An asymptotic expansion for

the incomplete beta function

Author:
B. G. S. Doman

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

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Abstract | References | Similar Articles | Additional Information

Abstract: A new asymptotic expansion is derived for the incomplete beta function , which is suitable for large , small and . This expansion is of the form

where is the incomplete Gamma function ratio and . This form has some advantages over previous asymptotic expansions in this region in which depends on as well as on and .

**1.**M. Abramowitz and I. A. Stegun,*Handbook of Mathematical Functions*, Dover, New York, 1970.**2.**A. R. Didonato and A. H. Morris,*Computation of the Incomplete Gamma Function Ratios and their Inverse*, ACM Trans. Math. Software**12**(1986), 377-393.**3.**------,*Significant Digit Computation of the Incomplete Beta Function Ratios*, ACM Trans. Math. Software**18**(1992), 360-373.**4.**B. G. S. Doman, C. J. Pursglove and W. M. Coen,*A Set of Ada Packages for High Precision Calculations*, ACM Trans. Math. Software**21**(1995), 416-431.**5.**J. L. Fields,*A Note on the Asymptotic Expansion of the Ratio of Two Gamma Functions*, Proc. Edinburgh Math. Soc.**15**(1966), 43-55. MR**34:379****6.**C. L. Frenzen,*Error Bounds for Asymptotic Expansions of the Ratio of Two Gamma Functions*, SIAM J. Math. Anal.**18**(1987), 890-896. MR**88d:33001****7.**Y. L. Luke,*The Special Functions and their Approximations*, Vol. I, Academic Press, New York, 1969. MR**39:3039****8.**E. C. Molina,*Expansions for Laplacian Integrals in Terms of Incomplete Gamma Functions*, International Congress of Mathematicians, Zurich, Bell System Technical Journal**11**(1932), 563-575 and Monograph B704.**9.**N. M. Temme,*Incomplete Laplace Integrals: Uniform Asymptotic Expansion with Application to the Incomplete Beta Function*, SIAM J. Math. Anal**18**(1987), 1638-1663. MR**89f:41036****10.**M. E. Wise,*The use of the Negative Binomial Distribution in an Industrial Sampling Problem*, Suppl. J. Roy. Statist. Soc.**8**(1946), 202-211. MR**9:49c****11.**------,*The Incomplete Beta Function as a Contour Integral and a Quickly Converging Series for its Inverse*, Biometrika**37**(1950), 208-218. MR**12:724e**

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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

DOI:
https://doi.org/10.1090/S0025-5718-96-00729-6

Keywords:
Gamma function ratio,
incomplete Beta function,
Chi-square distribution,
Student's distribution,
$F$ distribution

Received by editor(s):
March 16, 1995

Received by editor(s) in revised form:
June 26, 1995

Article copyright:
© Copyright 1996
American Mathematical Society