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Statements
cn
2
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1
2
z
,
k
)
=
-
k
′
2
+
dn
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z
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k
)
+
k
2
cn
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z
,
k
)
k
2
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1
+
cn
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z
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k
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)
=
k
′
2
(
1
-
dn
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z
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k
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)
k
2
(
dn
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z
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k
)
-
cn
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z
,
k
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)
=
k
′
2
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1
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cn
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z
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k
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)
k
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dn
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z
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-
k
2
cn
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z
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k
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,
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)},}}
cn
(
z
,
k
)
Jacobi-elliptic-cn
𝑧
𝑘
{\displaystyle{\displaystyle\operatorname{cn}\left(\NVar{z},\NVar{k}\right)}}
dn
(
z
,
k
)
Jacobi-elliptic-dn
𝑧
𝑘
{\displaystyle{\displaystyle\operatorname{dn}\left(\NVar{z},\NVar{k}\right)}}
z
𝑧
{\displaystyle{\displaystyle z}}
k
𝑘
{\displaystyle{\displaystyle k}}
k
′
superscript
𝑘
′
{\displaystyle{\displaystyle k^{\prime}}}
