DLMF:22.16.E24 (Q7144)

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DLMF:22.16.E24
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    Statements

    Defining formula
    ℰ ⁡ ( x , k ) = - ∫ 0 x ( ns 2 ⁡ ( t , k ) - t - 2 ) ⁢ d t + x - 1 + x - cn ⁡ ( x , k ) ⁢ ds ⁡ ( x , k ) , Jacobi-Epsilon 𝑥 𝑘 superscript subscript 0 𝑥 Jacobi-elliptic-ns 2 𝑡 𝑘 superscript 𝑡 2 𝑡 superscript 𝑥 1 𝑥 Jacobi-elliptic-cn 𝑥 𝑘 Jacobi-elliptic-ds 𝑥 𝑘 {\displaystyle{\displaystyle\mathcal{E}\left(x,k\right)=-\int_{0}^{x}\left({% \operatorname{ns}^{2}}\left(t,k\right)-t^{-2}\right)\mathrm{d}t+x^{-1}+x-% \operatorname{cn}\left(x,k\right)\operatorname{ds}\left(x,k\right),}}
    0 references
    DLMF id
    DLMF:22.16.E24
    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.m2ajdec
    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.m2abdec
    0 references
    Jacobian elliptic function
    Defining formula
    ds ⁡ ( z , k ) Jacobi-elliptic-ds 𝑧 𝑘 {\displaystyle{\displaystyle\operatorname{ds}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E7.m3adec
    0 references
    Jacobian elliptic function
    Defining formula
    ns ⁡ ( z , k ) Jacobi-elliptic-ns 𝑧 𝑘 {\displaystyle{\displaystyle\operatorname{ns}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E4.m3adec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1akdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3akdec
    0 references
    Q12003
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C22.S1.XMD1.m1wdec
    0 references
    Q11984
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C22.S1.XMD4.m1udec
    0 references
     
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