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11: 35.8 Generalized Hypergeometric Functions of Matrix Argument
Euler Integral
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 ) .
12: 35.4 Partitions and Zonal Polynomials
13: 6.2 Definitions and Interrelations
6.2.7 Ei ( ± x ) = - Ein ( x ) + ln x + γ .
6.2.13 Ci ( z ) = - Cin ( z ) + ln z + γ .
6.2.16 Chi ( z ) = γ + ln z + 0 z cosh t - 1 t d t .
14: 24.17 Mathematical Applications
15: 31.9 Orthogonality
31.9.2 ζ ( 1 + , 0 + , 1 - , 0 - ) t γ - 1 ( 1 - t ) δ - 1 ( t - a ) ϵ - 1 w m ( t ) w k ( t ) d t = δ m , k θ m .
16: 25.2 Definition and Expansions
25.2.8 ζ ( s ) = k = 1 N 1 k s + N 1 - s s - 1 - s N x - x x s + 1 d x , s > 0 , N = 1 , 2 , 3 , .
25.2.9 ζ ( s ) = k = 1 N 1 k s + N 1 - s s - 1 - 1 2 N - s + k = 1 n ( s + 2 k - 2 2 k - 1 ) B 2 k 2 k N 1 - s - 2 k - ( s + 2 n 2 n + 1 ) N B ~ 2 n + 1 ( x ) x s + 2 n + 1 d x , s > - 2 n ; n , N = 1 , 2 , 3 , .
25.2.10 ζ ( s ) = 1 s - 1 + 1 2 + k = 1 n ( s + 2 k - 2 2 k - 1 ) B 2 k 2 k - ( s + 2 n 2 n + 1 ) 1 B ~ 2 n + 1 ( x ) x s + 2 n + 1 d x , s > - 2 n , n = 1 , 2 , 3 , .
17: 19.11 Addition Theorems
19.11.6_5 R C ( γ - δ , γ ) = - 1 δ arctan ( δ sin θ sin ϕ sin ψ α 2 - 1 - α 2 cos θ cos ϕ cos ψ ) .
18: 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 | .
19: 25.5 Integral Representations
25.5.1 ζ ( s ) = 1 Γ ( s ) 0 x s - 1 e x - 1 d x , s > 1 .
25.5.3 ζ ( s ) = 1 ( 1 - 2 1 - s ) Γ ( s ) 0 x s - 1 e x + 1 d x , s > 0 .
25.5.6 ζ ( s ) = 1 2 + 1 s - 1 + 1 Γ ( s ) 0 ( 1 e x - 1 - 1 x + 1 2 ) x s - 1 e x d x , s > - 1 .
25.5.8 ζ ( s ) = 1 2 ( 1 - 2 - s ) Γ ( s ) 0 x s - 1 sinh x d x , s > 1 .
25.5.9 ζ ( s ) = 2 s - 1 Γ ( s + 1 ) 0 x s ( sinh x ) 2 d x , s > 1 .
20: 19.18 Derivatives and Differential Equations
and two similar equations obtained by permuting x , y , z in (19.18.10). More concisely, if v = R - a ( b ; z ) , then each of (19.16.14)–(19.16.18) and (19.16.20)–(19.16.23) satisfies Euler’s homogeneity relation: … The next four differential equations apply to the complete case of R F and R G in the form R - a ( 1 2 , 1 2 ; z 1 , z 2 ) (see (19.16.20) and (19.16.23)). The function w = R - a ( 1 2 , 1 2 ; x + y , x - y ) satisfies an Euler–Poisson–Darboux equation: …