DLMF:22.11.E7 (Q7031)

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DLMF:22.11.E7
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    Statements

    Defining formula
    ns ⁡ ( z , k ) - π 2 ⁢ K ⁢ csc ⁡ ζ = 2 ⁢ π K ⁢ ∑ n = 0 ∞ q 2 ⁢ n + 1 ⁢ sin ⁡ ( ( 2 ⁢ n + 1 ) ⁢ ζ ) 1 - q 2 ⁢ n + 1 , Jacobi-elliptic-ns 𝑧 𝑘 𝜋 2 𝐾 𝜁 2 𝜋 𝐾 superscript subscript 𝑛 0 superscript 𝑞 2 𝑛 1 2 𝑛 1 𝜁 1 superscript 𝑞 2 𝑛 1 {\displaystyle{\displaystyle\operatorname{ns}\left(z,k\right)-\frac{\pi}{2K}% \csc\zeta=\frac{2\pi}{K}\sum_{n=0}^{\infty}\frac{q^{2n+1}\sin\left((2n+1)\zeta% \right)}{1-q^{2n+1}},}}
    0 references
    DLMF id
    DLMF:22.11.E7
    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.m2afdec
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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.m1afdec
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    cosecant function
    Defining formula
    csc ⁡ z 𝑧 {\displaystyle{\displaystyle\csc\NVar{z}}}
    xml-id
    C4.S14.E5.m2adec
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    nome
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C22.S2.E1.m2afdec
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    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2abdec
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    Q11985
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C22.S1.XMD3.m1fdec
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    Q11984
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C22.S1.XMD4.m1fdec
    0 references
    Q11986
    Defining formula
    ζ 𝜁 {\displaystyle{\displaystyle\zeta}}
    xml-id
    C22.S2.XMD2.m1fdec
    0 references
     
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