DLMF:22.14.E18 (Q7093)

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DLMF:22.14.E18
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    Statements

    test
    ∫ 0 K ⁡ ( k ) ln ⁡ ( dn ⁡ ( t , k ) ) ⁢ d t = 1 2 ⁢ K ⁡ ( k ) ⁢ ln ⁡ k ′ . superscript subscript 0 complete-elliptic-integral-first-kind-K 𝑘 Jacobi-elliptic-dn 𝑡 𝑘 𝑡 1 2 complete-elliptic-integral-first-kind-K 𝑘 superscript 𝑘 ′ {\displaystyle{\displaystyle\int_{0}^{K\left(k\right)}\ln\left(\operatorname{% dn}\left(t,k\right)\right)\mathrm{d}t=\tfrac{1}{2}K\left(k\right)\ln k^{\prime% }.}}
    0 references
    DLMF id
    DLMF:22.14.E18
    0 references
    Symbols used
    Jacobian elliptic function
    test
    dn ⁡ ( z , k ) Jacobi-elliptic-dn 𝑧 𝑘 {\displaystyle{\displaystyle\operatorname{dn}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E6.m2afdec
    0 references
    Q11600
    test
    K ⁡ ( k ) complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle K\left(\NVar{k}\right)}}
    xml-id
    C19.S2.E8.m1abdec
    0 references
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aqdec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aqdec
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    principal branch of logarithm function
    test
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2aldec
    0 references
    Q11984
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C22.S1.XMD4.m1qdec
    0 references
     
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