Uniform representation of the incomplete beta function in terms of elementary functions
Ver/
Fecha
2018Versión
Acceso abierto / Sarbide irekia
Tipo
Artículo / Artikulua
Versión
Versión aceptada / Onetsi den bertsioa
Identificador del proyecto
Impacto
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10.1553/etna_vol48s450
Resumen
We consider the incomplete beta function Bz(a, b) in the maximum domain of analyticity of its three variables: a, b, z ∈ C, −a /∈ N, z /∈ [1, ∞). For <b ≤ 1 we derive a convergent expansion of z−aBz(a, b) in terms of the function (1 − z) b
and of rational functions of z that is uniformly valid for z in any compact set in C \ [1, ∞). When −b ∈ N ∪ {0}, the expansion also contains a logarithmic te ...
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We consider the incomplete beta function Bz(a, b) in the maximum domain of analyticity of its three variables: a, b, z ∈ C, −a /∈ N, z /∈ [1, ∞). For <b ≤ 1 we derive a convergent expansion of z−aBz(a, b) in terms of the function (1 − z) b
and of rational functions of z that is uniformly valid for z in any compact set in C \ [1, ∞). When −b ∈ N ∪ {0}, the expansion also contains a logarithmic term of the form log(1 − z). For <b ≥ 1 we derive a convergent expansion of z−a(1 − z) bBz(a, b) in terms of the function (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 |z − 1| = r for arbitrary r > 0. The expansions are accompanied by realistic error bounds. Some numerical experiments show the accuracy of
the approximations. [--]
Materias
Incomplete beta function,
Convergent expansions,
Uniform expansions
Editor
Kent State University Johann Radon Institute (RICAM)
Publicado en
Electronic Transactions on Numerical Analysis, Volume 48, pp. 450–461, 2018.
Departamento
Universidad Pública de Navarra. Departamento de Ingeniería Matemática e Informática /
Nafarroako Unibertsitate Publikoa. Matematika eta Informatika Ingeniaritza Saila /
Universidad Pública de Navarra/Nafarroako Unibertsitate Publikoa. Institute for Advanced Materials and Mathematics - INAMAT2
Versión del editor
Entidades Financiadoras
This research was supported by Ministerio de Economía, Industria
y Competitividad, Gobierno de España, project MTM2017-83490-P, Gobierno de Aragón and
European Social Fund (group E24-17R).