DLMF:20.10.E1 (Q6837)

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DLMF:20.10.E1
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    Statements

    Defining formula
    ∫ 0 ∞ x s - 1 ⁢ θ 2 ⁡ ( 0 | i ⁢ x 2 ) ⁢ d x = 2 s ⁢ ( 1 - 2 - s ) ⁢ π - s / 2 ⁢ Γ ⁡ ( 1 2 ⁢ s ) ⁢ ζ ⁡ ( s ) , superscript subscript 0 superscript 𝑥 𝑠 1 Jacobi-theta-tau 2 0 𝑖 superscript 𝑥 2 𝑥 superscript 2 𝑠 1 superscript 2 𝑠 superscript 𝜋 𝑠 2 Euler-Gamma 1 2 𝑠 Riemann-zeta 𝑠 {\displaystyle{\displaystyle\int_{0}^{\infty}x^{s-1}\theta_{2}\left(0\middle|% ix^{2}\right)\mathrm{d}x=2^{s}(1-2^{-s})\pi^{-s/2}\Gamma\left(\tfrac{1}{2}s% \right)\zeta\left(s\right),}}
    0 references
    DLMF id
    DLMF:20.10.E1
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
    0 references
    Q11946
    Defining formula
    θ j ⁡ ( z | τ ) Jacobi-theta-tau 𝑗 𝑧 𝜏 {\displaystyle{\displaystyle\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}% \right)}}
    xml-id
    C20.S2.SS1.m1adec
    0 references
    Riemann zeta function
    Defining formula
    ζ ⁡ ( s ) Riemann-zeta 𝑠 {\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}
    xml-id
    C25.S2.E1.m2adec
    0 references
     
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