DLMF:22.12.E2 (Q7040)

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DLMF:22.12.E2
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    Statements

    Defining formula
    2 ⁢ K ⁢ k ⁢ sn ⁡ ( 2 ⁢ K ⁢ t , k ) = ∑ n = - ∞ ∞ π sin ⁡ ( π ⁢ ( t - ( n + 1 2 ) ⁢ τ ) ) = ∑ n = - ∞ ∞ ( ∑ m = - ∞ ∞ ( - 1 ) m t - m - ( n + 1 2 ) ⁢ τ ) , 2 𝐾 𝑘 Jacobi-elliptic-sn 2 𝐾 𝑡 𝑘 superscript subscript 𝑛 𝜋 𝜋 𝑡 𝑛 1 2 𝜏 superscript subscript 𝑛 superscript subscript 𝑚 superscript 1 𝑚 𝑡 𝑚 𝑛 1 2 𝜏 {\displaystyle{\displaystyle 2Kk\operatorname{sn}\left(2Kt,k\right)=\sum_{n=-% \infty}^{\infty}\frac{\pi}{\sin\left(\pi(t-(n+\frac{1}{2})\tau)\right)}=\sum_{% n=-\infty}^{\infty}\left(\sum_{m=-\infty}^{\infty}\frac{(-1)^{m}}{t-m-(n+\frac% {1}{2})\tau}\right),}}
    0 references
    DLMF id
    DLMF:22.12.E2
    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.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.m2adec
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