# Definition:Zeraoulia Function

## Definition

The Zeraoulia function is a new special function see [1] , [2] proposed by Zeraoulia Rafik in 17/02/2017 and have been studied by Zeraoulia Rafik,Alvaro Humberto Salas Davide L.Ocampo and published in International Journal of Mathematics and Mathematical Sciences , [3], it behave like more than error function and it is defined as :

$\displaystyle \map T a = \int_0^a \paren {e^{-x^2} }^{\map \erf x} \rd x$

Mathematica gives for the first $100$ digits:

$\map T \infty = 0 \cdotp 9721069927691785931510778754423911755542721833855699009722910408441888759958220033410678218401258734$

with $\map T 0 = 0$

## Series expansion of Zeraoulia Function

Series expansion of zeraoulia function defined by $T(x)=\int_{0}^{x}e^{-\xi^2 \erf (\xi)}d\xi$ defined by this identity [4] (for proof see 4-page 5) :

$$\label{eq_4} \int\limits_{0}^{x}e^{-\xi ^{2} \erf (\xi )d\xi }=\sum\limits_{n=0}^{\infty }\lim_{\varepsilon ->0}\left( \sum\limits _{\substack{ k_{1}+2k_{2}+\cdots +nk_{n}=n \\ k_{1}\geq 0,k_{2}\geq 0,...,k_{n}\geq 0}}\prod\limits_{j=1}^{n}\frac{^{A_{j,\varepsilon }^{k_{j}}}% }{k_{j}!}\right) \frac{x^{n+1}}{n+1}$$ where: \begin{eqnarray*} A_{j,\epsilon } &=&\frac{2(-1)^{(j-1)/2}}{(j-2)(\frac{1}{2}(j-3))!\sqrt{\pi }% }\text{ if \ }j\geq 3\text{ and }j\text{ an odd integer;} \\ A_{j,\epsilon } &=&\varepsilon \text{ otherwise \ }(0<\varepsilon <1)\text{. } \end{eqnarray*}

## Source of Name

This entry was named for Zeraoulia Rafik.

## Sources

• E. W. Ng and M. Geller, “A table of integrals of the error functions,” Journal of Research of the National Bureau of Standards, vol. 73B, pp. 1–20, 1969
• E. W. Ng and M. Geller, “A table of integrals of the error functions,” Journal of Research of the National Bureau of Standards, vol. 73B, pp. 1–20, 1969. View at Google Scholar · View at MathSciNet
• Zeraoulia Rafik, Alvaro H. Salas, and David L. Ocampo, “A New Special Function and Its Application in Probability,” International Journal of Mathematics and Mathematical Sciences, vol. 2018, Article ID 5146794, 12 pages, 2018. https://doi.org/10.1155/2018/5146794