DLMF:22.11.E5 (Q7029)

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DLMF:22.11.E5
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    Statements

    test
    sd ⁡ ( z , k ) = 2 ⁢ π K ⁢ k ⁢ k ′ ⁢ ∑ n = 0 ∞ ( - 1 ) n ⁢ q n + 1 2 ⁢ sin ⁡ ( ( 2 ⁢ n + 1 ) ⁢ ζ ) 1 + q 2 ⁢ n + 1 , Jacobi-elliptic-sd 𝑧 𝑘 2 𝜋 𝐾 𝑘 superscript 𝑘 ′ superscript subscript 𝑛 0 superscript 1 𝑛 superscript 𝑞 𝑛 1 2 2 𝑛 1 𝜁 1 superscript 𝑞 2 𝑛 1 {\displaystyle{\displaystyle\operatorname{sd}\left(z,k\right)=\frac{2\pi}{Kkk^% {\prime}}\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{n+\frac{1}{2}}\sin\left((2n+1)% \zeta\right)}{1+q^{2n+1}},}}
    0 references
    DLMF id
    DLMF:22.11.E5
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    Symbols used
    Jacobian elliptic function
    test
    sd ⁡ ( z , k ) Jacobi-elliptic-sd 𝑧 𝑘 {\displaystyle{\displaystyle\operatorname{sd}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E7.m2adec
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    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2addec
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    Q11600
    test
    K ⁡ ( k ) complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle K\left(\NVar{k}\right)}}
    xml-id
    C19.S2.E8.m1addec
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    nome
    test
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C22.S2.E1.m2addec
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