DLMF:22.6.E20 (Q6954): Difference between revisions

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DLMF:22.6.E20
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    cn 2 ( 1 2 z , k ) = - k 2 + dn ( z , k ) + k 2 cn ( z , k ) k 2 ( 1 + cn ( z , k ) ) = k 2 ( 1 - dn ( z , k ) ) k 2 ( dn ( z , k ) - cn ( z , k ) ) = k 2 ( 1 + cn ( z , k ) ) k 2 + dn ( z , k ) - k 2 cn ( z , k ) , Jacobi-elliptic-cn 2 1 2 𝑧 𝑘 superscript superscript 𝑘 2 Jacobi-elliptic-dn 𝑧 𝑘 superscript 𝑘 2 Jacobi-elliptic-cn 𝑧 𝑘 superscript 𝑘 2 1 Jacobi-elliptic-cn 𝑧 𝑘 superscript superscript 𝑘 2 1 Jacobi-elliptic-dn 𝑧 𝑘 superscript 𝑘 2 Jacobi-elliptic-dn 𝑧 𝑘 Jacobi-elliptic-cn 𝑧 𝑘 superscript superscript 𝑘 2 1 Jacobi-elliptic-cn 𝑧 𝑘 superscript superscript 𝑘 2 Jacobi-elliptic-dn 𝑧 𝑘 superscript 𝑘 2 Jacobi-elliptic-cn 𝑧 𝑘 {\displaystyle{\displaystyle{\operatorname{cn}^{2}}\left(\tfrac{1}{2}z,k\right% )=\frac{-{k^{\prime}}^{2}+\operatorname{dn}\left(z,k\right)+k^{2}\operatorname% {cn}\left(z,k\right)}{k^{2}(1+\operatorname{cn}\left(z,k\right))}=\frac{{k^{% \prime}}^{2}(1-\operatorname{dn}\left(z,k\right))}{k^{2}(\operatorname{dn}% \left(z,k\right)-\operatorname{cn}\left(z,k\right))}=\frac{{k^{\prime}}^{2}(1+% \operatorname{cn}\left(z,k\right))}{{k^{\prime}}^{2}+\operatorname{dn}\left(z,% k\right)-k^{2}\operatorname{cn}\left(z,k\right)},}}
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    DLMF:22.6.E20
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