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Statements
2
i
K
k
′
nd
(
2
K
t
,
k
)
=
lim
N
→
∞
∑
n
=
-
N
N
(
-
1
)
n
π
tan
(
π
(
t
+
1
2
-
(
n
+
1
2
)
τ
)
)
=
lim
N
→
∞
∑
n
=
-
N
N
(
-
1
)
n
lim
M
→
∞
(
∑
m
=
-
M
M
1
t
+
1
2
-
m
-
(
n
+
1
2
)
τ
)
,
2
𝑖
𝐾
superscript
𝑘
′
Jacobi-elliptic-nd
2
𝐾
𝑡
𝑘
subscript
→
𝑁
superscript
subscript
𝑛
𝑁
𝑁
superscript
1
𝑛
𝜋
𝜋
𝑡
1
2
𝑛
1
2
𝜏
subscript
→
𝑁
superscript
subscript
𝑛
𝑁
𝑁
superscript
1
𝑛
subscript
→
𝑀
superscript
subscript
𝑚
𝑀
𝑀
1
𝑡
1
2
𝑚
𝑛
1
2
𝜏
{\displaystyle{\displaystyle 2iKk^{\prime}\operatorname{nd}\left(2Kt,k\right)=%
\lim_{N\to\infty}\sum_{n=-N}^{N}(-1)^{n}\frac{\pi}{\tan\left(\pi(t+\frac{1}{2}%
-(n+\frac{1}{2})\tau)\right)}=\lim_{N\to\infty}\sum_{n=-N}^{N}(-1)^{n}\lim_{M%
\to\infty}\left(\sum_{m=-M}^{M}\frac{1}{t+\frac{1}{2}-m-(n+\frac{1}{2})\tau}%
\right),}}
nd
(
z
,
k
)
Jacobi-elliptic-nd
𝑧
𝑘
{\displaystyle{\displaystyle\operatorname{nd}\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}}}
tan
z
𝑧
{\displaystyle{\displaystyle\tan\NVar{z}}}
k
𝑘
{\displaystyle{\displaystyle k}}
k
′
superscript
𝑘
′
{\displaystyle{\displaystyle k^{\prime}}}
τ
𝜏
{\displaystyle{\displaystyle\tau}}
