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Statements
2
i
K
k
cn
(
2
K
t
,
k
)
=
∑
n
=
-
∞
∞
(
-
1
)
n
π
sin
(
π
(
t
-
(
n
+
1
2
)
τ
)
)
=
∑
n
=
-
∞
∞
(
∑
m
=
-
∞
∞
(
-
1
)
m
+
n
t
-
m
-
(
n
+
1
2
)
τ
)
,
2
𝑖
𝐾
𝑘
Jacobi-elliptic-cn
2
𝐾
𝑡
𝑘
superscript
subscript
𝑛
superscript
1
𝑛
𝜋
𝜋
𝑡
𝑛
1
2
𝜏
superscript
subscript
𝑛
superscript
subscript
𝑚
superscript
1
𝑚
𝑛
𝑡
𝑚
𝑛
1
2
𝜏
{\displaystyle{\displaystyle 2iKk\operatorname{cn}\left(2Kt,k\right)=\sum_{n=-%
\infty}^{\infty}\frac{(-1)^{n}\pi}{\sin\left(\pi(t-(n+\frac{1}{2})\tau)\right)%
}=\sum_{n=-\infty}^{\infty}\left(\sum_{m=-\infty}^{\infty}\frac{(-1)^{m+n}}{t-%
m-(n+\frac{1}{2})\tau}\right),}}
cn
(
z
,
k
)
Jacobi-elliptic-cn
𝑧
𝑘
{\displaystyle{\displaystyle\operatorname{cn}\left(\NVar{z},\NVar{k}\right)}}
π
{\displaystyle{\displaystyle\pi}}
K
(
k
)
complete-elliptic-integral-first-kind-K
𝑘
{\displaystyle{\displaystyle K\left(\NVar{k}\right)}}
i
imaginary-unit
{\displaystyle{\displaystyle\mathrm{i}}}
sin
z
𝑧
{\displaystyle{\displaystyle\sin\NVar{z}}}
k
𝑘
{\displaystyle{\displaystyle k}}
τ
𝜏
{\displaystyle{\displaystyle\tau}}
