DLMF:22.6.E21 (Q6955)

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DLMF:22.6.E21
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    Statements

    Defining formula
    dn 2 ⁡ ( 1 2 ⁢ z , k ) = k 2 ⁢ cn ⁡ ( z , k ) + dn ⁡ ( z , k ) + k ′ 2 1 + dn ⁡ ( z , k ) = k ′ 2 ⁢ ( 1 - cn ⁡ ( z , k ) ) dn ⁡ ( z , k ) - cn ⁡ ( z , k ) = k ′ 2 ⁢ ( 1 + dn ⁡ ( z , k ) ) k ′ 2 + dn ⁡ ( z , k ) - k 2 ⁢ cn ⁡ ( z , k ) . Jacobi-elliptic-dn 2 1 2 𝑧 𝑘 superscript 𝑘 2 Jacobi-elliptic-cn 𝑧 𝑘 Jacobi-elliptic-dn 𝑧 𝑘 superscript superscript 𝑘 ′ 2 1 Jacobi-elliptic-dn 𝑧 𝑘 superscript superscript 𝑘 ′ 2 1 Jacobi-elliptic-cn 𝑧 𝑘 Jacobi-elliptic-dn 𝑧 𝑘 Jacobi-elliptic-cn 𝑧 𝑘 superscript superscript 𝑘 ′ 2 1 Jacobi-elliptic-dn 𝑧 𝑘 superscript superscript 𝑘 ′ 2 Jacobi-elliptic-dn 𝑧 𝑘 superscript 𝑘 2 Jacobi-elliptic-cn 𝑧 𝑘 {\displaystyle{\displaystyle{\operatorname{dn}^{2}}\left(\tfrac{1}{2}z,k\right% )=\frac{k^{2}\operatorname{cn}\left(z,k\right)+\operatorname{dn}\left(z,k% \right)+{k^{\prime}}^{2}}{1+\operatorname{dn}\left(z,k\right)}=\frac{{k^{% \prime}}^{2}(1-\operatorname{cn}\left(z,k\right))}{\operatorname{dn}\left(z,k% \right)-\operatorname{cn}\left(z,k\right)}=\frac{{k^{\prime}}^{2}(1+% \operatorname{dn}\left(z,k\right))}{{k^{\prime}}^{2}+\operatorname{dn}\left(z,% k\right)-k^{2}\operatorname{cn}\left(z,k\right)}.}}
    0 references
    DLMF id
    DLMF:22.6.E21
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    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.m2ahdec
    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.m2ahdec
    0 references
     
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