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1: 7.14 Integrals
7.14.2 0 e a t erf ( b t ) d t = 1 a e a 2 / ( 4 b 2 ) erfc ( a 2 b ) , a > 0 , | ph b | < 1 4 π ,
7.14.3 0 e a t erf b t d t = 1 a b a + b , a > 0 , b > 0 ,
7.14.4 0 e ( a b ) t erfc ( a t + c t ) d t = e 2 ( a c + b c ) b ( a + b ) , | ph a | < 1 2 π , b > 0 , c 0 .
7.14.5 0 e a t C ( t ) d t = 1 a f ( a π ) , a > 0 ,
7.14.6 0 e a t S ( t ) d t = 1 a g ( a π ) , a > 0 ,
2: 8.15 Sums
8.15.2 a k = 1 ( e 2 π i k ( z + h ) ( 2 π i k ) a + 1 Γ ( a , 2 π i k z ) + e 2 π i k ( z + h ) ( 2 π i k ) a + 1 Γ ( a , 2 π i k z ) ) = ζ ( a , z + h ) + z a + 1 a + 1 + ( h 1 2 ) z a , h [ 0 , 1 ] .
3: 1.4 Calculus of One Variable
Integration by Parts
4: 6.16 Mathematical Applications
By integration by parts
5: 2.3 Integrals of a Real Variable
§2.3(i) Integration by Parts
6: 18.17 Integrals
18.17.34 0 e x z L n ( α ) ( x ) e x x α d x = Γ ( α + n + 1 ) z n n ! ( z + 1 ) α + n + 1 , z > 1 .
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 .
18.17.38 0 1 P 2 n ( x ) x z 1 d x = ( 1 ) n ( 1 2 1 2 z ) n 2 ( 1 2 z ) n + 1 , z > 0 ,
18.17.39 0 1 P 2 n + 1 ( x ) x z 1 d x = ( 1 ) n ( 1 1 2 z ) n 2 ( 1 2 + 1 2 z ) n + 1 , z > 1 .
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 .
7: 9.12 Scorer Functions
9.12.31 0 z Hi ( t ) d t 1 π ln z + 2 γ + ln 3 3 π + 1 π k = 1 ( 1 ) k 1 ( 3 k 1 ) ! k ! ( 3 z 3 ) k , | ph z | 2 3 π δ ,
8: 25.5 Integral Representations
25.5.2 ζ ( s ) = 1 Γ ( s + 1 ) 0 e x x s ( e x 1 ) 2 d x , s > 1 .
25.5.4 ζ ( s ) = 1 ( 1 2 1 s ) Γ ( s + 1 ) 0 e x x s ( e x + 1 ) 2 d x , s > 0 .
9: 2.4 Contour Integrals
Then by integration by parts the integral … If, in addition, the corresponding integrals with Q and F replaced by their derivatives Q ( j ) and F ( j ) , j = 1 , 2 , , m , converge uniformly, then by repeated integrations by partsCases in which p ( t 0 ) 0 are usually handled by deforming the integration path in such a way that the minimum of ( z p ( t ) ) is attained at a saddle point or at an endpoint. …
10: 25.2 Definition and Expansions
25.2.9 ζ ( s ) = k = 1 N 1 k s + N 1 s s 1 1 2 N s + k = 1 n ( s + 2 k 2 2 k 1 ) B 2 k 2 k N 1 s 2 k ( s + 2 n 2 n + 1 ) N B ~ 2 n + 1 ( x ) x s + 2 n + 1 d x , s > 2 n ; n , N = 1 , 2 , 3 , .