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Euler-product representation

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1: 27.4 Euler Products and Dirichlet Series
The completely multiplicative function f ( n ) = n - s gives the Euler product representation of the Riemann zeta function ζ ( s ) 25.2(i)): …
2: 25.11 Hurwitz Zeta Function
25.11.37 k = 1 ( - 1 ) k k ζ ( n k , a ) = - n ln Γ ( a ) + ln ( j = 0 n - 1 Γ ( a - e ( 2 j + 1 ) π i / n ) ) , n = 2 , 3 , 4 , , a 1 .
3: 16.17 Definition
Then the Meijer G -function is defined via the Mellin--Barnes integral representation: …where the integration path L separates the poles of the factors Γ ( b - s ) from those of the factors Γ ( 1 - a + s ) . …
  • (ii)

    L is a loop that starts at infinity on a line parallel to the positive real axis, encircles the poles of the Γ ( b - s ) once in the negative sense and returns to infinity on another line parallel to the positive real axis. The integral converges for all z ( 0 ) if p < q , and for 0 < | z | < 1 if p = q 1 .

  • (iii)

    L is a loop that starts at infinity on a line parallel to the negative real axis, encircles the poles of the Γ ( 1 - a + s ) once in the positive sense and returns to infinity on another line parallel to the negative real axis. The integral converges for all z if p > q , and for | z | > 1 if p = q 1 .

  • where * indicates that the entry 1 + b k - b k is omitted. …
    4: 16.8 Differential Equations
    is a value z 0 of z at which all the coefficients f j ( z ) , j = 0 , 1 , , n - 1 , are analytic. If z 0 is not an ordinary point but ( z - z 0 ) n - j f j ( z ) , j = 0 , 1 , , n - 1 , are analytic at z = z 0 , then z 0 is a regular singularity. … where * indicates that the entry 1 + b j - b j is omitted. … where * indicates that the entry 1 - a j + a j is omitted. … In this reference it is also explained that in general when q > 1 no simple representations in terms of generalized hypergeometric functions are available for the fundamental solutions near z = 1 . …
    5: 5.14 Multidimensional Integrals
    §5.14 Multidimensional Integrals
    5.14.4 [ 0 , 1 ] n t 1 t 2 t m | Δ ( t 1 , , t n ) | 2 c k = 1 n t k a - 1 ( 1 - t k ) b - 1 d t k = 1 ( Γ ( 1 + c ) ) n k = 1 m a + ( n - k ) c a + b + ( 2 n - k - 1 ) c k = 1 n Γ ( a + ( n - k ) c ) Γ ( b + ( n - k ) c ) Γ ( 1 + k c ) Γ ( a + b + ( 2 n - k - 1 ) c ) ,
    provided that a , b > 0 , c > - min ( 1 / n , a / ( n - 1 ) , b / ( n - 1 ) ) . … when a > 0 , c > - min ( 1 / n , a / ( n - 1 ) ) . …
    5.14.7 1 ( 2 π ) n [ - π , π ] n 1 j < k n | e i θ j - e i θ k | 2 b d θ 1 d θ n = Γ ( 1 + b n ) ( Γ ( 1 + b ) ) n , b > - 1 / n .
    6: 16.11 Asymptotic Expansions
    Explicit representations for the coefficients c k are given in Volkmer and Wood (2014). …
    Case p = q - 1
    with the same conventions on the phases of z e π i .
    Case p q - 2
    with the same conventions on the phases of z e π i . …
    7: 5.13 Integrals
    §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 .