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Pochhammer integral

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1: 7.12 Asymptotic Expansions
7.12.2 f ( z ) 1 π z m = 0 ( - 1 ) m ( 1 2 ) 2 m ( π z 2 / 2 ) 2 m ,
7.12.3 g ( z ) 1 π z m = 0 ( - 1 ) m ( 1 2 ) 2 m + 1 ( π z 2 / 2 ) 2 m + 1 ,
7.12.4 f ( z ) = 1 π z m = 0 n - 1 ( - 1 ) m ( 1 2 ) 2 m ( π z 2 / 2 ) 2 m + R n ( f ) ( z ) ,
7.12.5 g ( z ) = 1 π z m = 0 n - 1 ( - 1 ) m ( 1 2 ) 2 m + 1 ( π z 2 / 2 ) 2 m + 1 , + R n ( g ) ( z ) ,
2: 17.13 Integrals
17.13.1 - c d ( - q x / c ; q ) ( q x / d ; q ) ( - a x / c ; q ) ( b x / d ; q ) d q x = ( 1 - q ) ( q ; q ) ( a b ; q ) c d ( - c / d ; q ) ( - d / c ; q ) ( a ; q ) ( b ; q ) ( c + d ) ( - b c / d ; q ) ( - a d / c ; q ) ,
17.13.2 - c d ( - q x / c ; q ) ( q x / d ; q ) ( - x q α / c ; q ) ( x q β / d ; q ) d q x = Γ q ( α ) Γ q ( β ) Γ q ( α + β ) c d c + d ( - c / d ; q ) ( - d / c ; q ) ( - q β c / d ; q ) ( - q α d / c ; q ) .
17.13.3 0 t α - 1 ( - t q α + β ; q ) ( - t ; q ) d t = Γ ( α ) Γ ( 1 - α ) Γ q ( β ) Γ q ( 1 - α ) Γ q ( α + β ) ,
17.13.4 0 t α - 1 ( - c t q α + β ; q ) ( - c t ; q ) d q t = Γ q ( α ) Γ q ( β ) ( - c q α ; q ) ( - q 1 - α / c ; q ) Γ q ( α + β ) ( - c ; q ) ( - q / c ; q ) .
3: 5.12 Beta Function
Pochhammer’s Integral
When a , b
See accompanying text
Figure 5.12.3: t -plane. Contour for Pochhammer’s integral. Magnify
4: 18.28 Askey–Wilson Class
18.28.8 1 2 π 0 π Q n ( cos θ ; a , b | q ) Q m ( cos θ ; a , b | q ) | ( e 2 i θ ; q ) ( a e i θ , b e i θ ; q ) | 2 d θ = δ n , m ( q n + 1 , a b q n ; q ) , a , b or a = b ¯ ; | a b | < 1 ; | a | , | b | 1 .
18.28.15 1 2 π 0 π C n ( cos θ ; β | q ) C m ( cos θ ; β | q ) | ( e 2 i θ ; q ) ( β e 2 i θ ; q ) | 2 d θ = ( β , β q ; q ) ( β 2 , q ; q ) ( 1 - β ) ( β 2 ; q ) n ( 1 - β q n ) ( q ; q ) n δ n , m , - 1 < β < 1 .
5: 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 .
6: 19.12 Asymptotic Approximations
19.12.1 K ( k ) = m = 0 ( 1 2 ) m ( 1 2 ) m m ! m ! k 2 m ( ln ( 1 k ) + d ( m ) ) , 0 < | k | < 1 ,
19.12.2 E ( k ) = 1 + 1 2 m = 0 ( 1 2 ) m ( 3 2 ) m ( 2 ) m m ! k 2 m + 2 ( ln ( 1 k ) + d ( m ) - 1 ( 2 m + 1 ) ( 2 m + 2 ) ) , | k | < 1 ,
7: 25.5 Integral Representations
25.5.7 ζ ( s ) = 1 2 + 1 s - 1 + m = 1 n B 2 m ( 2 m ) ! ( s ) 2 m - 1 + 1 Γ ( s ) 0 ( 1 e x - 1 - 1 x + 1 2 - m = 1 n B 2 m ( 2 m ) ! x 2 m - 1 ) x s - 1 e x d x , s > - ( 2 n + 1 ) , n = 1 , 2 , 3 , .
8: 8.20 Asymptotic Expansions of E p ( z )
8.20.1 E p ( z ) = e - z z ( k = 0 n - 1 ( - 1 ) k ( p ) k z k + ( - 1 ) n ( p ) n e z z n - 1 E n + p ( z ) ) , n = 1 , 2 , 3 , .
8.20.3 E p ( z ) ± 2 π i Γ ( p ) e p π i z p - 1 + e - z z k = 0 ( - 1 ) k ( p ) k z k , 1 2 π + δ ± ph z 7 2 π - δ ,
9: 8.21 Generalized Sine and Cosine Integrals
8.21.16 Si ( a , z ) = z a k = 0 ( 2 k + 3 2 ) ( 1 - 1 2 a ) k ( 1 2 + 1 2 a ) k + 1 j 2 k + 1 ( z ) , a - 1 , - 3 , - 5 , ,
8.21.17 Ci ( a , z ) = z a k = 0 ( 2 k + 1 2 ) ( 1 2 - 1 2 a ) k ( 1 2 a ) k + 1 j 2 k ( z ) , a 0 , - 2 , - 4 , .
10: 25.11 Hurwitz Zeta Function
25.11.28 ζ ( s , a ) = 1 2 a - s + a 1 - s s - 1 + k = 1 n B 2 k ( 2 k ) ! ( s ) 2 k - 1 a 1 - s - 2 k + 1 Γ ( s ) 0 ( 1 e x - 1 - 1 x + 1 2 - k = 1 n B 2 k ( 2 k ) ! x 2 k - 1 ) x s - 1 e - a x d x , s > - ( 2 n + 1 ) , s 1 , a > 0 .