About the Project
NIST

beta integrals

AdvancedHelp

(0.003 seconds)

1—10 of 98 matching pages

1: 5.13 Integrals
§5.13 Integrals
Barnes’ Beta Integral
Ramanujan’s Beta Integral
de Branges–Wilson Beta Integral
2: 5.12 Beta Function
Euler’s Beta Integral
5.12.1 B ( a , b ) = 0 1 t a - 1 ( 1 - t ) b - 1 d t = Γ ( a ) Γ ( b ) Γ ( a + b ) .
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
3: 35.4 Partitions and Zonal Polynomials
4: 19.7 Connection Formulas
F ( ϕ , k 1 ) = k F ( β , k ) ,
E ( ϕ , k 1 ) = ( E ( β , k ) - k 2 F ( β , k ) ) / k ,
Π ( ϕ , α 2 , k 1 ) = k Π ( β , k 2 α 2 , k ) , k 1 = 1 / k , sin β = k 1 sin ϕ 1 .
5: Bibliography I
  • M. E. H. Ismail and D. R. Masson (1994) q -Hermite polynomials, biorthogonal rational functions, and q -beta integrals. Trans. Amer. Math. Soc. 346 (1), pp. 63–116.
  • 6: 29.18 Mathematical Applications
    β = K + i β ,
    0 β 2 K ,
    β = K + i β , 0 β 2 K , 0 γ 4 K ,
    7: 5.14 Multidimensional Integrals
    §5.14 Multidimensional Integrals
    8: 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 ) .
    9: 5.18 q -Gamma and q -Beta Functions
    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 .
    10: 35.6 Confluent Hypergeometric Functions of Matrix Argument
    35.6.6 B m ( b 1 , b 2 ) | T | b 1 + b 2 - 1 2 ( m + 1 ) F 1 1 ( a 1 + a 2 b 1 + b 2 ; T ) = 0 < X < T | X | b 1 - 1 2 ( m + 1 ) F 1 1 ( a 1 b 1 ; X ) | T - X | b 2 - 1 2 ( m + 1 ) F 1 1 ( a 2 b 2 ; T - X ) d X , ( b 1 ) , ( b 2 ) > 1 2 ( m - 1 ) .