DLMF:22.11.E6 (Q7030): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: Q11988 / rank
 
Normal rank
Property / Symbols used: Q11988 / qualifier
 
Defining formula:

k ′ superscript 𝑘 ′ {\displaystyle{\displaystyle k^{\prime}}}

k^{\prime}
Property / Symbols used: Q11988 / qualifier
 
xml-id: C22.S1.XMD5.m1adec

Revision as of 15:12, 2 January 2020

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DLMF:22.11.E6
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    Statements

    Defining formula
    nd ⁡ ( z , k ) = π 2 ⁢ K ⁢ k ′ + 2 ⁢ π K ⁢ k ′ ⁢ ∑ n = 1 ∞ ( - 1 ) n ⁢ q n ⁢ cos ⁡ ( 2 ⁢ n ⁢ ζ ) 1 + q 2 ⁢ n . Jacobi-elliptic-nd 𝑧 𝑘 𝜋 2 𝐾 superscript 𝑘 ′ 2 𝜋 𝐾 superscript 𝑘 ′ superscript subscript 𝑛 1 superscript 1 𝑛 superscript 𝑞 𝑛 2 𝑛 𝜁 1 superscript 𝑞 2 𝑛 {\displaystyle{\displaystyle\operatorname{nd}\left(z,k\right)=\frac{\pi}{2Kk^{% \prime}}+\frac{2\pi}{Kk^{\prime}}\sum_{n=1}^{\infty}\frac{(-1)^{n}q^{n}\cos% \left(2n\zeta\right)}{1+q^{2n}}.}}
    0 references
    DLMF id
    DLMF:22.11.E6
    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.m1aedec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2acdec
    0 references
    nome
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C22.S2.E1.m2aedec
    0 references
    Q11985
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C22.S1.XMD3.m1edec
    0 references
    Q11984
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C22.S1.XMD4.m1edec
    0 references
    Q11988
    Defining formula
    k ′ superscript 𝑘 ′ {\displaystyle{\displaystyle k^{\prime}}}
    xml-id
    C22.S1.XMD5.m1adec
    0 references
     
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