DLMF:25.11.E7 (Q7681)

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DLMF:25.11.E7
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    Statements

    Defining formula
    ζ ⁡ ( s , a ) = 1 a s + 1 ( 1 + a ) s ⁢ ( 1 2 + 1 + a s - 1 ) + ∑ k = 1 n ( s + 2 ⁢ k - 2 2 ⁢ k - 1 ) ⁢ B 2 ⁢ k 2 ⁢ k ⁢ 1 ( 1 + a ) s + 2 ⁢ k - 1 - ( s + 2 ⁢ n 2 ⁢ n + 1 ) ⁢ ∫ 1 ∞ B ~ 2 ⁢ n + 1 ⁡ ( x ) ( x + a ) s + 2 ⁢ n + 1 ⁢ d x , Hurwitz-zeta 𝑠 𝑎 1 superscript 𝑎 𝑠 1 superscript 1 𝑎 𝑠 1 2 1 𝑎 𝑠 1 superscript subscript 𝑘 1 𝑛 binomial 𝑠 2 𝑘 2 2 𝑘 1 Bernoulli-number-B 2 𝑘 2 𝑘 1 superscript 1 𝑎 𝑠 2 𝑘 1 binomial 𝑠 2 𝑛 2 𝑛 1 superscript subscript 1 periodic-Bernoulli-polynomial-B 2 𝑛 1 𝑥 superscript 𝑥 𝑎 𝑠 2 𝑛 1 𝑥 {\displaystyle{\displaystyle\zeta\left(s,a\right)=\frac{1}{a^{s}}+\frac{1}{(1+% a)^{s}}\left(\frac{1}{2}+\frac{1+a}{s-1}\right)+\sum_{k=1}^{n}\genfrac{(}{)}{0% .0pt}{}{s+2k-2}{2k-1}\frac{B_{2k}}{2k}\frac{1}{(1+a)^{s+2k-1}}-\genfrac{(}{)}{% 0.0pt}{}{s+2n}{2n+1}\int_{1}^{\infty}\frac{\widetilde{B}_{2n+1}\left(x\right)}% {(x+a)^{s+2n+1}}\mathrm{d}x,}}
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    DLMF id
    DLMF:25.11.E7
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    constraint
    ℜ ⁡ s > - 2 ⁢ n 𝑠 2 𝑛 {\displaystyle{\displaystyle\Re s>-2n}}
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