## 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: $a,b,z\in\mathbb{C}$, $-a\notin\mathbb{N}$, $z\notin[1,\infty)$. 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 functions of $z$ that is uniformly valid for $z$ in any compact set in $\mathbb{C}\setminus[1,\infty)$. When $-b\in \mathbb{N}\cup\{0\}$, the expansion also contains a logarithmic term of the form $\log(1-z)$. For $\Re b\ge 1$ we derive a convergent expansion of $z^{-a}(1-z)^bB_{z}(a,b)$ 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 $\vert z-1\vert=r$ for arbitrary $r>0$. The expansions are accompanied by realistic error bounds. Some numerical experiments show the accuracy of the approximations.

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### Key words

incomplete beta function, convergent expansions, uniform expansions

### AMS subject classifications

33B20, 41A58, 41A80

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