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1: 17.13 Integrals
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 ) .
2: 5.2 Definitions
Euler’s Integral
5.2.1 Γ ( z ) = 0 e - t t z - 1 d t , z > 0 .
3: 6.11 Relations to Other Functions
6.11.1 E 1 ( z ) = Γ ( 0 , z ) .
4: 35.3 Multivariate Gamma and Beta Functions
35.3.2 Γ 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 , ( s j ) > 1 2 ( j - 1 ) , j = 1 , , m .
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 ) .
5: 24.13 Integrals
§24.13(ii) Euler Polynomials
24.13.7 E n ( t ) d t = E n + 1 ( t ) n + 1 + const. ,
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 ,
24.13.10 0 1 / 2 E 2 n - 1 ( t ) d t = E 2 n n 2 2 n + 1 , n = 1 , 2 , .
§24.13(iii) Compendia
6: 35.6 Confluent Hypergeometric Functions of Matrix Argument
35.6.2 Ψ ( a ; b ; T ) = 1 Γ 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 .
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 ) .
35.6.8 Ω | T | c - 1 2 ( m + 1 ) Ψ ( a ; b ; T ) d T = Γ m ( c ) Γ m ( a - c ) Γ m ( c - b + 1 2 ( m + 1 ) ) Γ m ( a ) Γ m ( a - b + 1 2 ( m + 1 ) ) , ( a ) > ( c ) + 1 2 ( m - 1 ) > m - 1 , ( c - b ) > - 1 .
7: 8.22 Mathematical Applications
8.22.1 F p ( z ) = Γ ( p ) 2 π z 1 - p E p ( z ) = Γ ( p ) 2 π Γ ( 1 - p , z ) ,
8.22.2 ζ x ( s ) = 1 Γ ( s ) 0 x t s - 1 e t - 1 d t , s > 1 ,
8: 24.7 Integral Representations
§24.7(i) Bernoulli and Euler Numbers
24.7.5 B 2 n = ( - 1 ) n 2 n ( 2 n - 1 ) π 0 t 2 n - 2 ln ( 1 - e - 2 π t ) d t .
24.7.6 E 2 n = ( - 1 ) n 2 2 n + 1 0 t 2 n sech ( π t ) d t .
§24.7(ii) Bernoulli and Euler Polynomials
24.7.9 E 2 n ( x ) = ( - 1 ) n 4 0 sin ( π x ) cosh ( π t ) cosh ( 2 π t ) - cos ( 2 π x ) t 2 n d t ,
9: 6.6 Power Series
6.6.1 Ei ( x ) = γ + ln x + n = 1 x n n ! n , x > 0 .
6.6.2 E 1 ( z ) = - γ - ln z - n = 1 ( - 1 ) n z n n ! n .
6.6.6 Ci ( z ) = γ + ln z + n = 1 ( - 1 ) n z 2 n ( 2 n ) ! ( 2 n ) .
10: 8.4 Special Values
8.4.1 γ ( 1 2 , z 2 ) = 2 0 z e - t 2 d t = π erf ( z ) ,
8.4.4 Γ ( 0 , z ) = z t - 1 e - t d t = E 1 ( z ) ,
8.4.6 Γ ( 1 2 , z 2 ) = 2 z e - t 2 d t = π erfc ( z ) .
8.4.13 Γ ( 1 - n , z ) = z 1 - n E n ( z ) ,
8.4.15 Γ ( - 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 ) .