DLMF:25.11.E26 (Q7700)

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DLMF:25.11.E26
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    Statements

    Defining formula
    ζ ⁡ ( s , a ) = - s ⁢ ∫ - a ∞ x - ⌊ x ⌋ - 1 2 ( x + a ) s + 1 ⁢ d x , Hurwitz-zeta 𝑠 𝑎 𝑠 superscript subscript 𝑎 𝑥 𝑥 1 2 superscript 𝑥 𝑎 𝑠 1 𝑥 {\displaystyle{\displaystyle\zeta\left(s,a\right)=-s\int_{-a}^{\infty}\frac{x-% \left\lfloor x\right\rfloor-\frac{1}{2}}{(x+a)^{s+1}}\mathrm{d}x,}}
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    DLMF id
    DLMF:25.11.E26
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    constraint
    - 1 < ℜ ⁡ s < 0 1 𝑠 0 {\displaystyle{\displaystyle-1<\Re s<0}}
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    - 1 < ℜ ⁡ s < 0 1 𝑠 0 {\displaystyle{\displaystyle-1<\Re s<0}}
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    0 < a ≤ 1 0 𝑎 1 {\displaystyle{\displaystyle 0<a\leq 1}}
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    Symbols used
    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.m2aaadec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1afdec
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    Q11169
    Defining formula
    ⌊ x ⌋ 𝑥 {\displaystyle{\displaystyle\left\lfloor\NVar{x}\right\rfloor}}
    xml-id
    introduction.Sx4.p1.t1.r17.m4aadec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3afdec
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    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1ajdec
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    Q12152
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C25.S1.XMD5.m1hdec
    0 references
    Q12161
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C25.S1.XMD6.m1udec
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