.

• G. P. M. Poppe and C. M. J. Wijers, More efficient computation of the complex

error function, ACM Transact. Math. Software, 16 (1990) 38-46.

• Interactive Chi-Square Distribution Web Applet (Java)
• ,A quick refresher for Counting techniques and Probability

• ,Probability Theory The Logic of Science

• , Differential equation for Harry Batmen
• note on infinit series
• Hypergeometric series

• special function

• computation of hypergeometric function
• Mathworld hypergeometric function

ISBN 9781483267449 Search this book on . gbook

zeraoulia function

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