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Statements
4
π
R
G
(
0
,
a
0
2
,
g
0
2
)
=
1
M
(
a
0
,
g
0
)
(
a
0
2
-
∑
n
=
0
∞
2
n
-
1
c
n
2
)
=
1
M
(
a
0
,
g
0
)
(
a
1
2
-
∑
n
=
2
∞
2
n
-
1
c
n
2
)
,
4
𝜋
Carlson-integral-RG
0
superscript
subscript
𝑎
0
2
superscript
subscript
𝑔
0
2
1
arithmetic-geometric-mean
subscript
𝑎
0
subscript
𝑔
0
superscript
subscript
𝑎
0
2
superscript
subscript
𝑛
0
superscript
2
𝑛
1
superscript
subscript
𝑐
𝑛
2
1
arithmetic-geometric-mean
subscript
𝑎
0
subscript
𝑔
0
superscript
subscript
𝑎
1
2
superscript
subscript
𝑛
2
superscript
2
𝑛
1
superscript
subscript
𝑐
𝑛
2
{\displaystyle{\displaystyle\frac{4}{\pi}R_{G}\left(0,a_{0}^{2},g_{0}^{2}%
\right)=\frac{1}{M\left(a_{0},g_{0}\right)}\left(a_{0}^{2}-\sum_{n=0}^{\infty}%
2^{n-1}c_{n}^{2}\right)=\frac{1}{M\left(a_{0},g_{0}\right)}\left(a_{1}^{2}-%
\sum_{n=2}^{\infty}2^{n-1}c_{n}^{2}\right),}}
R
G
(
x
,
y
,
z
)
Carlson-integral-RG
𝑥
𝑦
𝑧
{\displaystyle{\displaystyle R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)}}
M
(
a
,
g
)
arithmetic-geometric-mean
𝑎
𝑔
{\displaystyle{\displaystyle M\left(\NVar{a},\NVar{g}\right)}}
π
{\displaystyle{\displaystyle\pi}}
n
𝑛
{\displaystyle{\displaystyle n}}
a
n
subscript
𝑎
𝑛
{\displaystyle{\displaystyle a_{n}}}
g
n
subscript
𝑔
𝑛
{\displaystyle{\displaystyle g_{n}}}
Identifiers
