Uniform representations of the incomplete beta function in terms of elementary functions

Ferreira, Chelo (Universidad de Zaragoza) ; López, José L. ; Pérez Sinusía, Ester (Universidad de Zaragoza)

Resumen: We consider the incomplete beta function $B_{z}(a,b)$ in the maximum domain ofanalyticity of its three variables: $a,b,z\\in\\mathbb{C}$, $-a\\notin\\mathbb{N}$,$z\\notin[1,\\infty)$. For $\\Re b\\le 1$ we derive a convergent expansion of$z^{-a}B_{z}(a,b)$ in terms of the function $(1-z)^b$ and of rational functionsof $z$ that is uniformly valid for $z$ in any compact set in$\\mathbb{C}\\setminus[1,\\infty)$. When $-b\\in \\mathbb{N}\\cup\\{0\\}$, the expansionalso contains a logarithmic term of the form $\\log(1-z)$. For $\\Re b\\ge 1$ wederive a convergent expansion of $z^{-a}(1-z)^bB_{z}(a,b)$ in terms of thefunction $(1-z)^b$ and of rational functions of $z$ that is uniformly valid for$z$ in any compact set in the exterior of the circle $\\vert z-1\\vert=r$ forarbitrary $r>0$. The expansions are accompanied by realistic error bounds. Somenumerical experiments show the accuracy of the approximations.

Idioma: Inglés

DOI: 10.1553/etna_vol48s450

Año: 2018

Publicado en: ELECTRONIC TRANSACTIONS ON NUMERICAL ANALYSIS 48 (2018), 450-461
ISSN: 1068-9613

Factor impacto JCR: 1.475 (2018)

Categ. JCR: MATHEMATICS, APPLIED rank: 76 / 254 = 0.299 (2018) - Q2 - T1

Factor impacto SCIMAGO: 0.953 - Analysis (Q2)

Financiación: info:eu-repo/grantAgreement/ES/DGA-FEDER/E24-17R
Financiación: info:eu-repo/grantAgreement/ES/IUMA/MTM2017-83490-P

Tipo y forma: Artículo (Versión definitiva)

Área (Departamento): Área Matemática Aplicada (Dpto. Matemática Aplicada)