DLMF:22.14.E17 (Q7092)

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DLMF:22.14.E17
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    Statements

    Defining formula
    ∫ 0 K ⁡ ( k ) ln ⁡ ( cn ⁡ ( t , k ) ) ⁢ d t = - 1 4 ⁢ K ′ ⁡ ( k ) + 1 2 ⁢ K ⁡ ( k ) ⁢ ln ⁡ ( k ′ / k ) , superscript subscript 0 complete-elliptic-integral-first-kind-K 𝑘 Jacobi-elliptic-cn 𝑡 𝑘 𝑡 1 4 complementary-complete-elliptic-integral-first-kind-K 𝑘 1 2 complete-elliptic-integral-first-kind-K 𝑘 superscript 𝑘 ′ 𝑘 {\displaystyle{\displaystyle\int_{0}^{K\left(k\right)}\ln\left(\operatorname{% cn}\left(t,k\right)\right)\mathrm{d}t=-\tfrac{1}{4}{K^{\prime}}\left(k\right)+% \tfrac{1}{2}K\left(k\right)\ln\left(k^{\prime}/k\right),}}
    0 references
    DLMF id
    DLMF:22.14.E17
    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.m2aedec
    0 references
    Q11821
    Defining formula
    K ′ ⁡ ( k ) complementary-complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle{K^{\prime}}\left(\NVar{k}\right)}}
    xml-id
    C19.S2.E9.m1aadec
    0 references
    Q11600
    Defining formula
    K ⁡ ( k ) complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle K\left(\NVar{k}\right)}}
    xml-id
    C19.S2.E8.m1aadec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1apdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3apdec
    0 references
     
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