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Statements
dn
2
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1
2
z
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k
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=
k
2
cn
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z
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k
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+
dn
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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
(
1
-
cn
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z
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k
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)
dn
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z
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k
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-
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
′
2
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dn
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k
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cn
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z
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.
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)}.}}
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}}}
Identifiers
