DLMF:22.11.E11 (Q7035)

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DLMF:22.11.E11
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    Statements

    Defining formula
    nc ⁡ ( z , k ) - π 2 ⁢ K ⁢ k ′ ⁢ sec ⁡ ζ = - 2 ⁢ π K ⁢ k ′ ⁢ ∑ n = 0 ∞ ( - 1 ) n ⁢ q 2 ⁢ n + 1 ⁢ cos ⁡ ( ( 2 ⁢ n + 1 ) ⁢ ζ ) 1 + q 2 ⁢ n + 1 , Jacobi-elliptic-nc 𝑧 𝑘 𝜋 2 𝐾 superscript 𝑘 ′ 𝜁 2 𝜋 𝐾 superscript 𝑘 ′ superscript subscript 𝑛 0 superscript 1 𝑛 superscript 𝑞 2 𝑛 1 2 𝑛 1 𝜁 1 superscript 𝑞 2 𝑛 1 {\displaystyle{\displaystyle\operatorname{nc}\left(z,k\right)-\frac{\pi}{2Kk^{% \prime}}\sec\zeta=-\frac{2\pi}{Kk^{\prime}}\sum_{n=0}^{\infty}\frac{(-1)^{n}q^% {2n+1}\cos\left((2n+1)\zeta\right)}{1+q^{2n+1}},}}
    0 references
    DLMF id
    DLMF:22.11.E11
    0 references
    Symbols used
    Jacobian elliptic function
    Defining formula
    nc ⁡ ( z , k ) Jacobi-elliptic-nc 𝑧 𝑘 {\displaystyle{\displaystyle\operatorname{nc}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E5.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.m2ajdec
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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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