DLMF:22.14.E14 (Q7089)

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DLMF:22.14.E14
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    Statements

    Defining formula
    ∫ cn ⁡ ( x , k ) ⁢ d x sn ⁡ ( x , k ) = 1 2 ⁢ ln ⁡ ( 1 - dn ⁡ ( x , k ) 1 + dn ⁡ ( x , k ) ) , Jacobi-elliptic-cn 𝑥 𝑘 𝑥 Jacobi-elliptic-sn 𝑥 𝑘 1 2 1 Jacobi-elliptic-dn 𝑥 𝑘 1 Jacobi-elliptic-dn 𝑥 𝑘 {\displaystyle{\displaystyle\int\frac{\operatorname{cn}\left(x,k\right)\mathrm% {d}x}{\operatorname{sn}\left(x,k\right)}=\frac{1}{2}\ln\left(\frac{1-% \operatorname{dn}\left(x,k\right)}{1+\operatorname{dn}\left(x,k\right)}\right)% ,}}
    0 references
    DLMF id
    DLMF:22.14.E14
    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.m2acdec
    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.m2addec
    0 references
    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.m2acdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1amdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3amdec
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    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2aidec
    0 references
     
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