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Statements
Π
(
α
2
,
k
)
=
π
4
M
(
1
,
k
′
)
(
2
+
α
2
1
-
α
2
∑
n
=
0
∞
Q
n
)
,
complete-elliptic-integral-third-kind-Pi
superscript
𝛼
2
𝑘
𝜋
4
arithmetic-geometric-mean
1
superscript
𝑘
′
2
superscript
𝛼
2
1
superscript
𝛼
2
superscript
subscript
𝑛
0
subscript
𝑄
𝑛
{\displaystyle{\displaystyle\Pi\left(\alpha^{2},k\right)=\frac{\pi}{4M\left(1,%
k^{\prime}\right)}\left(2+\frac{\alpha^{2}}{1-\alpha^{2}}\sum_{n=0}^{\infty}Q_%
{n}\right),}}
-
∞
<
α
2
<
1
superscript
𝛼
2
1
{\displaystyle{\displaystyle-\infty<\alpha^{2}<1}}
M
(
a
,
g
)
arithmetic-geometric-mean
𝑎
𝑔
{\displaystyle{\displaystyle M\left(\NVar{a},\NVar{g}\right)}}
π
{\displaystyle{\displaystyle\pi}}
Π
(
α
2
,
k
)
complete-elliptic-integral-third-kind-Pi
superscript
𝛼
2
𝑘
{\displaystyle{\displaystyle\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)}}
n
𝑛
{\displaystyle{\displaystyle n}}
k
𝑘
{\displaystyle{\displaystyle k}}
k
′
superscript
𝑘
′
{\displaystyle{\displaystyle k^{\prime}}}
α
2
superscript
𝛼
2
{\displaystyle{\displaystyle\alpha^{2}}}
Q
n
subscript
𝑄
𝑛
{\displaystyle{\displaystyle Q_{n}}}
Identifiers
