DLMF:22.6.E19 (Q6953): 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:

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

\Jacobiellcnk@{\NVar{z}}{\NVar{k}}
Property / Symbols used: Jacobian elliptic function / qualifier
 
xml-id: C22.S2.E5.m2afdec
Property / Symbols used
 
Property / Symbols used: Jacobian elliptic function / rank
 
Normal rank
Property / Symbols used: Jacobian elliptic function / qualifier
 
test:

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

\Jacobielldnk@{\NVar{z}}{\NVar{k}}
Property / Symbols used: Jacobian elliptic function / qualifier
 
xml-id: C22.S2.E6.m2afdec
Property / Symbols used
 
Property / Symbols used: Jacobian elliptic function / rank
 
Normal rank
Property / Symbols used: Jacobian elliptic function / qualifier
 
test:

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

\Jacobiellsnk@{\NVar{z}}{\NVar{k}}
Property / Symbols used: Jacobian elliptic function / qualifier
 
xml-id: C22.S2.E4.m2afdec
Property / Symbols used
 
Property / Symbols used: Q11985 / rank
 
Normal rank
Property / Symbols used: Q11985 / qualifier
 
test:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11985 / qualifier
 
xml-id: C22.S1.XMD3.m1rdec
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.m1rdec
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.m1ldec

Latest revision as of 15:00, 2 January 2020

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DLMF:22.6.E19
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    Statements

    test
    sn 2 ⁡ ( 1 2 ⁢ z , k ) = 1 - cn ⁡ ( z , k ) 1 + dn ⁡ ( z , k ) = 1 - dn ⁡ ( z , k ) k 2 ⁢ ( 1 + cn ⁡ ( z , k ) ) = dn ⁡ ( z , k ) - k 2 ⁢ cn ⁡ ( z , k ) - k ′ 2 k 2 ⁢ ( dn ⁡ ( z , k ) - cn ⁡ ( z , k ) ) , Jacobi-elliptic-sn 2 1 2 𝑧 𝑘 1 Jacobi-elliptic-cn 𝑧 𝑘 1 Jacobi-elliptic-dn 𝑧 𝑘 1 Jacobi-elliptic-dn 𝑧 𝑘 superscript 𝑘 2 1 Jacobi-elliptic-cn 𝑧 𝑘 Jacobi-elliptic-dn 𝑧 𝑘 superscript 𝑘 2 Jacobi-elliptic-cn 𝑧 𝑘 superscript superscript 𝑘 ′ 2 superscript 𝑘 2 Jacobi-elliptic-dn 𝑧 𝑘 Jacobi-elliptic-cn 𝑧 𝑘 {\displaystyle{\displaystyle{\operatorname{sn}^{2}}\left(\tfrac{1}{2}z,k\right% )=\frac{1-\operatorname{cn}\left(z,k\right)}{1+\operatorname{dn}\left(z,k% \right)}=\frac{1-\operatorname{dn}\left(z,k\right)}{k^{2}(1+\operatorname{cn}% \left(z,k\right))}=\frac{\operatorname{dn}\left(z,k\right)-k^{2}\operatorname{% cn}\left(z,k\right)-{k^{\prime}}^{2}}{k^{2}(\operatorname{dn}\left(z,k\right)-% \operatorname{cn}\left(z,k\right))},}}
    0 references
    DLMF id
    DLMF:22.6.E19
    0 references
    Symbols used
    Jacobian elliptic function
    test
    cn ⁡ ( z , k ) Jacobi-elliptic-cn 𝑧 𝑘 {\displaystyle{\displaystyle\operatorname{cn}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E5.m2afdec
    0 references
    Jacobian elliptic function
    test
    dn ⁡ ( z , k ) Jacobi-elliptic-dn 𝑧 𝑘 {\displaystyle{\displaystyle\operatorname{dn}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E6.m2afdec
    0 references
    Jacobian elliptic function
    test
    sn ⁡ ( z , k ) Jacobi-elliptic-sn 𝑧 𝑘 {\displaystyle{\displaystyle\operatorname{sn}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E4.m2afdec
    0 references
    Q11985
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C22.S1.XMD3.m1rdec
    0 references
    Q11984
    test
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C22.S1.XMD4.m1rdec
    0 references
    Q11988
    test
    k ′ superscript 𝑘 ′ {\displaystyle{\displaystyle k^{\prime}}}
    xml-id
    C22.S1.XMD5.m1ldec
    0 references
     
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