An asymptotic expansion for the incomplete beta function
Author:
B. G. S. Doman
Journal:
Math. Comp. 65 (1996), 12831288
MSC (1991):
Primary 33B20; Secondary 65D20
MathSciNet review:
1344611
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Abstract 
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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 .
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 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), 377393.
 3.
 , Significant Digit Computation of the Incomplete Beta Function Ratios, ACM Trans. Math. Software 18 (1992), 360373.
 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), 416431.
 5.
 J. L. Fields, A Note on the Asymptotic Expansion of the Ratio of Two Gamma Functions, Proc. Edinburgh Math. Soc. 15 (1966), 4355. 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), 890896. MR 88d:33001
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 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), 563575 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), 16381663. MR 89f:41036
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 M. E. Wise, The use of the Negative Binomial Distribution in an Industrial Sampling Problem, Suppl. J. Roy. Statist. Soc. 8 (1946), 202211. MR 9:49c
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 , The Incomplete Beta Function as a Contour Integral and a Quickly Converging Series for its Inverse, Biometrika 37 (1950), 208218. 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:
http://dx.doi.org/10.1090/S0025571896007296
PII:
S 00255718(96)007296
Keywords:
Gamma function ratio,
incomplete Beta function,
Chisquare 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
