DLMF:22.14.E5 (Q7080)

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DLMF:22.14.E5
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    Statements

    Defining formula
    ∫ sd ⁡ ( x , k ) ⁢ d x = ( k ⁢ k ′ ) - 1 ⁢ Arcsin ⁡ ( - k ⁢ cd ⁡ ( x , k ) ) , Jacobi-elliptic-sd 𝑥 𝑘 𝑥 superscript 𝑘 superscript 𝑘 ′ 1 multivalued-inverse-sine 𝑘 Jacobi-elliptic-cd 𝑥 𝑘 {\displaystyle{\displaystyle\int\operatorname{sd}\left(x,k\right)\mathrm{d}x=(% kk^{\prime})^{-1}\operatorname{Arcsin}\left(-k\operatorname{cd}\left(x,k\right% )\right),}}
    0 references
    DLMF id
    DLMF:22.14.E5
    0 references
    Symbols used
    Jacobian elliptic function
    Defining formula
    cd ⁡ ( z , k ) Jacobi-elliptic-cd 𝑧 𝑘 {\displaystyle{\displaystyle\operatorname{cd}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E8.m2aadec
    0 references
    Jacobian elliptic function
    Defining formula
    sd ⁡ ( z , k ) Jacobi-elliptic-sd 𝑧 𝑘 {\displaystyle{\displaystyle\operatorname{sd}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E7.m2aadec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1addec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3addec
    0 references
    general arcsine function
    Defining formula
    Arcsin ⁡ z multivalued-inverse-sine 𝑧 {\displaystyle{\displaystyle\operatorname{Arcsin}\NVar{z}}}
    xml-id
    C4.S23.E1.m2aadec
    0 references
    Q12003
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C22.S1.XMD1.m1ddec
    0 references
    Q11984
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C22.S1.XMD4.m1ddec
    0 references
    Q11988
    Defining formula
    k ′ superscript 𝑘 ′ {\displaystyle{\displaystyle k^{\prime}}}
    xml-id
    C22.S1.XMD5.m1dec
    0 references
     
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