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1: 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 ) .
2: 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 ) .
3: 35.4 Partitions and Zonal Polynomials
4: 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 ) .
5.12.3 0 t a - 1 d t ( 1 + t ) a + b = B ( a , b ) .
5.12.4 0 1 t a - 1 ( 1 - t ) b - 1 ( t + z ) a + b d t = B ( a , b ) ( 1 + z ) - a z - b , | ph z | < π .
5.12.7 0 cosh ( 2 b t ) ( cosh t ) 2 a d t = 4 a - 1 B ( a + b , a - b ) , a > | b | .
5: 35.7 Gaussian Hypergeometric Function of Matrix Argument
6: 35.8 Generalized Hypergeometric Functions of Matrix Argument
35.8.13 0 < X < I | X | a 1 - 1 2 ( m + 1 ) | I - X | b 1 - a 1 - 1 2 ( m + 1 ) F q p ( a 2 , , a p + 1 b 2 , , b q + 1 ; T X ) d X = 1 B m ( b 1 - a 1 , a 1 ) F q + 1 p + 1 ( a 1 , , a p + 1 b 1 , , b q + 1 ; T ) , ( b 1 - a 1 ) , ( a 1 ) > 1 2 ( m - 1 ) .
7: 19.28 Integrals of Elliptic Integrals
19.28.1 0 1 t σ - 1 R F ( 0 , t , 1 ) d t = 1 2 ( B ( σ , 1 2 ) ) 2 ,
19.28.2 0 1 t σ - 1 R G ( 0 , t , 1 ) d t = σ 4 σ + 2 ( B ( σ , 1 2 ) ) 2 ,
19.28.3 0 1 t σ - 1 ( 1 - t ) R D ( 0 , t , 1 ) d t = 3 4 σ + 2 ( B ( σ , 1 2 ) ) 2 .
8: 5.13 Integrals
§5.13 Integrals
Barnes’ Beta Integral
Ramanujan’s Beta Integral
de Branges–Wilson Beta Integral
9: 19.33 Triaxial Ellipsoids
19.33.11 U = 1 2 ( α β γ ) 2 R F ( α 2 , β 2 , γ 2 ) 0 ( g ( r ) ) 2 d r ,
10: 16.15 Integral Representations and Integrals
16.15.1 F 1 ( α ; β , β ; γ ; x , y ) = Γ ( γ ) Γ ( α ) Γ ( γ - α ) 0 1 u α - 1 ( 1 - u ) γ - α - 1 ( 1 - u x ) β ( 1 - u y ) β d u , α > 0 , ( γ - α ) > 0 ,
16.15.2 F 2 ( α ; β , β ; γ , γ ; x , y ) = Γ ( γ ) Γ ( γ ) Γ ( β ) Γ ( β ) Γ ( γ - β ) Γ ( γ - β ) 0 1 0 1 u β - 1 v β - 1 ( 1 - u ) γ - β - 1 ( 1 - v ) γ - β - 1 ( 1 - u x - v y ) α d u d v , γ > β > 0 , γ > β > 0 ,
16.15.3 F 3 ( α , α ; β , β ; γ ; x , y ) = Γ ( γ ) Γ ( β ) Γ ( β ) Γ ( γ - β - β ) Δ u β - 1 v β - 1 ( 1 - u - v ) γ - β - β - 1 ( 1 - u x ) α ( 1 - v y ) α d u d v , ( γ - β - β ) > 0 , β > 0 , β > 0 ,
16.15.4 F 4 ( α , β ; γ , γ ; x ( 1 - y ) , y ( 1 - x ) ) = Γ ( γ ) Γ ( γ ) Γ ( α ) Γ ( β ) Γ ( γ - α ) Γ ( γ - β ) 0 1 0 1 u α - 1 v β - 1 ( 1 - u ) γ - α - 1 ( 1 - v ) γ - β - 1 ( 1 - u x ) γ + γ - α - 1 ( 1 - v y ) γ + γ - β - 1 ( 1 - u x - v y ) α + β - γ - γ + 1 d u d v , γ > α > 0 , γ > β > 0 .