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Statements
𝖯
l
(
cos
θ
1
cos
θ
2
+
sin
θ
1
sin
θ
2
cos
(
ϕ
1
-
ϕ
2
)
)
=
4
π
2
l
+
1
∑
m
=
-
l
l
Y
l
,
m
(
θ
1
,
ϕ
1
)
¯
Y
l
,
m
(
θ
2
,
ϕ
2
)
.
shorthand-Ferrers-Legendre-P-first-kind
𝑙
subscript
𝜃
1
subscript
𝜃
2
subscript
𝜃
1
subscript
𝜃
2
subscript
italic-ϕ
1
subscript
italic-ϕ
2
4
𝜋
2
𝑙
1
superscript
subscript
𝑚
𝑙
𝑙
spherical-harmonic-Y
𝑙
𝑚
subscript
𝜃
1
subscript
italic-ϕ
1
spherical-harmonic-Y
𝑙
𝑚
subscript
𝜃
2
subscript
italic-ϕ
2
{\displaystyle{\displaystyle\mathsf{P}_{l}\left(\cos\theta_{1}\cos\theta_{2}+%
\sin\theta_{1}\sin\theta_{2}\cos\left(\phi_{1}-\phi_{2}\right)\right)=\frac{4%
\pi}{2l+1}\sum_{m=-l}^{l}\overline{Y_{{l},{m}}\left(\theta_{1},\phi_{1}\right)%
}Y_{{l},{m}}\left(\theta_{2},\phi_{2}\right).}}
π
{\displaystyle{\displaystyle\pi}}
z
¯
𝑧
{\displaystyle{\displaystyle\overline{\NVar{z}}}}
cos
z
𝑧
{\displaystyle{\displaystyle\cos\NVar{z}}}
𝖯
ν
(
x
)
=
𝖯
ν
0
(
x
)
shorthand-Ferrers-Legendre-P-first-kind
𝜈
𝑥
Ferrers-Legendre-P-first-kind
0
𝜈
𝑥
{\displaystyle{\displaystyle\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=%
\mathsf{P}^{0}_{\nu}\left(x\right)}}
sin
z
𝑧
{\displaystyle{\displaystyle\sin\NVar{z}}}
Y
l
,
m
(
θ
,
ϕ
)
spherical-harmonic-Y
𝑙
𝑚
𝜃
italic-ϕ
{\displaystyle{\displaystyle Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},%
\NVar{\phi}\right)}}
m
𝑚
{\displaystyle{\displaystyle m}}
l
𝑙
{\displaystyle{\displaystyle l}}
Identifiers
