DLMF:22.11.E2 (Q7026)

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DLMF:22.11.E2
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    Statements

    Defining formula
    cn ⁡ ( z , k ) = 2 ⁢ π K ⁢ k ⁢ ∑ n = 0 ∞ q n + 1 2 ⁢ cos ⁡ ( ( 2 ⁢ n + 1 ) ⁢ ζ ) 1 + q 2 ⁢ n + 1 , Jacobi-elliptic-cn 𝑧 𝑘 2 𝜋 𝐾 𝑘 superscript subscript 𝑛 0 superscript 𝑞 𝑛 1 2 2 𝑛 1 𝜁 1 superscript 𝑞 2 𝑛 1 {\displaystyle{\displaystyle\operatorname{cn}\left(z,k\right)=\frac{2\pi}{Kk}% \sum_{n=0}^{\infty}\frac{q^{n+\frac{1}{2}}\cos\left((2n+1)\zeta\right)}{1+q^{2% n+1}},}}
    0 references
    DLMF id
    DLMF:22.11.E2
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    Symbols used
    Jacobian elliptic function
    Defining formula
    cn ⁡ ( z , k ) Jacobi-elliptic-cn 𝑧 𝑘 {\displaystyle{\displaystyle\operatorname{cn}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E5.m2adec
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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.m2aadec
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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.m1aadec
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    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2adec
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