DLMF:25.5.E16 (Q7629)

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DLMF:25.5.E16
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    Statements

    Defining formula
    ζ ⁡ ( s ) = 1 s - 1 + sin ⁡ ( π ⁢ s ) π ⁢ ( s - 1 ) ⁢ ∫ 0 ∞ ( 1 1 + x - ψ ′ ⁡ ( 1 + x ) ) ⁢ x 1 - s ⁢ d x , Riemann-zeta 𝑠 1 𝑠 1 𝜋 𝑠 𝜋 𝑠 1 superscript subscript 0 1 1 𝑥 diffop digamma 1 1 𝑥 superscript 𝑥 1 𝑠 𝑥 {\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{s-1}+\frac{\sin\left(% \pi s\right)}{\pi(s-1)}\*\int_{0}^{\infty}\left(\frac{1}{1+x}-\psi'\left(1+x% \right)\right)x^{1-s}\mathrm{d}x,}}
    0 references
    DLMF id
    DLMF:25.5.E16
    0 references
    Symbols used
    Riemann zeta function
    Defining formula
    ζ ⁡ ( s ) Riemann-zeta 𝑠 {\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}
    xml-id
    C25.S2.E1.m2andec
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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.m2afdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1andec
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    psi (or digamma) function
    Defining formula
    ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E2.m2aadec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3andec
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    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2acdec
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    Q12152
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C25.S1.XMD5.m1odec
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    Q12149
    Defining formula
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C25.S1.XMD7.m1ndec
    0 references
     
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