DLMF:22.12.E11 (Q7049): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Jacobian elliptic function / rank
 
Normal rank
Property / Symbols used: Jacobian elliptic function / qualifier
 
Defining formula:

ns ⁡ ( z , k ) Jacobi-elliptic-ns 𝑧 𝑘 {\displaystyle{\displaystyle\operatorname{ns}\left(\NVar{z},\NVar{k}\right)}}

\Jacobiellnsk@{\NVar{z}}{\NVar{k}}
Property / Symbols used: Jacobian elliptic function / qualifier
 
xml-id: C22.S2.E4.m3adec
Property / Symbols used
 
Property / Symbols used: the ratio of the circumference of a circle to its diameter / rank
 
Normal rank
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
Defining formula:

π {\displaystyle{\displaystyle\pi}}

\cpi
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
xml-id: C3.S12.E1.m2aidec
Property / Symbols used
 
Property / Symbols used: Q11600 / rank
 
Normal rank
Property / Symbols used: Q11600 / qualifier
 
Defining formula:

K ⁡ ( k ) complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle K\left(\NVar{k}\right)}}

\compellintKk@{\NVar{k}}
Property / Symbols used: Q11600 / qualifier
 
xml-id: C19.S2.E8.m1ajdec
Property / Symbols used
 
Property / Symbols used: sine function / rank
 
Normal rank
Property / Symbols used: sine function / qualifier
 
Defining formula:

sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}

\sin@@{\NVar{z}}
Property / Symbols used: sine function / qualifier
 
xml-id: C4.S14.E1.m2afdec
Property / Symbols used
 
Property / Symbols used: Q11984 / rank
 
Normal rank
Property / Symbols used: Q11984 / qualifier
 
Defining formula:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q11984 / qualifier
 
xml-id: C22.S1.XMD4.m1jdec
Property / Symbols used
 
Property / Symbols used: Q11987 / rank
 
Normal rank
Property / Symbols used: Q11987 / qualifier
 
Defining formula:

τ 𝜏 {\displaystyle{\displaystyle\tau}}

\tau
Property / Symbols used: Q11987 / qualifier
 
xml-id: C22.S1.XMD6.m1jdec

Latest revision as of 15:17, 2 January 2020

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DLMF:22.12.E11
No description defined

    Statements

    Defining formula
    2 ⁢ K ⁢ ns ⁡ ( 2 ⁢ K ⁢ t , k ) = ∑ n = - ∞ ∞ π sin ⁡ ( π ⁢ ( t - n ⁢ τ ) ) = ∑ n = - ∞ ∞ ( ∑ m = - ∞ ∞ ( - 1 ) m t - m - n ⁢ τ ) , 2 𝐾 Jacobi-elliptic-ns 2 𝐾 𝑡 𝑘 superscript subscript 𝑛 𝜋 𝜋 𝑡 𝑛 𝜏 superscript subscript 𝑛 superscript subscript 𝑚 superscript 1 𝑚 𝑡 𝑚 𝑛 𝜏 {\displaystyle{\displaystyle 2K\operatorname{ns}\left(2Kt,k\right)=\sum_{n=-% \infty}^{\infty}\frac{\pi}{\sin\left(\pi(t-n\tau)\right)}=\sum_{n=-\infty}^{% \infty}\left(\sum_{m=-\infty}^{\infty}\frac{(-1)^{m}}{t-m-n\tau}\right),}}
    0 references
    DLMF id
    DLMF:22.12.E11
    0 references
    Symbols used
    Jacobian elliptic function
    Defining formula
    ns ⁡ ( z , k ) Jacobi-elliptic-ns 𝑧 𝑘 {\displaystyle{\displaystyle\operatorname{ns}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E4.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.m2aidec
    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.m1ajdec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2afdec
    0 references
    Q11984
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C22.S1.XMD4.m1jdec
    0 references
    Q11987
    Defining formula
    τ 𝜏 {\displaystyle{\displaystyle\tau}}
    xml-id
    C22.S1.XMD6.m1jdec
    0 references
     
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