DLMF:25.13.E2 (Q7738)

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DLMF:25.13.E2
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    Statements

    Defining formula
    F ⁡ ( x , s ) = Γ ⁡ ( 1 - s ) ( 2 ⁢ π ) 1 - s ⁢ ( e π ⁢ i ⁢ ( 1 - s ) / 2 ⁢ ζ ⁡ ( 1 - s , x ) + e π ⁢ i ⁢ ( s - 1 ) / 2 ⁢ ζ ⁡ ( 1 - s , 1 - x ) ) , periodic-zeta 𝑥 𝑠 Euler-Gamma 1 𝑠 superscript 2 𝜋 1 𝑠 superscript 𝑒 𝜋 𝑖 1 𝑠 2 Hurwitz-zeta 1 𝑠 𝑥 superscript 𝑒 𝜋 𝑖 𝑠 1 2 Hurwitz-zeta 1 𝑠 1 𝑥 {\displaystyle{\displaystyle F\left(x,s\right)=\frac{\Gamma\left(1-s\right)}{(% 2\pi)^{1-s}}\*\left(e^{\pi i(1-s)/2}\zeta\left(1-s,x\right)+e^{\pi i(s-1)/2}% \zeta\left(1-s,1-x\right)\right),}}
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    DLMF id
    DLMF:25.13.E2
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    constraint
    ℜ ⁡ s > 1 𝑠 1 {\displaystyle{\displaystyle\Re s>1}}
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    0 < x < 1 0 𝑥 1 {\displaystyle{\displaystyle 0<x<1}}
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    ℜ ⁡ s > 1 𝑠 1 {\displaystyle{\displaystyle\Re s>1}}
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