# Definition:Zeraoulia Function

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## Definition

The **Zeraoulia function** is a special function defined as:

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

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 $\ds \map T x = \int_0^x e^{-\xi^2 \map \erf \xi} \rd \xi$ defined by:

- $\ds \int_0^x e^{-\xi^e \map \erf \xi} \rd \xi = \sum_{n \mathop = 0}^\infty \lim_{\varepsilon \mathop \to 0} \paren {\sum_{\substack {k_1 + 2 k_2 + \cdots + n k_n \mathop = n \\ k_1, k_2, \ldots, k_n \mathop \ge 0} } \prod_{j \mathop = 1}^n \frac {A_{j, \varepsilon}^{k_j} } {k_j!} } \frac {x^{n + 1} } {n + 1}$

where:

- $A_{j, \epsilon} = \begin {cases} \dfrac {2 \paren {-1}^{\paren {j - 1} / 2} } {\paren {j - 2} \paren {\paren {j - 3} / 2}! \sqrt \pi} & : j \ge 3 \text { and } j \text { odd} \\ \varepsilon & : \text{ otherwise} \paren {0 < \varepsilon < 1} \end {cases}$

## Source of Name

This entry was named for Zeraoulia Rafik.

## Sources

- 1969: Edward W. Ng and Murray Geller:
*A table of integrals of the Error functions*(*J. Res. Natl. Bur. Stand.***Vol. 73B**: pp. 1 – 20)

- 2018: Zeraoulia Rafik, Alvaro H. Salas and David L. Ocampo:
*A New Special Function and Its Application in Probability*(*Int. J. Math. Math. Sci.***Vol. 2018**: p. 12 pages)