Definition:Zeraoulia Function
Jump to navigation
Jump to search
 | This page has been identified as a candidate for refactoring of advanced complexity. In particular: and a general restructure and rewrite to match house style Until this has been finished, please leave {{Refactor}} in the code.
New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Refactor}} from the code. |
 | This article needs to be tidied. Please fix formatting and $\LaTeX$ errors and inconsistencies. It may also need to be brought up to our standard house style. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Tidy}} from the code. |
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)