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11: 13.10 Integrals
13.10.10 0 t λ - 1 M ( a , b , - t ) d t = Γ ( λ ) Γ ( a - λ ) Γ ( a ) Γ ( b - λ ) , 0 < λ < a ,
13.10.11 0 t λ - 1 U ( a , b , t ) d t = Γ ( λ ) Γ ( a - λ ) Γ ( λ - b + 1 ) Γ ( a ) Γ ( a - b + 1 ) , max ( b - 1 , 0 ) < λ < a .
12: 21.5 Modular Transformations
The modular transformations form a group under the composition of such transformations, the modular group, which is generated by simpler transformations, for which ξ ( Γ ) is determinate: …
13: 20.10 Integrals
20.10.1 0 x s - 1 θ 2 ( 0 | i x 2 ) d x = 2 s ( 1 - 2 - s ) π - s / 2 Γ ( 1 2 s ) ζ ( s ) ,
20.10.2 0 x s - 1 ( θ 3 ( 0 | i x 2 ) - 1 ) d x = π - s / 2 Γ ( 1 2 s ) ζ ( s ) ,
20.10.3 0 x s - 1 ( 1 - θ 4 ( 0 | i x 2 ) ) d x = ( 1 - 2 1 - s ) π - s / 2 Γ ( 1 2 s ) ζ ( s ) .
14: 16.5 Integral Representations and Integrals
16.5.2 F q + 1 p + 1 ( a 0 , , a p b 0 , , b q ; z ) = Γ ( b 0 ) Γ ( a 0 ) Γ ( b 0 - a 0 ) 0 1 t a 0 - 1 ( 1 - t ) b 0 - a 0 - 1 F q p ( a 1 , , a p b 1 , , b q ; z t ) d t , b 0 > a 0 > 0 ,
15: 15.6 Integral Representations
15.6.2 F ( a , b ; c ; z ) = Γ ( 1 + b - c ) 2 π i Γ ( b ) 0 ( 1 + ) t b - 1 ( t - 1 ) c - b - 1 ( 1 - z t ) a d t , | ph ( 1 - z ) | < π ; c - b 1 , 2 , 3 , , b > 0 .
15.6.3 F ( a , b ; c ; z ) = e - b π i Γ ( 1 - b ) 2 π i Γ ( c - b ) ( 0 + ) t b - 1 ( t + 1 ) a - c ( t - z t + 1 ) a d t , | ph ( 1 - z ) | < π ; b 1 , 2 , 3 , , ( c - b ) > 0 .
15.6.4 F ( a , b ; c ; z ) = e - b π i Γ ( 1 - b ) 2 π i Γ ( c - b ) 1 ( 0 + ) t b - 1 ( 1 - t ) c - b - 1 ( 1 - z t ) a d t , | ph ( 1 - z ) | < π ; b 1 , 2 , 3 , , ( c - b ) > 0 .
15.6.5 F ( a , b ; c ; z ) = e - c π i Γ ( 1 - b ) Γ ( 1 + b - c ) 1 4 π 2 A ( 0 + , 1 + , 0 - , 1 - ) t b - 1 ( 1 - t ) c - b - 1 ( 1 - z t ) a d t , | ph ( 1 - z ) | < π ; b , c - b 1 , 2 , 3 , .
15.6.8 F ( a , b ; c ; z ) = 1 Γ ( c - d ) 0 1 F ( a , b ; d ; z t ) t d - 1 ( 1 - t ) c - d - 1 d t , | ph ( 1 - z ) | < π ; c > d > 0 .
16: 8.19 Generalized Exponential Integral
8.19.4 E p ( z ) = z p - 1 e - z Γ ( p ) 0 t p - 1 e - z t 1 + t d t , | ph z | < 1 2 π , p > 0 .
17: 18.17 Integrals
18.17.32 0 e - a x x ν - 1 - 2 n L 2 n ( ν - 1 - 2 n ) ( a x ) cos ( x y ) d x = ( - 1 ) n Γ ( ν ) 2 ( 2 n ) ! y 2 n ( ( a + i y ) - ν + ( a - i y ) - ν ) , ν > 2 n , a > 0 .
18.17.36 - 1 1 ( 1 - x ) z - 1 ( 1 + x ) β P n ( α , β ) ( x ) d x = 2 β + z Γ ( z ) Γ ( 1 + β + n ) ( 1 + α - z ) n n ! Γ ( 1 + β + z + n ) , z > 0 .
18.17.37 0 1 ( 1 - x 2 ) λ - 1 2 C n ( λ ) ( x ) x z - 1 d x = π  2 1 - 2 λ - z Γ ( n + 2 λ ) Γ ( z ) n ! Γ ( λ ) Γ ( 1 2 + 1 2 n + λ + 1 2 z ) Γ ( 1 2 + 1 2 z - 1 2 n ) , z > 0 .
18.17.40 0 e - a x L n ( α ) ( b x ) x z - 1 d x = Γ ( z + n ) n ! ( a - b ) n a - n - z F 1 2 ( - n , 1 + α - z 1 - n - z ; a a - b ) , a > 0 , z > 0 .
18.17.41 0 e - a x He n ( x ) x z - 1 d x = Γ ( z + n ) a - n - 2 F 2 2 ( - 1 2 n , - 1 2 n + 1 2 - 1 2 z - 1 2 n , - 1 2 z - 1 2 n + 1 2 ; - 1 2 a 2 ) , a > 0 . Also, z > 0 , n even; z > - 1 , n odd.
18: 10.43 Integrals
10.43.19 0 t μ - 1 K ν ( t ) d t = 2 μ - 2 Γ ( 1 2 μ - 1 2 ν ) Γ ( 1 2 μ + 1 2 ν ) , | ν | < μ .
10.43.22 0 t μ - 1 e - a t K ν ( t ) d t = { ( 1 2 π ) 1 2 Γ ( μ - ν ) Γ ( μ + ν ) ( 1 - a 2 ) - 1 2 μ + 1 4 P ν - 1 2 - μ + 1 2 ( a ) , - 1 < a < 1 , ( 1 2 π ) 1 2 Γ ( μ - ν ) Γ ( μ + ν ) ( a 2 - 1 ) - 1 2 μ + 1 4 P ν - 1 2 - μ + 1 2 ( a ) , a 0 , a 1 .
10.43.26 0 K μ ( a t ) J ν ( b t ) t λ d t = b ν Γ ( 1 2 ν - 1 2 λ + 1 2 μ + 1 2 ) Γ ( 1 2 ν - 1 2 λ - 1 2 μ + 1 2 ) 2 λ + 1 a ν - λ + 1 F ( ν - λ + μ + 1 2 , ν - λ - μ + 1 2 ; ν + 1 ; - b 2 a 2 ) , ( ν + 1 - λ ) > | μ | , a > | b | .
10.43.27 0 t μ + ν + 1 K μ ( a t ) J ν ( b t ) d t = ( 2 a ) μ ( 2 b ) ν Γ ( μ + ν + 1 ) ( a 2 + b 2 ) μ + ν + 1 , ( ν + 1 ) > | μ | , a > | b | .
19: 16.15 Integral Representations and Integrals
16.15.1 F 1 ( α ; β , β ; γ ; x , y ) = Γ ( γ ) Γ ( α ) Γ ( γ - α ) 0 1 u α - 1 ( 1 - u ) γ - α - 1 ( 1 - u x ) β ( 1 - u y ) β d u , α > 0 , ( γ - α ) > 0 ,
16.15.2 F 2 ( α ; β , β ; γ , γ ; x , y ) = Γ ( γ ) Γ ( γ ) Γ ( β ) Γ ( β ) Γ ( γ - β ) Γ ( γ - β ) 0 1 0 1 u β - 1 v β - 1 ( 1 - u ) γ - β - 1 ( 1 - v ) γ - β - 1 ( 1 - u x - v y ) α d u d v , γ > β > 0 , γ > β > 0 ,
16.15.3 F 3 ( α , α ; β , β ; γ ; x , y ) = Γ ( γ ) Γ ( β ) Γ ( β ) Γ ( γ - β - β ) Δ u β - 1 v β - 1 ( 1 - u - v ) γ - β - β - 1 ( 1 - u x ) α ( 1 - v y ) α d u d v , ( γ - β - β ) > 0 , β > 0 , β > 0 ,
16.15.4 F 4 ( α , β ; γ , γ ; x ( 1 - y ) , y ( 1 - x ) ) = Γ ( γ ) Γ ( γ ) Γ ( α ) Γ ( β ) Γ ( γ - α ) Γ ( γ - β ) 0 1 0 1 u α - 1 v β - 1 ( 1 - u ) γ - α - 1 ( 1 - v ) γ - β - 1 ( 1 - u x ) γ + γ - α - 1 ( 1 - v y ) γ + γ - β - 1 ( 1 - u x - v y ) α + β - γ - γ + 1 d u d v , γ > α > 0 , γ > β > 0 .
20: 16.4 Argument Unity
§16.4(iii) Identities
Balanced F 3 4 ( 1 ) series have transformation formulas and three-term relations. The basic transformation is given by … A different type of transformation is that of Whipple: … See Bailey (1964, §§4.3(7) and 7.6(1)) for the transformation formulas and Wilson (1978) for contiguous relations. …