An asymptotic expansion for the incomplete beta function
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- by B. G. S. Doman PDF
- Math. Comp. 65 (1996), 1283-1288 Request permission
Abstract:
A new asymptotic expansion is derived for the incomplete beta function $I(a,b,x)$, which is suitable for large $a$, small $b$ and $x > 0.5$. This expansion is of the form \begin{equation*}I(a,b,x) \quad \sim \quad Q(b, -\gamma \log x) + {\frac {\Gamma (a + b)}{\Gamma (a) \Gamma (b)}} x^{\gamma } \sum ^{\infty }_{n=0}T_{n}(b,x)/ \gamma ^{n+1} , \end{equation*} where $Q$ is the incomplete Gamma function ratio and $\gamma = a + (b - 1)/2$ . This form has some advantages over previous asymptotic expansions in this region in which $T_{n}$ depends on $a$ as well as on $b$ and $x$.References
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Additional Information
- B. G. S. Doman
- Affiliation: Department of Mathematical Sciences, University of Liverpool, PO Box 147, Liverpool L69 3BX, England
- Email: doman@liv.ac.uk
- Received by editor(s): March 16, 1995
- Received by editor(s) in revised form: June 26, 1995
- © Copyright 1996 American Mathematical Society
- Journal: Math. Comp. 65 (1996), 1283-1288
- MSC (1991): Primary 33B20; Secondary 65D20
- DOI: https://doi.org/10.1090/S0025-5718-96-00729-6
- MathSciNet review: 1344611