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Statements
2
K
k
′
nc
(
2
K
t
,
k
)
=
∑
n
=
-
∞
∞
(
-
1
)
n
π
sin
(
π
(
t
+
1
2
-
n
τ
)
)
=
∑
n
=
-
∞
∞
(
∑
m
=
-
∞
∞
(
-
1
)
m
+
n
t
+
1
2
-
m
-
n
τ
)
,
2
𝐾
superscript
𝑘
′
Jacobi-elliptic-nc
2
𝐾
𝑡
𝑘
superscript
subscript
𝑛
superscript
1
𝑛
𝜋
𝜋
𝑡
1
2
𝑛
𝜏
superscript
subscript
𝑛
superscript
subscript
𝑚
superscript
1
𝑚
𝑛
𝑡
1
2
𝑚
𝑛
𝜏
{\displaystyle{\displaystyle 2Kk^{\prime}\operatorname{nc}\left(2Kt,k\right)=%
\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}\pi}{\sin\left(\pi(t+\frac{1}{2}-n\tau)%
\right)}=\sum_{n=-\infty}^{\infty}\left(\sum_{m=-\infty}^{\infty}\frac{(-1)^{m%
+n}}{t+\frac{1}{2}-m-n\tau}\right),}}
nc
(
z
,
k
)
Jacobi-elliptic-nc
𝑧
𝑘
{\displaystyle{\displaystyle\operatorname{nc}\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)}}
sin
z
𝑧
{\displaystyle{\displaystyle\sin\NVar{z}}}
k
𝑘
{\displaystyle{\displaystyle k}}
k
′
superscript
𝑘
′
{\displaystyle{\displaystyle k^{\prime}}}
τ
𝜏
{\displaystyle{\displaystyle\tau}}
