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Statements
k
2
sn
2
(
z
,
k
)
=
E
′
K
′
-
(
π
2
K
′
)
2
∑
n
=
-
∞
∞
(
sech
2
(
π
2
K
′
(
z
-
2
n
K
)
)
)
,
superscript
𝑘
2
Jacobi-elliptic-sn
2
𝑧
𝑘
complementary-complete-elliptic-integral-second-kind-E
𝑘
complementary-complete-elliptic-integral-first-kind-K
𝑘
superscript
𝜋
2
complementary-complete-elliptic-integral-first-kind-K
2
superscript
subscript
𝑛
2
𝜋
2
complementary-complete-elliptic-integral-first-kind-K
𝑘
𝑧
2
𝑛
complete-elliptic-integral-first-kind-K
𝑘
{\displaystyle{\displaystyle k^{2}{\operatorname{sn}^{2}}\left(z,k\right)=%
\frac{{E^{\prime}}}{{K^{\prime}}}-\left(\frac{\pi}{2{K^{\prime}}}\right)^{2}%
\sum_{n=-\infty}^{\infty}\left({\operatorname{sech}^{2}}\left(\frac{\pi}{2{K^{%
\prime}}}(z-2nK)\right)\right),}}
sn
(
z
,
k
)
Jacobi-elliptic-sn
𝑧
𝑘
{\displaystyle{\displaystyle\operatorname{sn}\left(\NVar{z},\NVar{k}\right)}}
π
{\displaystyle{\displaystyle\pi}}
K
′
(
k
)
complementary-complete-elliptic-integral-first-kind-K
𝑘
{\displaystyle{\displaystyle{K^{\prime}}\left(\NVar{k}\right)}}
E
′
(
k
)
complementary-complete-elliptic-integral-second-kind-E
𝑘
{\displaystyle{\displaystyle{E^{\prime}}\left(\NVar{k}\right)}}
K
(
k
)
complete-elliptic-integral-first-kind-K
𝑘
{\displaystyle{\displaystyle K\left(\NVar{k}\right)}}
sech
z
𝑧
{\displaystyle{\displaystyle\operatorname{sech}\NVar{z}}}
z
𝑧
{\displaystyle{\displaystyle z}}
k
𝑘
{\displaystyle{\displaystyle k}}
