DLMF:22.12.E8 (Q7046)

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DLMF:22.12.E8
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    2 ⁢ K ⁢ dc ⁡ ( 2 ⁢ K ⁢ t , k ) = ∑ n = - ∞ ∞ π sin ⁡ ( π ⁢ ( t + 1 2 - n ⁢ τ ) ) = ∑ n = - ∞ ∞ ( ∑ m = - ∞ ∞ ( - 1 ) m t + 1 2 - m - n ⁢ τ ) , 2 𝐾 Jacobi-elliptic-dc 2 𝐾 𝑡 𝑘 superscript subscript 𝑛 𝜋 𝜋 𝑡 1 2 𝑛 𝜏 superscript subscript 𝑛 superscript subscript 𝑚 superscript 1 𝑚 𝑡 1 2 𝑚 𝑛 𝜏 {\displaystyle{\displaystyle 2K\operatorname{dc}\left(2Kt,k\right)=\sum_{n=-% \infty}^{\infty}\frac{\pi}{\sin\left(\pi(t+\frac{1}{2}-n\tau)\right)}=\sum_{n=% -\infty}^{\infty}\left(\sum_{m=-\infty}^{\infty}\frac{(-1)^{m}}{t+\frac{1}{2}-% m-n\tau}\right),}}
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    DLMF:22.12.E8
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