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1: 8.17 Incomplete Beta Functions
§8.17 Incomplete Beta Functions
§8.17(ii) Hypergeometric Representations
§8.17(iii) Integral Representation
§8.17(iv) Recurrence Relations
§8.17(vi) Sums
2: 8.23 Statistical Applications
§8.23 Statistical Applications
The function B x ( a , b ) and its normalization I x ( a , b ) play a similar role in statistics in connection with the beta distribution; see Johnson et al. (1995, pp. 210–275). …
3: 8.24 Physical Applications
§8.24 Physical Applications
§8.24(ii) Incomplete Beta Functions
4: 8.1 Special Notation
Unless otherwise indicated, primes denote derivatives with respect to the argument. The functions treated in this chapter are the incomplete gamma functions γ ( a , z ) , Γ ( a , z ) , γ * ( a , z ) , P ( a , z ) , and Q ( a , z ) ; the incomplete beta functions B x ( a , b ) and I x ( a , b ) ; the generalized exponential integral E p ( z ) ; the generalized sine and cosine integrals si ( a , z ) , ci ( a , z ) , Si ( a , z ) , and Ci ( a , z ) . Alternative notations include: Prym’s functions P z ( a ) = γ ( a , z ) , Q z ( a ) = Γ ( a , z ) , Nielsen (1906a, pp. 25–26), Batchelder (1967, p. 63); ( a , z ) ! = γ ( a + 1 , z ) , [ a , z ] ! = Γ ( a + 1 , z ) , Dingle (1973); B ( a , b , x ) = B x ( a , b ) , I ( a , b , x ) = I x ( a , b ) , Magnus et al. (1966); Si ( a , x ) Si ( 1 - a , x ) , Ci ( a , x ) Ci ( 1 - a , x ) , Luke (1975).
5: 8.26 Tables
§8.26(iii) Incomplete Beta Functions
6: 8.18 Asymptotic Expansions of I x ( a , b )
8.18.1 I x ( a , b ) = Γ ( a + b ) x a ( 1 - x ) b - 1 ( k = 0 n - 1 1 Γ ( a + k + 1 ) Γ ( b - k ) ( x 1 - x ) k + O ( 1 Γ ( a + n + 1 ) ) ) ,
Symmetric Case
General Case
Inverse Function
8.18.18 I x ( a , b ) = p , 0 p 1 ,
7: 8.28 Software
§8.28(iv) Incomplete Beta Functions for Real Argument and Parameters
§8.28(v) Incomplete Beta Functions for Complex Argument and Parameters
8: Bibliography P
  • K. Pearson (Ed.) (1968) Tables of the Incomplete Beta-function. 2nd edition, Published for the Biometrika Trustees at the Cambridge University Press, Cambridge.
  • H. N. Phien (1990) A note on the computation of the incomplete beta function. Adv. Eng. Software 12 (1), pp. 39–44.
  • 9: Bibliography N
  • G. Nemes and A. B. Olde Daalhuis (2016) Uniform asymptotic expansion for the incomplete beta function. SIGMA Symmetry Integrability Geom. Methods Appl. 12, pp. 101, 5 pages.
  • 10: Bibliography D
  • A. R. DiDonato and A. H. Morris (1992) Algorithm 708: Significant digit computation of the incomplete beta function ratios. ACM Trans. Math. Software 18 (3), pp. 360–373.
  • J. Dutka (1981) The incomplete beta function—a historical profile. Arch. Hist. Exact Sci. 24 (1), pp. 11–29.