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DLMF:22.12.E10 - MaRDI portal

DLMF:22.12.E10 (Q7741)

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DLMF:22.12.E10
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    Statements

    DLMF defining formula
    - 2 ⁢ K ⁢ k ′ ⁢ sc ⁡ ( 2 ⁢ K ⁢ t , k ) = lim N → ∞ ⁡ ∑ n = - N N ( - 1 ) n ⁢ π tan ⁡ ( π ⁢ ( t + 1 2 - n ⁢ τ ) ) = lim N → ∞ ⁡ ∑ n = - N N ( - 1 ) n ⁢ ( lim M → ∞ ⁡ ∑ m = - M M 1 t + 1 2 - m - n ⁢ τ ) , 2 𝐾 superscript 𝑘 ′ Jacobi-elliptic-sc 2 𝐾 𝑡 𝑘 subscript → 𝑁 superscript subscript 𝑛 𝑁 𝑁 superscript 1 𝑛 𝜋 𝜋 𝑡 1 2 𝑛 𝜏 subscript → 𝑁 superscript subscript 𝑛 𝑁 𝑁 superscript 1 𝑛 subscript → 𝑀 superscript subscript 𝑚 𝑀 𝑀 1 𝑡 1 2 𝑚 𝑛 𝜏 {\displaystyle{\displaystyle-2Kk^{\prime}\operatorname{sc}\left(2Kt,k\right)=% \lim_{N\to\infty}\sum_{n=-N}^{N}(-1)^{n}\frac{\pi}{\tan\left(\pi(t+\frac{1}{2}% -n\tau)\right)}=\lim_{N\to\infty}\sum_{n=-N}^{N}(-1)^{n}\left(\lim_{M\to\infty% }\sum_{m=-M}^{M}\frac{1}{t+\frac{1}{2}-m-n\tau}\right),}}
    0 references
    Symbols used
    Jacobian elliptic function
    DLMF defining formula
    sc ⁡ ( z , k ) Jacobi-elliptic-sc 𝑧 𝑘 {\displaystyle{\displaystyle\operatorname{sc}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E9.m2adec
    0 references
    the ratio of the circumference of a circle to its diameter
    DLMF defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2ahdec
    0 references
    Legendre’s complete elliptic integral of the first kind
    DLMF defining formula
    K ⁡ ( k ) complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle K\left(\NVar{k}\right)}}
    xml-id
    C19.S2.E8.m1aidec
    0 references
    tangent function
    DLMF defining formula
    tan ⁡ z 𝑧 {\displaystyle{\displaystyle\tan\NVar{z}}}
    xml-id
    C4.S14.E4.m2abdec
    0 references
    modulus
    DLMF defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C22.S1.XMD4.m1idec
    0 references
    complementary modulus
    DLMF defining formula
    k ′ superscript 𝑘 ′ {\displaystyle{\displaystyle k^{\prime}}}
    xml-id
    C22.S1.XMD5.m1cdec
    0 references
    ratio
    DLMF defining formula
    τ 𝜏 {\displaystyle{\displaystyle\tau}}
    xml-id
    C22.S1.XMD6.m1idec
    0 references

    Identifiers

     
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