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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: 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 .
6: 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
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 ) .