DLMF:8.22.E2 (Q2751)

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DLMF:8.22.E2
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    Statements

    test
    ζ x ⁢ ( s ) = 1 Γ ⁡ ( s ) ⁢ ∫ 0 x t s - 1 e t - 1 ⁢ d t , subscript 𝜁 𝑥 𝑠 1 Euler-Gamma 𝑠 superscript subscript 0 𝑥 superscript 𝑡 𝑠 1 superscript 𝑒 𝑡 1 𝑡 {\displaystyle{\displaystyle\zeta_{x}(s)=\frac{1}{\Gamma\left(s\right)}\int_{0% }^{x}\frac{t^{s-1}}{e^{t}-1}\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:8.22.E2
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    constraint
    ℜ ⁡ s > 1 𝑠 1 {\displaystyle{\displaystyle\Re s>1}}
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    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aadec
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    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
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    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
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    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
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    Q10811
    test
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1adec
    0 references
    Q11332
    test
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C8.S1.XMD1.m1dec
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