DLMF:25.5.E4 (Q7617)

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DLMF:25.5.E4
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    Statements

    Defining formula
    ζ ⁡ ( s ) = 1 ( 1 - 2 1 - s ) ⁢ Γ ⁡ ( s + 1 ) ⁢ ∫ 0 ∞ e x ⁢ x s ( e x + 1 ) 2 ⁢ d x , Riemann-zeta 𝑠 1 1 superscript 2 1 𝑠 Euler-Gamma 𝑠 1 superscript subscript 0 superscript 𝑒 𝑥 superscript 𝑥 𝑠 superscript superscript 𝑒 𝑥 1 2 𝑥 {\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{(1-2^{1-s})\Gamma% \left(s+1\right)}\int_{0}^{\infty}\frac{e^{x}x^{s}}{(e^{x}+1)^{2}}\mathrm{d}x,}}
    0 references
    DLMF id
    DLMF:25.5.E4
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    constraint
    ℜ ⁡ s > 0 𝑠 0 {\displaystyle{\displaystyle\Re s>0}}
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    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2acdec
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    Riemann zeta function
    Defining formula
    ζ ⁡ ( s ) Riemann-zeta 𝑠 {\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}
    xml-id
    C25.S2.E1.m2acdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1acdec
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