DLMF:22.14.E16 (Q7091)

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DLMF:22.14.E16
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    Statements

    Defining formula
    ∫ 0 K ⁡ ( k ) ln ⁡ ( sn ⁡ ( t , k ) ) ⁢ d t = - 1 4 ⁢ K ′ ⁡ ( k ) - 1 2 ⁢ K ⁡ ( k ) ⁢ ln ⁡ k , superscript subscript 0 complete-elliptic-integral-first-kind-K 𝑘 Jacobi-elliptic-sn 𝑡 𝑘 𝑡 1 4 complementary-complete-elliptic-integral-first-kind-K 𝑘 1 2 complete-elliptic-integral-first-kind-K 𝑘 𝑘 {\displaystyle{\displaystyle\int_{0}^{K\left(k\right)}\ln\left(\operatorname{% sn}\left(t,k\right)\right)\mathrm{d}t=-\tfrac{1}{4}{K^{\prime}}\left(k\right)-% \tfrac{1}{2}K\left(k\right)\ln k,}}
    0 references
    DLMF id
    DLMF:22.14.E16
    0 references
    Symbols used
    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.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.m1adec
    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.m1adec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aodec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aodec
    0 references
    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2ajdec
    0 references
    Q11984
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C22.S1.XMD4.m1odec
    0 references
     
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