Volume 48, pp. 450-461, 2018.

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

Chelo Ferreira, José L. López, and Ester Pérez Sinusía

Abstract

We consider the incomplete beta function $B_z(a,b)$ in the maximum domain of analyticity of its three variables: $a,b,z\in \mathbb{C}$, $-a\notin \mathbb{N}$, $z\notin [1,\mathrm{\infty })$. For $\mathrm{\Re }b\le 1$ we derive a convergent expansion of $z^{-a}{B}_z(a,b)$ in terms of the function $(1-z{)}^b$ and of rational functions of $z$ that is uniformly valid for $z$ in any compact set in $\mathbb{C}\setminus [1,\mathrm{\infty })$. When $-b\in \mathbb{N}\cup \{0\}$, the expansion also contains a logarithmic term of the form $\log (1-z)$. For $\mathrm{\Re }b\ge 1$ we derive a convergent expansion of $z^{-a}(1-z{)}^b{B}_z(a,b)$ 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.

Full Text (PDF) [439 KB], BibTeX , DOI: 10.1553/etna_vol48s450

Key words

incomplete beta function, convergent expansions, uniform expansions

AMS subject classifications

33B20, 41A58, 41A80

ETNA articles which cite this article

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