DLMF:22.16.E16 (Q7136)

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DLMF:22.16.E16
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    Statements

    Defining formula
    ℰ ⁡ ( x , k ) = k 2 ⁢ ∫ 0 x cn 2 ⁡ ( t , k ) ⁢ d t + k ′ 2 ⁢ x , Jacobi-Epsilon 𝑥 𝑘 superscript 𝑘 2 superscript subscript 0 𝑥 Jacobi-elliptic-cn 2 𝑡 𝑘 𝑡 superscript superscript 𝑘 ′ 2 𝑥 {\displaystyle{\displaystyle\mathcal{E}\left(x,k\right)=k^{2}\int_{0}^{x}{% \operatorname{cn}^{2}}\left(t,k\right)\mathrm{d}t+{k^{\prime}}^{2}x,}}
    0 references
    DLMF id
    DLMF:22.16.E16
    0 references
    Symbols used
    Jacobi’s epsilon function
    Defining formula
    ℰ ⁡ ( x , k ) Jacobi-Epsilon 𝑥 𝑘 {\displaystyle{\displaystyle\mathcal{E}\left(\NVar{x},\NVar{k}\right)}}
    xml-id
    C22.S16.E14.m2abdec
    0 references
    Jacobian elliptic function
    Defining formula
    cn ⁡ ( z , k ) Jacobi-elliptic-cn 𝑧 𝑘 {\displaystyle{\displaystyle\operatorname{cn}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E5.m2aadec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1acdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3acdec
    0 references
    Q12003
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C22.S1.XMD1.m1odec
    0 references
    Q11984
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C22.S1.XMD4.m1mdec
    0 references
    Q11988
    Defining formula
    k ′ superscript 𝑘 ′ {\displaystyle{\displaystyle k^{\prime}}}
    xml-id
    C22.S1.XMD5.m1adec
    0 references
     
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