DLMF:25.5.E18 (Q7631)

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DLMF:25.5.E18
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    Statements

    Defining formula
    ζ ⁡ ( 1 + s ) = sin ⁡ ( π ⁢ s ) π ⁢ s ⁢ ∫ 0 ∞ ψ ′ ⁡ ( 1 + x ) ⁢ x - s ⁢ d x , Riemann-zeta 1 𝑠 𝜋 𝑠 𝜋 𝑠 superscript subscript 0 diffop digamma 1 1 𝑥 superscript 𝑥 𝑠 𝑥 {\displaystyle{\displaystyle\zeta\left(1+s\right)=\frac{\sin\left(\pi s\right)% }{\pi s}\int_{0}^{\infty}\psi'\left(1+x\right)x^{-s}\mathrm{d}x,}}
    0 references
    DLMF id
    DLMF:25.5.E18
    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.m2apdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2ahdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1apdec
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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.m2acdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3apdec
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