DLMF:25.11.E34 (Q7708)

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DLMF:25.11.E34
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    Statements

    Defining formula
    n ⁢ ∫ 0 a ζ ′ ⁡ ( 1 - n , x ) ⁢ d x = ζ ′ ⁡ ( - n , a ) - ζ ′ ⁡ ( - n ) + B n + 1 - B n + 1 ⁡ ( a ) n ⁢ ( n + 1 ) , 𝑛 superscript subscript 0 𝑎 diffop Hurwitz-zeta 1 1 𝑛 𝑥 𝑥 diffop Hurwitz-zeta 1 𝑛 𝑎 diffop Riemann-zeta 1 𝑛 Bernoulli-number-B 𝑛 1 Bernoulli-polynomial-B 𝑛 1 𝑎 𝑛 𝑛 1 {\displaystyle{\displaystyle n\int_{0}^{a}\zeta'\left(1-n,x\right)\mathrm{d}x=% \zeta'\left(-n,a\right)-\zeta'\left(-n\right)+\frac{B_{n+1}-B_{n+1}\left(a% \right)}{n(n+1)},}}
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    DLMF id
    DLMF:25.11.E34
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    constraint
    ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}}
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    n = 1 , 2 , … 𝑛 1 2 … {\displaystyle{\displaystyle n=1,2,\dots}}
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    ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}}
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    Symbols used
    Q11109
    Defining formula
    B n Bernoulli-number-B 𝑛 {\displaystyle{\displaystyle B_{\NVar{n}}}}
    xml-id
    C24.S2.SS1.m1aidec
    0 references
    Q11288
    Defining formula
    B n ⁡ ( x ) Bernoulli-polynomial-B 𝑛 𝑥 {\displaystyle{\displaystyle B_{\NVar{n}}\left(\NVar{x}\right)}}
    xml-id
    C24.S2.SS1.m2abdec
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    Hurwitz zeta function
    Defining formula
    ζ ⁡ ( s , a ) Hurwitz-zeta 𝑠 𝑎 {\displaystyle{\displaystyle\zeta\left(\NVar{s},\NVar{a}\right)}}
    xml-id
    C25.S11.E1.m2aahdec
    0 references
    Riemann zeta function
    Defining formula
    ζ ⁡ ( s ) Riemann-zeta 𝑠 {\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}
    xml-id
    C25.S2.E1.m2ahdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1amdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3amdec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aqdec
    0 references
    Q12148
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C25.S1.XMD3.m1qdec
    0 references
    Q12152
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C25.S1.XMD5.m1ndec
    0 references
    Q12161
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C25.S1.XMD6.m1abdec
    0 references
     
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