DLMF:25.11.E20 (Q7694)

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DLMF:25.11.E20
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    Statements

    Defining formula
    ( - 1 ) k ⁢ ζ ( k ) ⁡ ( s , a ) = ( ln ⁡ a ) k a s ⁢ ( 1 2 + a s - 1 ) + k ! ⁢ a 1 - s ⁢ ∑ r = 0 k - 1 ( ln ⁡ a ) r r ! ⁢ ( s - 1 ) k - r + 1 - s ⁢ ( s + 1 ) 2 ⁢ ∫ 0 ∞ ( B ~ 2 ⁡ ( x ) - B 2 ) ⁢ ( ln ⁡ ( x + a ) ) k ( x + a ) s + 2 ⁢ d x + k ⁢ ( 2 ⁢ s + 1 ) 2 ⁢ ∫ 0 ∞ ( B ~ 2 ⁡ ( x ) - B 2 ) ⁢ ( ln ⁡ ( x + a ) ) k - 1 ( x + a ) s + 2 ⁢ d x - k ⁢ ( k - 1 ) 2 ⁢ ∫ 0 ∞ ( B ~ 2 ⁡ ( x ) - B 2 ) ⁢ ( ln ⁡ ( x + a ) ) k - 2 ( x + a ) s + 2 ⁢ d x , superscript 1 𝑘 Hurwitz-zeta 𝑘 𝑠 𝑎 superscript 𝑎 𝑘 superscript 𝑎 𝑠 1 2 𝑎 𝑠 1 𝑘 superscript 𝑎 1 𝑠 superscript subscript 𝑟 0 𝑘 1 superscript 𝑎 𝑟 𝑟 superscript 𝑠 1 𝑘 𝑟 1 𝑠 𝑠 1 2 superscript subscript 0 periodic-Bernoulli-polynomial-B 2 𝑥 Bernoulli-number-B 2 superscript 𝑥 𝑎 𝑘 superscript 𝑥 𝑎 𝑠 2 𝑥 𝑘 2 𝑠 1 2 superscript subscript 0 periodic-Bernoulli-polynomial-B 2 𝑥 Bernoulli-number-B 2 superscript 𝑥 𝑎 𝑘 1 superscript 𝑥 𝑎 𝑠 2 𝑥 𝑘 𝑘 1 2 superscript subscript 0 periodic-Bernoulli-polynomial-B 2 𝑥 Bernoulli-number-B 2 superscript 𝑥 𝑎 𝑘 2 superscript 𝑥 𝑎 𝑠 2 𝑥 {\displaystyle{\displaystyle(-1)^{k}{\zeta^{(k)}}\left(s,a\right)=\frac{(\ln a% )^{k}}{a^{s}}\left(\frac{1}{2}+\frac{a}{s-1}\right)+k!a^{1-s}\sum_{r=0}^{k-1}% \frac{(\ln a)^{r}}{r!(s-1)^{k-r+1}}-\frac{s(s+1)}{2}\int_{0}^{\infty}\frac{(% \widetilde{B}_{2}\left(x\right)-B_{2})(\ln\left(x+a\right))^{k}}{(x+a)^{s+2}}% \mathrm{d}x+\frac{k(2s+1)}{2}\int_{0}^{\infty}\frac{(\widetilde{B}_{2}\left(x% \right)-B_{2})(\ln\left(x+a\right))^{k-1}}{(x+a)^{s+2}}\mathrm{d}x-\frac{k(k-1% )}{2}\int_{0}^{\infty}\frac{(\widetilde{B}_{2}\left(x\right)-B_{2})(\ln\left(x% +a\right))^{k-2}}{(x+a)^{s+2}}\mathrm{d}x,}}
    0 references
    DLMF id
    DLMF:25.11.E20
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    constraint
    ℜ ⁡ s > - 1 𝑠 1 {\displaystyle{\displaystyle\Re s>-1}}
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    s ≠ 1 𝑠 1 {\displaystyle{\displaystyle s\neq 1}}
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    a > 0 𝑎 0 {\displaystyle{\displaystyle 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.m1acdec
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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.m2audec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1addec
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    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5abdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3addec
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    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2abdec
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    Q11110
    Defining formula
    B ~ n ⁡ ( x ) periodic-Bernoulli-polynomial-B 𝑛 𝑥 {\displaystyle{\displaystyle\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)}}
    xml-id
    C24.S2.SS3.m1acdec
    0 references
     
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