DLMF:22.6.E6 (Q6940): 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.m1bdec

Latest revision as of 14:57, 2 January 2020

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DLMF:22.6.E6
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    Statements

    Defining formula
    cn ⁡ ( 2 ⁢ z , k ) = cn 2 ⁡ ( z , k ) - sn 2 ⁡ ( z , k ) ⁢ dn 2 ⁡ ( z , k ) 1 - k 2 ⁢ sn 4 ⁡ ( z , k ) = cn 4 ⁡ ( z , k ) - k ′ 2 ⁢ sn 4 ⁡ ( z , k ) 1 - k 2 ⁢ sn 4 ⁡ ( z , k ) , Jacobi-elliptic-cn 2 𝑧 𝑘 Jacobi-elliptic-cn 2 𝑧 𝑘 Jacobi-elliptic-sn 2 𝑧 𝑘 Jacobi-elliptic-dn 2 𝑧 𝑘 1 superscript 𝑘 2 Jacobi-elliptic-sn 4 𝑧 𝑘 Jacobi-elliptic-cn 4 𝑧 𝑘 superscript superscript 𝑘 ′ 2 Jacobi-elliptic-sn 4 𝑧 𝑘 1 superscript 𝑘 2 Jacobi-elliptic-sn 4 𝑧 𝑘 {\displaystyle{\displaystyle\operatorname{cn}\left(2z,k\right)=\frac{{% \operatorname{cn}^{2}}\left(z,k\right)-{\operatorname{sn}^{2}}\left(z,k\right)% {\operatorname{dn}^{2}}\left(z,k\right)}{1-k^{2}{\operatorname{sn}^{4}}\left(z% ,k\right)}=\frac{{\operatorname{cn}^{4}}\left(z,k\right)-{k^{\prime}}^{2}{% \operatorname{sn}^{4}}\left(z,k\right)}{1-k^{2}{\operatorname{sn}^{4}}\left(z,% k\right)},}}
    0 references
    DLMF id
    DLMF:22.6.E6
    0 references
    Symbols used
    Jacobian elliptic function
    Defining formula
    cn ⁡ ( z , k ) Jacobi-elliptic-cn 𝑧 𝑘 {\displaystyle{\displaystyle\operatorname{cn}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E5.m2abdec
    0 references
    Jacobian elliptic function
    Defining formula
    dn ⁡ ( z , k ) Jacobi-elliptic-dn 𝑧 𝑘 {\displaystyle{\displaystyle\operatorname{dn}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E6.m2abdec
    0 references
    Jacobian elliptic function
    Defining formula
    sn ⁡ ( z , k ) Jacobi-elliptic-sn 𝑧 𝑘 {\displaystyle{\displaystyle\operatorname{sn}\left(\NVar{z},\NVar{k}\right)}}
    xml-id
    C22.S2.E4.m2abdec
    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.m1bdec
    0 references
     
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