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Statements
π―
1
2
β‘
(
cos
β‘
ΞΈ
)
=
2
Ο
β’
(
2
β’
E
β‘
(
sin
β‘
(
1
2
β’
ΞΈ
)
)
-
K
β‘
(
sin
β‘
(
1
2
β’
ΞΈ
)
)
)
,
shorthand-Ferrers-Legendre-P-first-kind
1
2
π
2
π
2
complete-elliptic-integral-second-kind-E
1
2
π
complete-elliptic-integral-first-kind-K
1
2
π
{\displaystyle{\displaystyle\mathsf{P}_{\frac{1}{2}}\left(\cos\theta\right)=%
\frac{2}{\pi}\left(2E\left(\sin\left(\tfrac{1}{2}\theta\right)\right)-K\left(%
\sin\left(\tfrac{1}{2}\theta\right)\right)\right),}}
Ο
{\displaystyle{\displaystyle\pi}}
K
β‘
(
k
)
complete-elliptic-integral-first-kind-K
π
{\displaystyle{\displaystyle K\left(\NVar{k}\right)}}
E
β‘
(
k
)
complete-elliptic-integral-second-kind-E
π
{\displaystyle{\displaystyle E\left(\NVar{k}\right)}}
cos
β‘
z
π§
{\displaystyle{\displaystyle\cos\NVar{z}}}
π―
Ξ½
β‘
(
x
)
=
π―
Ξ½
0
β‘
(
x
)
shorthand-Ferrers-Legendre-P-first-kind
π
π₯
Ferrers-Legendre-P-first-kind
0
π
π₯
{\displaystyle{\displaystyle\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=%
\mathsf{P}^{0}_{\nu}\left(x\right)}}
sin
β‘
z
π§
{\displaystyle{\displaystyle\sin\NVar{z}}}
0
<
ΞΈ
<
Ο
0
π
π
{\displaystyle{\displaystyle 0<\theta<\pi}}
Identifiers
