DLMF:20.10.E2 (Q6838)

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DLMF:20.10.E2
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    Statements

    Defining formula
    ∫ 0 ∞ x s - 1 ⁢ ( θ 3 ⁡ ( 0 | i ⁢ x 2 ) - 1 ) ⁢ d x = π - s / 2 ⁢ Γ ⁡ ( 1 2 ⁢ s ) ⁢ ζ ⁡ ( s ) , superscript subscript 0 superscript 𝑥 𝑠 1 Jacobi-theta-tau 3 0 𝑖 superscript 𝑥 2 1 𝑥 superscript 𝜋 𝑠 2 Euler-Gamma 1 2 𝑠 Riemann-zeta 𝑠 {\displaystyle{\displaystyle\int_{0}^{\infty}x^{s-1}(\theta_{3}\left(0\middle|% ix^{2}\right)-1)\mathrm{d}x=\pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta% \left(s\right),}}
    0 references
    DLMF id
    DLMF:20.10.E2
    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.m2aadec
    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.m1aadec
    0 references
    Riemann zeta function
    Defining formula
    ζ ⁡ ( s ) Riemann-zeta 𝑠 {\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}
    xml-id
    C25.S2.E1.m2aadec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aadec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aadec
    0 references
     
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