# DLMF:22.6.E21 (Q6955)

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DLMF:22.6.E21
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## Statements

${\displaystyle{\displaystyle{\operatorname{dn}^{2}}\left(\tfrac{1}{2}z,k\right% )=\frac{k^{2}\operatorname{cn}\left(z,k\right)+\operatorname{dn}\left(z,k% \right)+{k^{\prime}}^{2}}{1+\operatorname{dn}\left(z,k\right)}=\frac{{k^{% \prime}}^{2}(1-\operatorname{cn}\left(z,k\right))}{\operatorname{dn}\left(z,k% \right)-\operatorname{cn}\left(z,k\right)}=\frac{{k^{\prime}}^{2}(1+% \operatorname{dn}\left(z,k\right))}{{k^{\prime}}^{2}+\operatorname{dn}\left(z,% k\right)-k^{2}\operatorname{cn}\left(z,k\right)}.}}$
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DLMF:22.6.E21
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${\displaystyle{\displaystyle\operatorname{cn}\left(\NVar{z},\NVar{k}\right)}}$
C22.S2.E5.m2ahdec
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${\displaystyle{\displaystyle\operatorname{dn}\left(\NVar{z},\NVar{k}\right)}}$
C22.S2.E6.m2ahdec
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${\displaystyle{\displaystyle z}}$
C22.S1.XMD3.m1tdec
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${\displaystyle{\displaystyle k}}$
C22.S1.XMD4.m1tdec
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${\displaystyle{\displaystyle k^{\prime}}}$
C22.S1.XMD5.m1ndec
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