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1: 27.4 Euler Products and Dirichlet Series
27.4.3 ζ ( s ) = n = 1 n s = p ( 1 p s ) 1 , s > 1 .
27.4.5 n = 1 μ ( n ) n s = 1 ζ ( s ) , s > 1 ,
27.4.10 n = 1 d k ( n ) n s = ( ζ ( s ) ) k , s > 1 ,
27.4.11 n = 1 σ α ( n ) n s = ζ ( s ) ζ ( s α ) , s > max ( 1 , 1 + α ) ,
27.4.12 n = 1 Λ ( n ) n s = ζ ( s ) ζ ( s ) , s > 1 ,
2: 8.14 Integrals
8.14.1 0 e a x γ ( b , x ) Γ ( b ) d x = ( 1 + a ) b a , a > 0 , b > 1 ,
8.14.2 0 e a x Γ ( b , x ) d x = Γ ( b ) 1 ( 1 + a ) b a , a > 1 , b > 1 .
8.14.3 0 x a 1 γ ( b , x ) d x = Γ ( a + b ) a , a < 0 , ( a + b ) > 0 ,
8.14.4 0 x a 1 Γ ( b , x ) d x = Γ ( a + b ) a , a > 0 , ( a + b ) > 0 ,
8.14.6 0 x a 1 e s x Γ ( b , x ) d x = Γ ( a + b ) a ( 1 + s ) a + b F ( 1 , a + b ; 1 + a ; s / ( 1 + s ) ) , s > 1 , ( a + b ) > 0 , a > 0 .
3: 13.10 Integrals
13.10.3 0 e z t t b 1 𝐌 ( a , c , k t ) d t = Γ ( b ) z b 𝐅 1 2 ( a , b ; c ; k / z ) , b > 0 , z > max ( k , 0 ) ,
13.10.5 0 e t t b 1 𝐌 ( a , c , t ) d t = Γ ( b ) Γ ( c a b ) Γ ( c a ) Γ ( c b ) , ( c a ) > b > 0 ,
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 .
13.10.13 0 e t t b 1 1 2 ν 𝐌 ( a , b , t ) J ν ( 2 x t ) d t = x a + 1 2 ν e x 𝐌 ( ν b + 1 , ν a + 1 , x ) , x > 0 , 2 a < ν + 5 2 , b > 0 ,
4: 25.14 Lerch’s Transcendent
25.14.1 Φ ( z , s , a ) n = 0 z n ( a + n ) s , | z | < 1 ; s > 1 , | z | = 1 .
25.14.2 ζ ( s , a ) = Φ ( 1 , s , a ) , s > 1 , a 0 , 1 , 2 , ,
25.14.3 Li s ( z ) = z Φ ( z , s , 1 ) , s > 1 , | z | 1 .
25.14.5 Φ ( z , s , a ) = 1 Γ ( s ) 0 x s 1 e a x 1 z e x d x , s > 1 , a > 0 if z = 1 ; s > 0 , a > 0 if z [ 1 , ) .
25.14.6 Φ ( z , s , a ) = 1 2 a s + 0 z x ( a + x ) s d x 2 0 sin ( x ln z s arctan ( x / a ) ) ( a 2 + x 2 ) s / 2 ( e 2 π x 1 ) d x , a > 0 if | z | < 1 ; s > 1 , a > 0 if | z | = 1 .
5: 5.13 Integrals
5.13.1 1 2 π i c i c + i Γ ( s + a ) Γ ( b s ) z s d s = Γ ( a + b ) z a ( 1 + z ) a + b , ( a + b ) > 0 , a < c < b , | ph z | < π .
5.13.3 1 2 π Γ ( a + i t ) Γ ( b + i t ) Γ ( c i t ) Γ ( d i t ) d t = Γ ( a + c ) Γ ( a + d ) Γ ( b + c ) Γ ( b + d ) Γ ( a + b + c + d ) , a , b , c , d > 0 .
5.13.4 d t Γ ( a + t ) Γ ( b + t ) Γ ( c t ) Γ ( d t ) = Γ ( a + b + c + d 3 ) Γ ( a + c 1 ) Γ ( a + d 1 ) Γ ( b + c 1 ) Γ ( b + d 1 ) , ( a + b + c + d ) > 3 .
5.13.5 1 4 π k = 1 4 Γ ( a k + i t ) Γ ( a k i t ) Γ ( 2 i t ) Γ ( 2 i t ) d t = 1 j < k 4 Γ ( a j + a k ) Γ ( a 1 + a 2 + a 3 + a 4 ) , ( a k ) > 0 , k = 1 , 2 , 3 , 4 .
6: 25.13 Periodic Zeta Function
where s > 1 if x is an integer, s > 0 otherwise. …
25.13.2 F ( x , s ) = Γ ( 1 s ) ( 2 π ) 1 s ( e π i ( 1 s ) / 2 ζ ( 1 s , x ) + e π i ( s 1 ) / 2 ζ ( 1 s , 1 x ) ) , 0 < x < 1 , s > 1 ,
25.13.3 ζ ( 1 s , x ) = Γ ( s ) ( 2 π ) s ( e π i s / 2 F ( x , s ) + e π i s / 2 F ( x , s ) ) , s > 0 if 0 < x < 1 ; s > 1 if x = 1 .
7: 7.9 Continued Fractions
7.9.1 π e z 2 erfc z = z z 2 + 1 2 1 + 1 z 2 + 3 2 1 + 2 z 2 + , z > 0 ,
7.9.2 π e z 2 erfc z = 2 z 2 z 2 + 1 1 2 2 z 2 + 5 3 4 2 z 2 + 9 , z > 0 ,
8: 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 ,
9: 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 .
10: 15.14 Integrals
15.14.1 0 x s 1 𝐅 ( a , b c ; x ) d x = Γ ( s ) Γ ( a s ) Γ ( b s ) Γ ( a ) Γ ( b ) Γ ( c s ) , min ( a , b ) > s > 0 .