DLMF:22.12.E9 (Q7047): 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
 
test:

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

\Jacobiellnck@{\NVar{z}}{\NVar{k}}
Property / Symbols used: Jacobian elliptic function / qualifier
 
xml-id: C22.S2.E5.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
 
test:

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.m2agdec
Property / Symbols used
 
Property / Symbols used: Q11600 / rank
 
Normal rank
Property / Symbols used: Q11600 / qualifier
 
test:

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.m1ahdec
Property / Symbols used
 
Property / Symbols used: sine function / rank
 
Normal rank
Property / Symbols used: sine function / qualifier
 
test:

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

\sin@@{\NVar{z}}
Property / Symbols used: sine function / qualifier
 
xml-id: C4.S14.E1.m2aedec
Property / Symbols used
 
Property / Symbols used: Q11984 / rank
 
Normal rank
Property / Symbols used: Q11984 / qualifier
 
test:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q11984 / qualifier
 
xml-id: C22.S1.XMD4.m1hdec
Property / Symbols used
 
Property / Symbols used: Q11988 / rank
 
Normal rank
Property / Symbols used: Q11988 / qualifier
 
test:

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

k^{\prime}
Property / Symbols used: Q11988 / qualifier
 
xml-id: C22.S1.XMD5.m1bdec
Property / Symbols used
 
Property / Symbols used: Q11987 / rank
 
Normal rank
Property / Symbols used: Q11987 / qualifier
 
test:

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

\tau
Property / Symbols used: Q11987 / qualifier
 
xml-id: C22.S1.XMD6.m1hdec

Latest revision as of 14:17, 2 January 2020

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DLMF:22.12.E9
No description defined

    Statements

    test
    2 ⁢ K ⁢ k ′ ⁢ nc ⁡ ( 2 ⁢ K ⁢ t , k ) = ∑ n = - ∞ ∞ ( - 1 ) n ⁢ π sin ⁡ ( π ⁢ ( t + 1 2 - n ⁢ τ ) ) = ∑ n = - ∞ ∞ ( ∑ m = - ∞ ∞ ( - 1 ) m + n t + 1 2 - m - n ⁢ τ ) , 2 𝐾 superscript 𝑘 ′ Jacobi-elliptic-nc 2 𝐾 𝑡 𝑘 superscript subscript 𝑛 superscript 1 𝑛 𝜋 𝜋 𝑡 1 2 𝑛 𝜏 superscript subscript 𝑛 superscript subscript 𝑚 superscript 1 𝑚 𝑛 𝑡 1 2 𝑚 𝑛 𝜏 {\displaystyle{\displaystyle 2Kk^{\prime}\operatorname{nc}\left(2Kt,k\right)=% \sum_{n=-\infty}^{\infty}\frac{(-1)^{n}\pi}{\sin\left(\pi(t+\frac{1}{2}-n\tau)% \right)}=\sum_{n=-\infty}^{\infty}\left(\sum_{m=-\infty}^{\infty}\frac{(-1)^{m% +n}}{t+\frac{1}{2}-m-n\tau}\right),}}
    0 references
    DLMF id
    DLMF:22.12.E9
    0 references
    Symbols used
    Jacobian elliptic function
    test
    nc ⁡ ( z , k ) Jacobi-elliptic-nc 𝑧 𝑘 {\displaystyle{\displaystyle\operatorname{nc}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E5.m3adec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2agdec
    0 references
    Q11600
    test
    K ⁡ ( k ) complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle K\left(\NVar{k}\right)}}
    xml-id
    C19.S2.E8.m1ahdec
    0 references
    sine function
    test
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2aedec
    0 references
    Q11984
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C22.S1.XMD4.m1hdec
    0 references
    Q11988
    test
    k ′ superscript 𝑘 ′ {\displaystyle{\displaystyle k^{\prime}}}
    xml-id
    C22.S1.XMD5.m1bdec
    0 references
    Q11987
    test
    τ 𝜏 {\displaystyle{\displaystyle\tau}}
    xml-id
    C22.S1.XMD6.m1hdec
    0 references
     
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