DLMF:22.14.E1 (Q7076)

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DLMF:22.14.E1
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    Statements

    Defining formula
    ∫ sn ⁡ ( x , k ) ⁢ d x = k - 1 ⁢ ln ⁡ ( dn ⁡ ( x , k ) - k ⁢ cn ⁡ ( x , k ) ) , Jacobi-elliptic-sn 𝑥 𝑘 𝑥 superscript 𝑘 1 Jacobi-elliptic-dn 𝑥 𝑘 𝑘 Jacobi-elliptic-cn 𝑥 𝑘 {\displaystyle{\displaystyle\int\operatorname{sn}\left(x,k\right)\mathrm{d}x=k% ^{-1}\ln\left(\operatorname{dn}\left(x,k\right)-k\operatorname{cn}\left(x,k% \right)\right),}}
    0 references
    DLMF id
    DLMF:22.14.E1
    0 references
    Symbols used
    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.m2adec
    0 references
    Jacobian elliptic function
    Defining formula
    dn ⁡ ( z , k ) Jacobi-elliptic-dn 𝑧 𝑘 {\displaystyle{\displaystyle\operatorname{dn}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E6.m2adec
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    Jacobian elliptic function
    Defining formula
    sn ⁡ ( z , k ) Jacobi-elliptic-sn 𝑧 𝑘 {\displaystyle{\displaystyle\operatorname{sn}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E4.m2adec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
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    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2adec
    0 references
    Q12003
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C22.S1.XMD1.m1dec
    0 references
    Q11984
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C22.S1.XMD4.m1dec
    0 references
     
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