DLMF:23.6.E14 (Q7255)

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DLMF:23.6.E14
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    Statements

    Defining formula
    ℘ ⁡ ( u ) = ( π 2 ⁢ ω 1 ) 2 ⁢ ( θ 1 ′′′ ⁡ ( 0 , q ) 3 ⁢ θ 1 ′ ⁡ ( 0 , q ) - d 2 d z 2 ⁡ ln ⁡ θ 1 ⁡ ( z , q ) ) , Weierstrass-P-on-lattice 𝑢 𝕃 superscript 𝜋 2 subscript 𝜔 1 2 diffop Jacobi-theta 1 3 0 𝑞 3 diffop Jacobi-theta 1 1 0 𝑞 derivative 𝑧 2 Jacobi-theta 1 𝑧 𝑞 {\displaystyle{\displaystyle\wp\left(u\right)=\left(\frac{\pi}{2\omega_{1}}% \right)^{2}\left(\frac{\theta_{1}'''\left(0,q\right)}{3\theta_{1}'\left(0,q% \right)}-\frac{{\mathrm{d}}^{2}}{{\mathrm{d}z}^{2}}\ln\theta_{1}\left(z,q% \right)\right),}}
    0 references
    DLMF id
    DLMF:23.6.E14
    0 references
    Symbols used
    Q11945
    Defining formula
    θ j ⁡ ( z , q ) Jacobi-theta 𝑗 𝑧 𝑞 {\displaystyle{\displaystyle\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)}}
    xml-id
    C20.S2.SS1.m2aldec
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    DLMF:23.2.E4
    Defining formula
    ℘ ⁡ ( z ) Weierstrass-P-on-lattice 𝑧 𝕃 {\displaystyle{\displaystyle\wp\left(\NVar{z}\right)}}
    xml-id
    C23.S2.E4.m2acdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2amdec
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    derivative of $$f$$ with respect to $$x$$
    Defining formula
    d f d x derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}}}
    xml-id
    C1.S4.E4.m2aadec
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    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2aadec
    0 references
    Q12055
    Defining formula
    𝕃 𝕃 {\displaystyle{\displaystyle\mathbb{L}}}
    xml-id
    C23.S1.XMD1.m1kdec
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    Q12066
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C23.S1.XMD10.m1mdec
    0 references
    Q12056
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C23.S1.XMD7.m1edec
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    Q12049
    Defining formula
    ω 1 subscript 𝜔 1 {\displaystyle{\displaystyle\omega_{1}}}
    xml-id
    C23.S1.XMD8.m1mdec
    0 references
     
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