DLMF:22.12.E6 (Q7044)

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DLMF:22.12.E6
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    Defining formula
    - 2 ⁢ i ⁢ K ⁢ k ⁢ k ′ ⁢ sd ⁡ ( 2 ⁢ K ⁢ t , k ) = ∑ n = - ∞ ∞ ( - 1 ) n ⁢ π sin ⁡ ( π ⁢ ( t + 1 2 - ( n + 1 2 ) ⁢ τ ) ) = ∑ n = - ∞ ∞ ( ∑ m = - ∞ ∞ ( - 1 ) m + n t + 1 2 - m - ( n + 1 2 ) ⁢ τ ) , 2 𝑖 𝐾 𝑘 superscript 𝑘 ′ Jacobi-elliptic-sd 2 𝐾 𝑡 𝑘 superscript subscript 𝑛 superscript 1 𝑛 𝜋 𝜋 𝑡 1 2 𝑛 1 2 𝜏 superscript subscript 𝑛 superscript subscript 𝑚 superscript 1 𝑚 𝑛 𝑡 1 2 𝑚 𝑛 1 2 𝜏 {\displaystyle{\displaystyle-2iKkk^{\prime}\operatorname{sd}\left(2Kt,k\right)% =\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}\pi}{\sin\left(\pi(t+\frac{1}{2}-(n+% \frac{1}{2})\tau)\right)}=\sum_{n=-\infty}^{\infty}\left(\sum_{m=-\infty}^{% \infty}\frac{(-1)^{m+n}}{t+\frac{1}{2}-m-(n+\frac{1}{2})\tau}\right),}}
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    DLMF:22.12.E6
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