DLMF:22.12.E7 (Q7045)

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DLMF:22.12.E7
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    Statements

    Defining formula
    2 ⁢ i ⁢ K ⁢ k ′ ⁢ nd ⁡ ( 2 ⁢ K ⁢ t , k ) = lim N → ∞ ⁡ ∑ n = - N N ( - 1 ) n ⁢ π tan ⁡ ( π ⁢ ( t + 1 2 - ( n + 1 2 ) ⁢ τ ) ) = lim N → ∞ ⁡ ∑ n = - N N ( - 1 ) n ⁢ lim M → ∞ ⁡ ( ∑ m = - M M 1 t + 1 2 - m - ( n + 1 2 ) ⁢ τ ) , 2 𝑖 𝐾 superscript 𝑘 ′ Jacobi-elliptic-nd 2 𝐾 𝑡 𝑘 subscript → 𝑁 superscript subscript 𝑛 𝑁 𝑁 superscript 1 𝑛 𝜋 𝜋 𝑡 1 2 𝑛 1 2 𝜏 subscript → 𝑁 superscript subscript 𝑛 𝑁 𝑁 superscript 1 𝑛 subscript → 𝑀 superscript subscript 𝑚 𝑀 𝑀 1 𝑡 1 2 𝑚 𝑛 1 2 𝜏 {\displaystyle{\displaystyle 2iKk^{\prime}\operatorname{nd}\left(2Kt,k\right)=% \lim_{N\to\infty}\sum_{n=-N}^{N}(-1)^{n}\frac{\pi}{\tan\left(\pi(t+\frac{1}{2}% -(n+\frac{1}{2})\tau)\right)}=\lim_{N\to\infty}\sum_{n=-N}^{N}(-1)^{n}\lim_{M% \to\infty}\left(\sum_{m=-M}^{M}\frac{1}{t+\frac{1}{2}-m-(n+\frac{1}{2})\tau}% \right),}}
    0 references
    DLMF id
    DLMF:22.12.E7
    0 references
    Symbols used
    Jacobian elliptic function
    Defining formula
    nd ⁡ ( z , k ) Jacobi-elliptic-nd 𝑧 𝑘 {\displaystyle{\displaystyle\operatorname{nd}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E6.m3adec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aedec
    0 references
    Q11600
    Defining formula
    K ⁡ ( k ) complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle K\left(\NVar{k}\right)}}
    xml-id
    C19.S2.E8.m1afdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2addec
    0 references
    tangent function
    Defining formula
    tan ⁡ z 𝑧 {\displaystyle{\displaystyle\tan\NVar{z}}}
    xml-id
    C4.S14.E4.m2aadec
    0 references
    Q11984
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C22.S1.XMD4.m1fdec
    0 references
    Q11988
    Defining formula
    k ′ superscript 𝑘 ′ {\displaystyle{\displaystyle k^{\prime}}}
    xml-id
    C22.S1.XMD5.m1adec
    0 references
     
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