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1: 5.12 Beta Function
§5.12 Beta Function
Euler’s Beta Integral
See accompanying text
Figure 5.12.1: t -plane. Contour for first loop integral for the beta function. Magnify
See accompanying text
Figure 5.12.2: t -plane. Contour for second loop integral for the beta function. Magnify
Pochhammer’s Integral
2: 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
3: 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). …
4: 8.24 Physical Applications
§8.24 Physical Applications
§8.24(ii) Incomplete Beta Functions
5: 5.20 Physical Applications
§5.20 Physical Applications
Then the partition function (with β = 1 / ( k T ) ) is given by …
Elementary Particles
Veneziano (1968) identifies relationships between particle scattering amplitudes described by the beta function, an important early development in string theory. …
6: 8.26 Tables
§8.26(iii) Incomplete Beta Functions
7: 5.18 q -Gamma and q -Beta Functions
§5.18 q -Gamma and q -Beta Functions
§5.18(iii) q -Beta Function
5.18.11 B q ( a , b ) = Γ q ( a ) Γ q ( b ) Γ q ( a + b ) .
5.18.12 B q ( a , b ) = 0 1 t a - 1 ( t q ; q ) ( t q b ; q ) d q t , 0 < q < 1 , a > 0 , b > 0 .
8: 35.3 Multivariate Gamma and Beta Functions
§35.3 Multivariate Gamma and Beta Functions
35.3.3 B m ( a , b ) = 0 < X < I | X | a - 1 2 ( m + 1 ) | I - X | b - 1 2 ( m + 1 ) d X , ( a ) , ( b ) > 1 2 ( m - 1 ) .
35.3.6 Γ m ( a , , a ) = Γ m ( a ) .
35.3.7 B m ( a , b ) = Γ m ( a ) Γ m ( b ) Γ m ( a + b ) .
9: 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 ,
10: 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).