DLMF:25.5.E19 (Q7632)

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DLMF:25.5.E19
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    Statements

    Defining formula
    ζ ⁡ ( m + s ) = ( - 1 ) m - 1 ⁢ Γ ⁡ ( s ) ⁢ sin ⁡ ( π ⁢ s ) π ⁢ Γ ⁡ ( m + s ) ⁢ ∫ 0 ∞ ψ ( m ) ⁡ ( 1 + x ) ⁢ x - s ⁢ d x , Riemann-zeta 𝑚 𝑠 superscript 1 𝑚 1 Euler-Gamma 𝑠 𝜋 𝑠 𝜋 Euler-Gamma 𝑚 𝑠 superscript subscript 0 digamma 𝑚 1 𝑥 superscript 𝑥 𝑠 𝑥 {\displaystyle{\displaystyle\zeta\left(m+s\right)=(-1)^{m-1}\frac{\Gamma\left(% s\right)\sin\left(\pi s\right)}{\pi\Gamma\left(m+s\right)}\*\int_{0}^{\infty}{% \psi^{(m)}}\left(1+x\right)x^{-s}\mathrm{d}x,}}
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    DLMF id
    DLMF:25.5.E19
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    constraint
    m = 1 , 2 , 3 , … 𝑚 1 2 3 … {\displaystyle{\displaystyle m=1,2,3,\dots}}
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