DLMF:25.5.E20 (Q7633)

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DLMF:25.5.E20
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    Statements

    Defining formula
    ζ ⁡ ( s ) = Γ ⁡ ( 1 - s ) 2 ⁢ π ⁢ i ⁢ ∫ - ∞ ( 0 + ) z s - 1 e - z - 1 ⁢ d z , Riemann-zeta 𝑠 Euler-Gamma 1 𝑠 2 𝜋 𝑖 superscript subscript limit-from 0 superscript 𝑧 𝑠 1 superscript 𝑒 𝑧 1 𝑧 {\displaystyle{\displaystyle\zeta\left(s\right)=\frac{\Gamma\left(1-s\right)}{% 2\pi i}\int_{-\infty}^{(0+)}\frac{z^{s-1}}{e^{-z}-1}\mathrm{d}z,}}
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    DLMF id
    DLMF:25.5.E20
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    constraint
    s ≠ 1 , 2 , … 𝑠 1 2 … {\displaystyle{\displaystyle s\neq 1,2,\dots}}
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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.m2ajdec
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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.m2ardec
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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.m2ajdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1ardec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aidec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aadec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3ardec
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    Q12149
    Defining formula
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C25.S1.XMD7.m1rdec
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    Q12156
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C25.S1.XMD8.m1dec
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