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Statements
∑
p
=
0
∞
A
2
p
+
1
C
2
p
+
1
=
4
π
∫
0
K
(
𝐸𝑐
ν
2
m
+
1
(
x
,
k
2
)
)
2
d
x
.
superscript
subscript
𝑝
0
subscript
𝐴
2
𝑝
1
subscript
𝐶
2
𝑝
1
4
𝜋
superscript
subscript
0
complete-elliptic-integral-first-kind-K
𝑘
superscript
Lame-Ec
2
𝑚
1
𝜈
𝑥
superscript
𝑘
2
2
𝑥
{\displaystyle{\displaystyle\sum_{p=0}^{\infty}A_{2p+1}C_{2p+1}=\frac{4}{\pi}%
\int_{0}^{K}\left(\mathit{Ec}^{2m+1}_{\nu}\left(x,k^{2}\right)\right)^{2}%
\mathrm{d}x.}}
𝐸𝑐
ν
m
(
z
,
k
2
)
Lame-Ec
𝑚
𝜈
𝑧
superscript
𝑘
2
{\displaystyle{\displaystyle\mathit{Ec}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},%
\NVar{k^{2}}\right)}}
π
{\displaystyle{\displaystyle\pi}}
K
(
k
)
complete-elliptic-integral-first-kind-K
𝑘
{\displaystyle{\displaystyle K\left(\NVar{k}\right)}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
∫
{\displaystyle{\displaystyle\int}}
m
𝑚
{\displaystyle{\displaystyle m}}
p
𝑝
{\displaystyle{\displaystyle p}}
x
𝑥
{\displaystyle{\displaystyle x}}
k
𝑘
{\displaystyle{\displaystyle k}}
ν
𝜈
{\displaystyle{\displaystyle\nu}}
A
2
p
subscript
𝐴
2
𝑝
{\displaystyle{\displaystyle A_{2p}}}
C
2
p
+
1
subscript
𝐶
2
𝑝
1
{\displaystyle{\displaystyle C_{2p+1}}}
