DLMF:20.10.E5 (Q6841)

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DLMF:20.10.E5
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    Statements

    test
    ∫ 0 ∞ e - s ⁢ t ⁢ θ 3 ⁡ ( ( 1 + β ) ⁢ π 2 ⁢ ℓ | i ⁢ π ⁢ t ℓ 2 ) ⁢ d t = ∫ 0 ∞ e - s ⁢ t ⁢ θ 4 ⁡ ( β ⁢ π 2 ⁢ ℓ | i ⁢ π ⁢ t ℓ 2 ) ⁢ d t = ℓ s ⁢ cosh ⁡ ( β ⁢ s ) ⁢ csch ⁡ ( ℓ ⁢ s ) . superscript subscript 0 superscript 𝑒 𝑠 𝑡 Jacobi-theta-tau 3 1 𝛽 𝜋 2 ℓ 𝑖 𝜋 𝑡 superscript ℓ 2 𝑡 superscript subscript 0 superscript 𝑒 𝑠 𝑡 Jacobi-theta-tau 4 𝛽 𝜋 2 ℓ 𝑖 𝜋 𝑡 superscript ℓ 2 𝑡 ℓ 𝑠 𝛽 𝑠 ℓ 𝑠 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-st}\theta_{3}\left(\frac{(1+% \beta)\pi}{2\ell}\middle|\frac{i\pi t}{\ell^{2}}\right)\mathrm{d}t=\int_{0}^{% \infty}e^{-st}\theta_{4}\left(\frac{\beta\pi}{2\ell}\middle|\frac{i\pi t}{\ell% ^{2}}\right)\mathrm{d}t=\frac{\ell}{\sqrt{s}}\cosh\left(\beta\sqrt{s}\right)% \operatorname{csch}\left(\ell\sqrt{s}\right).}}
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    DLMF id
    DLMF:20.10.E5
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    Symbols used
    Q11946
    test
    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.m1addec
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    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2addec
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