DLMF:22.12.E11 (Q7049)

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DLMF:22.12.E11
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    Statements

    Defining formula
    2 ⁢ K ⁢ ns ⁡ ( 2 ⁢ K ⁢ t , k ) = ∑ n = - ∞ ∞ π sin ⁡ ( π ⁢ ( t - n ⁢ τ ) ) = ∑ n = - ∞ ∞ ( ∑ m = - ∞ ∞ ( - 1 ) m t - m - n ⁢ τ ) , 2 𝐾 Jacobi-elliptic-ns 2 𝐾 𝑡 𝑘 superscript subscript 𝑛 𝜋 𝜋 𝑡 𝑛 𝜏 superscript subscript 𝑛 superscript subscript 𝑚 superscript 1 𝑚 𝑡 𝑚 𝑛 𝜏 {\displaystyle{\displaystyle 2K\operatorname{ns}\left(2Kt,k\right)=\sum_{n=-% \infty}^{\infty}\frac{\pi}{\sin\left(\pi(t-n\tau)\right)}=\sum_{n=-\infty}^{% \infty}\left(\sum_{m=-\infty}^{\infty}\frac{(-1)^{m}}{t-m-n\tau}\right),}}
    0 references
    DLMF id
    DLMF:22.12.E11
    0 references
    Symbols used
    Jacobian elliptic function
    Defining formula
    ns ⁡ ( z , k ) Jacobi-elliptic-ns 𝑧 𝑘 {\displaystyle{\displaystyle\operatorname{ns}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E4.m3adec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aidec
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    Q11600
    Defining formula
    K ⁡ ( k ) complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle K\left(\NVar{k}\right)}}
    xml-id
    C19.S2.E8.m1ajdec
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    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2afdec
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    Q11984
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C22.S1.XMD4.m1jdec
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    Q11987
    Defining formula
    τ 𝜏 {\displaystyle{\displaystyle\tau}}
    xml-id
    C22.S1.XMD6.m1jdec
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