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Statements
∂
Π
(
α
2
,
k
)
∂
k
=
k
k
′
2
(
k
2
-
α
2
)
(
E
(
k
)
-
k
′
2
Π
(
α
2
,
k
)
)
.
partial-derivative
complete-elliptic-integral-third-kind-Pi
superscript
𝛼
2
𝑘
𝑘
𝑘
superscript
superscript
𝑘
′
2
superscript
𝑘
2
superscript
𝛼
2
complete-elliptic-integral-second-kind-E
𝑘
superscript
superscript
𝑘
′
2
complete-elliptic-integral-third-kind-Pi
superscript
𝛼
2
𝑘
{\displaystyle{\displaystyle\frac{\partial\Pi\left(\alpha^{2},k\right)}{%
\partial k}=\frac{k}{{k^{\prime}}^{2}(k^{2}-\alpha^{2})}(E\left(k\right)-{k^{%
\prime}}^{2}\Pi\left(\alpha^{2},k\right)).}}
E
(
k
)
complete-elliptic-integral-second-kind-E
𝑘
{\displaystyle{\displaystyle E\left(\NVar{k}\right)}}
Π
(
α
2
,
k
)
complete-elliptic-integral-third-kind-Pi
superscript
𝛼
2
𝑘
{\displaystyle{\displaystyle\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)}}
∂
f
∂
x
partial-derivative
𝑓
𝑥
{\displaystyle{\displaystyle\frac{\partial\NVar{f}}{\partial\NVar{x}}}}
∂
x
𝑥
{\displaystyle{\displaystyle\partial\NVar{x}}}
k
𝑘
{\displaystyle{\displaystyle k}}
k
′
superscript
𝑘
′
{\displaystyle{\displaystyle k^{\prime}}}
α
2
superscript
𝛼
2
{\displaystyle{\displaystyle\alpha^{2}}}
