DLMF:25.11.E35 (Q7709)

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DLMF:25.11.E35
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    Statements

    Defining formula
    ∑ n = 0 ∞ ( - 1 ) n ( n + a ) s = 1 Γ ⁡ ( s ) ⁢ ∫ 0 ∞ x s - 1 ⁢ e - a ⁢ x 1 + e - x ⁢ d x = 2 - s ⁢ ( ζ ⁡ ( s , 1 2 ⁢ a ) - ζ ⁡ ( s , 1 2 ⁢ ( 1 + a ) ) ) , superscript subscript 𝑛 0 superscript 1 𝑛 superscript 𝑛 𝑎 𝑠 1 Euler-Gamma 𝑠 superscript subscript 0 superscript 𝑥 𝑠 1 superscript 𝑒 𝑎 𝑥 1 superscript 𝑒 𝑥 𝑥 superscript 2 𝑠 Hurwitz-zeta 𝑠 1 2 𝑎 Hurwitz-zeta 𝑠 1 2 1 𝑎 {\displaystyle{\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+a)^{s}}=% \frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{1+e^{-x}}% \mathrm{d}x=2^{-s}\left(\zeta\left(s,\tfrac{1}{2}a\right)-\zeta\left(s,\tfrac{% 1}{2}(1+a)\right)\right),}}
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    DLMF id
    DLMF:25.11.E35
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    constraint
    ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}}
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