DLMF:20.10.E4 (Q6840)

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DLMF:20.10.E4
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    Statements

    Defining formula
    ∫ 0 ∞ e - s ⁢ t ⁢ θ 1 ⁡ ( β ⁢ π 2 ⁢ ℓ | i ⁢ π ⁢ t ℓ 2 ) ⁢ d t = ∫ 0 ∞ e - s ⁢ t ⁢ θ 2 ⁡ ( ( 1 + β ) ⁢ π 2 ⁢ ℓ | i ⁢ π ⁢ t ℓ 2 ) ⁢ d t = - ℓ s ⁢ sinh ⁡ ( β ⁢ s ) ⁢ sech ⁡ ( ℓ ⁢ s ) , superscript subscript 0 superscript 𝑒 𝑠 𝑡 Jacobi-theta-tau 1 𝛽 𝜋 2 ℓ 𝑖 𝜋 𝑡 superscript ℓ 2 𝑡 superscript subscript 0 superscript 𝑒 𝑠 𝑡 Jacobi-theta-tau 2 1 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_{1}\left(\frac{% \beta\pi}{2\ell}\middle|\frac{i\pi t}{\ell^{2}}\right)\mathrm{d}t=\int_{0}^{% \infty}e^{-st}\theta_{2}\left(\frac{(1+\beta)\pi}{2\ell}\middle|\frac{i\pi t}{% \ell^{2}}\right)\mathrm{d}t=-\frac{\ell}{\sqrt{s}}\sinh\left(\beta\sqrt{s}% \right)\operatorname{sech}\left(\ell\sqrt{s}\right),}}
    0 references
    DLMF id
    DLMF:20.10.E4
    0 references
    Symbols used
    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.m1acdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2acdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1acdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
    0 references
    hyperbolic secant function
    Defining formula
    sech ⁡ z 𝑧 {\displaystyle{\displaystyle\operatorname{sech}\NVar{z}}}
    xml-id
    C4.S28.E6.m2adec
    0 references
    hyperbolic sine function
    Defining formula
    sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}
    xml-id
    C4.S28.E1.m2adec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2acdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3acdec
    0 references
    Q11952
    Defining formula
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C20.S10.XMD1.m1dec
    0 references
    Q11953
    Defining formula
    β 𝛽 {\displaystyle{\displaystyle\beta}}
    xml-id
    C20.S10.XMD2.m1dec
    0 references
     
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