DLMF:25.11.E30 (Q7704)

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DLMF:25.11.E30
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    Statements

    Defining formula
    ζ ⁡ ( s , a ) = Γ ⁡ ( 1 - s ) 2 ⁢ π ⁢ i ⁢ ∫ - ∞ ( 0 + ) e a ⁢ z ⁢ z s - 1 1 - e z ⁢ d z , Hurwitz-zeta 𝑠 𝑎 Euler-Gamma 1 𝑠 2 𝜋 𝑖 superscript subscript limit-from 0 superscript 𝑒 𝑎 𝑧 superscript 𝑧 𝑠 1 1 superscript 𝑒 𝑧 𝑧 {\displaystyle{\displaystyle\zeta\left(s,a\right)=\frac{\Gamma\left(1-s\right)% }{2\pi i}\int_{-\infty}^{(0+)}\frac{e^{az}z^{s-1}}{1-e^{z}}\mathrm{d}z,}}
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    DLMF id
    DLMF:25.11.E30
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    constraint
    ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}}
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    s ≠ 1 𝑠 1 {\displaystyle{\displaystyle s\neq 1}}
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    ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}}
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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.m2agdec
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    Hurwitz zeta function
    Defining formula
    ζ ⁡ ( s , a ) Hurwitz-zeta 𝑠 𝑎 {\displaystyle{\displaystyle\zeta\left(\NVar{s},\NVar{a}\right)}}
    xml-id
    C25.S11.E1.m2aaedec
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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.m1ajdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2addec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3ajdec
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    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1andec
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    Q12161
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C25.S1.XMD6.m1ydec
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    Q12149
    Defining formula
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C25.S1.XMD7.m1vdec
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