DLMF:25.5.E17 (Q7630)

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DLMF:25.5.E17
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    Statements

    Defining formula
    ζ ⁡ ( 1 + s ) = sin ⁡ ( π ⁢ s ) π ⁢ ∫ 0 ∞ ( γ + ψ ⁡ ( 1 + x ) ) ⁢ x - s - 1 ⁢ d x , Riemann-zeta 1 𝑠 𝜋 𝑠 𝜋 superscript subscript 0 digamma 1 𝑥 superscript 𝑥 𝑠 1 𝑥 {\displaystyle{\displaystyle\zeta\left(1+s\right)=\frac{\sin\left(\pi s\right)% }{\pi}\int_{0}^{\infty}\left(\gamma+\psi\left(1+x\right)\right)x^{-s-1}\mathrm% {d}x,}}
    0 references
    DLMF id
    DLMF:25.5.E17
    0 references
    Symbols used
    Euler’s constant
    Defining formula
    γ {\displaystyle{\displaystyle\gamma}}
    xml-id
    C5.S2.E3.m2adec
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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.m2aodec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2agdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aodec
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