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Statements
E
(
k
)
=
K
(
k
)
+
2
π
2
K
(
k
)
∑
n
=
1
∞
(
-
1
)
n
n
2
q
n
2
1
+
2
∑
n
=
1
∞
(
-
1
)
n
q
n
2
,
complete-elliptic-integral-second-kind-E
𝑘
complete-elliptic-integral-first-kind-K
𝑘
2
superscript
𝜋
2
complete-elliptic-integral-first-kind-K
𝑘
superscript
subscript
𝑛
1
superscript
1
𝑛
superscript
𝑛
2
superscript
𝑞
superscript
𝑛
2
1
2
superscript
subscript
𝑛
1
superscript
1
𝑛
superscript
𝑞
superscript
𝑛
2
{\displaystyle{\displaystyle E\left(k\right)=K\left(k\right)+\frac{2\pi^{2}}{K%
\left(k\right)}\,\frac{\sum_{n=1}^{\infty}(-1)^{n}n^{2}q^{n^{2}}}{1+2\sum_{n=1%
}^{\infty}(-1)^{n}q^{n^{2}}},}}
|
q
|
<
1
𝑞
1
{\displaystyle{\displaystyle|q|<1}}
π
{\displaystyle{\displaystyle\pi}}
K
(
k
)
complete-elliptic-integral-first-kind-K
𝑘
{\displaystyle{\displaystyle K\left(\NVar{k}\right)}}
E
(
k
)
complete-elliptic-integral-second-kind-E
𝑘
{\displaystyle{\displaystyle E\left(\NVar{k}\right)}}
q
𝑞
{\displaystyle{\displaystyle q}}
n
𝑛
{\displaystyle{\displaystyle n}}
k
𝑘
{\displaystyle{\displaystyle k}}
Identifiers
