DLMF:22.11.E13 (Q7037)

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DLMF:22.11.E13
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    Statements

    Defining formula
    sn 2 ⁡ ( z , k ) = 1 k 2 ⁢ ( 1 - E ⁡ K ) - 2 ⁢ π 2 k 2 ⁢ K 2 ⁢ ∑ n = 1 ∞ n ⁢ q n 1 - q 2 ⁢ n ⁢ cos ⁡ ( 2 ⁢ n ⁢ ζ ) . Jacobi-elliptic-sn 2 𝑧 𝑘 1 superscript 𝑘 2 1 complete-elliptic-integral-second-kind-E 𝑘 𝐾 2 superscript 𝜋 2 superscript 𝑘 2 superscript 𝐾 2 superscript subscript 𝑛 1 𝑛 superscript 𝑞 𝑛 1 superscript 𝑞 2 𝑛 2 𝑛 𝜁 {\displaystyle{\displaystyle{\operatorname{sn}^{2}}\left(z,k\right)=\frac{1}{k% ^{2}}\left(1-\frac{E}{K}\right)-\frac{2\pi^{2}}{k^{2}K^{2}}\sum_{n=1}^{\infty}% \frac{nq^{n}}{1-q^{2n}}\cos\left(2n\zeta\right).}}
    0 references
    DLMF id
    DLMF:22.11.E13
    0 references
    Symbols used
    Jacobian elliptic function
    Defining formula
    sn ⁡ ( z , k ) Jacobi-elliptic-sn 𝑧 𝑘 {\displaystyle{\displaystyle\operatorname{sn}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E4.m2aadec
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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.m2aldec
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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.m1aldec
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    Q11601
    Defining formula
    E ⁡ ( k ) complete-elliptic-integral-second-kind-E 𝑘 {\displaystyle{\displaystyle E\left(\NVar{k}\right)}}
    xml-id
    C19.S2.E8.m2adec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2afdec
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    nome
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C22.S2.E1.m2aldec
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    Q11985
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C22.S1.XMD3.m1ldec
    0 references
    Q11984
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C22.S1.XMD4.m1ldec
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    Q11986
    Defining formula
    ζ 𝜁 {\displaystyle{\displaystyle\zeta}}
    xml-id
    C22.S2.XMD2.m1ldec
    0 references
     
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