Deprecated : $wgMWOAuthSharedUserIDs=false is deprecated, set $wgMWOAuthSharedUserIDs=true, $wgMWOAuthSharedUserSource='local' instead [Called from MediaWiki\HookContainer\HookContainer::run in /var/www/html/w/includes/HookContainer/HookContainer.php at line 135] in /var/www/html/w/includes/Debug/MWDebug.php on line 372 DLMF:19.4.E7 - MaRDI portal
Statements
∂
Π
(
ϕ
,
α
2
,
k
)
∂
k
=
k
k
′
2
(
k
2
-
α
2
)
(
E
(
ϕ
,
k
)
-
k
′
2
Π
(
ϕ
,
α
2
,
k
)
-
k
2
sin
ϕ
cos
ϕ
1
-
k
2
sin
2
ϕ
)
.
partial-derivative
elliptic-integral-third-kind-Pi
italic-ϕ
superscript
𝛼
2
𝑘
𝑘
𝑘
superscript
superscript
𝑘
′
2
superscript
𝑘
2
superscript
𝛼
2
elliptic-integral-second-kind-E
italic-ϕ
𝑘
superscript
superscript
𝑘
′
2
elliptic-integral-third-kind-Pi
italic-ϕ
superscript
𝛼
2
𝑘
superscript
𝑘
2
italic-ϕ
italic-ϕ
1
superscript
𝑘
2
2
italic-ϕ
{\displaystyle{\displaystyle\frac{\partial\Pi\left(\phi,\alpha^{2},k\right)}{%
\partial k}=\frac{k}{{k^{\prime}}^{2}(k^{2}-\alpha^{2})}\left({E\left(\phi,k%
\right)-{k^{\prime}}^{2}\Pi\left(\phi,\alpha^{2},k\right)}-\frac{k^{2}\sin\phi%
\cos\phi}{\sqrt{1-k^{2}{\sin^{2}}\phi}}\right).}}
cos
z
𝑧
{\displaystyle{\displaystyle\cos\NVar{z}}}
E
(
ϕ
,
k
)
elliptic-integral-second-kind-E
italic-ϕ
𝑘
{\displaystyle{\displaystyle E\left(\NVar{\phi},\NVar{k}\right)}}
Π
(
ϕ
,
α
2
,
k
)
elliptic-integral-third-kind-Pi
italic-ϕ
superscript
𝛼
2
𝑘
{\displaystyle{\displaystyle\Pi\left(\NVar{\phi},\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}}}
sin
z
𝑧
{\displaystyle{\displaystyle\sin\NVar{z}}}
ϕ
italic-ϕ
{\displaystyle{\displaystyle\phi}}
k
𝑘
{\displaystyle{\displaystyle k}}
k
′
superscript
𝑘
′
{\displaystyle{\displaystyle k^{\prime}}}
α
2
superscript
𝛼
2
{\displaystyle{\displaystyle\alpha^{2}}}
Identifiers
