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1: 17.13 Integrals
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17.13.2 c d ( q ⁒ x / c ; q ) ⁒ ( q ⁒ x / d ; q ) ( x ⁒ q Ξ± / c ; q ) ⁒ ( x ⁒ q Ξ² / d ; q ) ⁒ d q x = Ξ“ q ⁑ ( Ξ± ) ⁒ Ξ“ q ⁑ ( Ξ² ) Ξ“ q ⁑ ( Ξ± + Ξ² ) ⁒ c ⁒ d c + d ⁒ ( c / d ; q ) ⁒ ( d / c ; q ) ( q Ξ² ⁒ c / d ; q ) ⁒ ( q Ξ± ⁒ d / c ; q ) .
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17.13.3 0 t Ξ± 1 ⁒ ( t ⁒ q Ξ± + Ξ² ; q ) ( t ; q ) ⁒ d t = Ξ“ ⁑ ( Ξ± ) ⁒ Ξ“ ⁑ ( 1 Ξ± ) ⁒ Ξ“ q ⁑ ( Ξ² ) Ξ“ q ⁑ ( 1 Ξ± ) ⁒ Ξ“ q ⁑ ( Ξ± + Ξ² ) ,
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17.13.4 0 t Ξ± 1 ⁒ ( c ⁒ t ⁒ q Ξ± + Ξ² ; q ) ( c ⁒ t ; q ) ⁒ d q t = Ξ“ q ⁑ ( Ξ± ) ⁒ Ξ“ q ⁑ ( Ξ² ) ⁒ ( c ⁒ q Ξ± ; q ) ⁒ ( q 1 Ξ± / c ; q ) Ξ“ q ⁑ ( Ξ± + Ξ² ) ⁒ ( c ; q ) ⁒ ( q / c ; q ) .
2: 5.2 Definitions
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Euler’s Integral
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5.2.1 Ξ“ ⁑ ( z ) = 0 e t ⁒ t z 1 ⁒ d t , ⁑ z > 0 .
3: 6.11 Relations to Other Functions
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6.11.1 E 1 ⁑ ( z ) = Ξ“ ⁑ ( 0 , z ) .
4: 35.3 Multivariate Gamma and Beta Functions
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35.3.1 Ξ“ m ⁑ ( a ) = 𝛀 etr ⁑ ( 𝐗 ) ⁒ | 𝐗 | a 1 2 ⁒ ( m + 1 ) ⁒ d 𝐗 , ⁑ ( a ) > 1 2 ⁒ ( m 1 ) .
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35.3.2 Ξ“ m ⁑ ( s 1 , , s m ) = 𝛀 etr ⁑ ( 𝐗 ) ⁒ | 𝐗 | s m 1 2 ⁒ ( m + 1 ) ⁒ j = 1 m 1 | ( 𝐗 ) j | s j s j + 1 ⁒ d 𝐗 , s j β„‚ , ⁑ ( s j ) > 1 2 ⁒ ( j 1 ) , j = 1 , , m .
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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 ) .
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35.3.8 B m ⁑ ( a , b ) = 𝛀 | 𝐗 | a 1 2 ⁒ ( m + 1 ) ⁒ | 𝐈 + 𝐗 | ( a + b ) ⁒ d 𝐗 , ⁑ ( a ) , ⁑ ( b ) > 1 2 ⁒ ( m 1 ) .
5: 35.6 Confluent Hypergeometric Functions of Matrix Argument
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35.6.2 Ξ¨ ⁑ ( a ; b ; 𝐓 ) = 1 Ξ“ m ⁑ ( a ) ⁒ 𝛀 etr ⁑ ( 𝐓 ⁒ 𝐗 ) ⁒ | 𝐗 | a 1 2 ⁒ ( m + 1 ) ⁒ | 𝐈 + 𝐗 | b a 1 2 ⁒ ( m + 1 ) ⁒ d 𝐗 , ⁑ ( a ) > 1 2 ⁒ ( m 1 ) , 𝐓 𝛀 .
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35.6.4 F 1 1 ⁑ ( a b ; 𝐓 ) = 1 B m ⁑ ( a , b a ) ⁒ 𝟎 < 𝐗 < 𝐈 etr ⁑ ( 𝐓 ⁒ 𝐗 ) ⁒ | 𝐗 | a 1 2 ⁒ ( m + 1 ) ⁒ | 𝐈 𝐗 | b a 1 2 ⁒ ( m + 1 ) ⁒ d 𝐗 , ⁑ ( a ) , ⁑ ( b a ) > 1 2 ⁒ ( m 1 ) .
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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 ) .
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35.6.8 𝛀 | 𝐓 | c 1 2 ⁒ ( m + 1 ) ⁒ Ξ¨ ⁑ ( a ; b ; 𝐓 ) ⁒ d 𝐓 = Ξ“ m ⁑ ( c ) ⁒ Ξ“ m ⁑ ( a c ) ⁒ Ξ“ m ⁑ ( c b + 1 2 ⁒ ( m + 1 ) ) Ξ“ m ⁑ ( a ) ⁒ Ξ“ m ⁑ ( a b + 1 2 ⁒ ( m + 1 ) ) , ⁑ ( a ) > ⁑ ( c ) + 1 2 ⁒ ( m 1 ) > m 1 , ⁑ ( c b ) > 1 .
6: 24.13 Integrals
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§24.13(ii) Euler Polynomials
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24.13.7 E n ⁑ ( t ) ⁒ d t = E n + 1 ⁑ ( t ) n + 1 + const. ,
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24.13.8 0 1 E n ⁑ ( t ) ⁒ d t = 2 ⁒ E n + 1 ⁑ ( 0 ) n + 1 = 4 ⁒ ( 2 n + 2 1 ) ( n + 1 ) ⁒ ( n + 2 ) ⁒ B n + 2 ,
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24.13.10 0 1 / 2 E 2 ⁒ n 1 ⁑ ( t ) ⁒ d t = E 2 ⁒ n n ⁒ 2 2 ⁒ n + 1 , n = 1 , 2 , .
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§24.13(iii) Compendia
7: 8.22 Mathematical Applications
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8.22.1 F p ⁑ ( z ) = Ξ“ ⁑ ( p ) 2 ⁒ Ο€ ⁒ z 1 p ⁒ E p ⁑ ( z ) = Ξ“ ⁑ ( p ) 2 ⁒ Ο€ ⁒ Ξ“ ⁑ ( 1 p , z ) ,
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8.22.2 ΞΆ x ⁑ ( s ) = 1 Ξ“ ⁑ ( s ) ⁒ 0 x t s 1 e t 1 ⁒ d t , ⁑ s > 1 ,
8: 24.7 Integral Representations
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§24.7(i) Bernoulli and Euler Numbers
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24.7.5 B 2 ⁒ n = ( 1 ) n ⁒ 2 ⁒ n ⁒ ( 2 ⁒ n 1 ) Ο€ ⁒ 0 t 2 ⁒ n 2 ⁒ ln ⁑ ( 1 e 2 ⁒ Ο€ ⁒ t ) ⁒ d t .
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24.7.6 E 2 ⁒ n = ( 1 ) n ⁒ 2 2 ⁒ n + 1 ⁒ 0 t 2 ⁒ n ⁒ sech ⁑ ( Ο€ ⁒ t ) ⁒ d t .
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§24.7(ii) Bernoulli and Euler Polynomials
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24.7.9 E 2 ⁒ n ⁑ ( x ) = ( 1 ) n ⁒ 4 ⁒ 0 sin ⁑ ( Ο€ ⁒ x ) ⁒ cosh ⁑ ( Ο€ ⁒ t ) cosh ⁑ ( 2 ⁒ Ο€ ⁒ t ) cos ⁑ ( 2 ⁒ Ο€ ⁒ x ) ⁒ t 2 ⁒ n ⁒ d t ,
9: 6.6 Power Series
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6.6.1 Ei ⁑ ( x ) = γ + ln ⁑ x + n = 1 x n n ! ⁒ n , x > 0 .
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6.6.2 E 1 ⁑ ( z ) = γ ln ⁑ z n = 1 ( 1 ) n ⁒ z n n ! ⁒ n .
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6.6.6 Ci ⁑ ( z ) = γ + ln ⁑ z + n = 1 ( 1 ) n ⁒ z 2 ⁒ n ( 2 ⁒ n ) ! ⁒ ( 2 ⁒ n ) .
10: 8.4 Special Values
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8.4.1 Ξ³ ⁑ ( 1 2 , z 2 ) = 2 ⁒ 0 z e t 2 ⁒ d t = Ο€ ⁒ erf ⁑ ( z ) ,
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8.4.4 Ξ“ ⁑ ( 0 , z ) = z t 1 ⁒ e t ⁒ d t = E 1 ⁑ ( z ) ,
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8.4.6 Ξ“ ⁑ ( 1 2 , z 2 ) = 2 ⁒ z e t 2 ⁒ d t = Ο€ ⁒ erfc ⁑ ( z ) .
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8.4.13 Ξ“ ⁑ ( 1 n , z ) = z 1 n ⁒ E n ⁑ ( z ) ,
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8.4.15 Ξ“ ⁑ ( n , z ) = ( 1 ) n n ! ⁒ ( E 1 ⁑ ( z ) e z ⁒ k = 0 n 1 ( 1 ) k ⁒ k ! z k + 1 ) = ( 1 ) n n ! ⁒ ( ψ ⁑ ( n + 1 ) ln ⁑ z ) z n ⁒ k = 0 k n ( z ) k k ! ⁒ ( k n ) .