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Statements
P
n
(
cos
θ
)
=
(
2
sin
θ
)
1
2
∑
m
=
0
M
-
1
(
-
1
2
m
)
(
m
-
1
2
n
)
cos
α
n
,
m
(
2
sin
θ
)
m
+
O
(
1
n
M
+
1
2
)
,
Legendre-spherical-polynomial
𝑛
𝜃
superscript
2
𝜃
1
2
superscript
subscript
𝑚
0
𝑀
1
binomial
1
2
𝑚
binomial
𝑚
1
2
𝑛
subscript
𝛼
𝑛
𝑚
superscript
2
𝜃
𝑚
Big-O
1
superscript
𝑛
𝑀
1
2
{\displaystyle{\displaystyle P_{n}\left(\cos\theta\right)=\left(\frac{2}{\sin%
\theta}\right)^{\frac{1}{2}}\sum_{m=0}^{M-1}\genfrac{(}{)}{0.0pt}{}{-\tfrac{1}%
{2}}{m}\genfrac{(}{)}{0.0pt}{}{m-\frac{1}{2}}{n}\frac{\cos\alpha_{n,m}}{(2\sin%
\theta)^{m}}+O\left(\frac{1}{n^{M+\frac{1}{2}}}\right),}}
O
(
x
)
Big-O
𝑥
{\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
P
n
(
x
)
Legendre-spherical-polynomial
𝑛
𝑥
{\displaystyle{\displaystyle P_{\NVar{n}}\left(\NVar{x}\right)}}
(
m
n
)
binomial
𝑚
𝑛
{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}}}
cos
z
𝑧
{\displaystyle{\displaystyle\cos\NVar{z}}}
sin
z
𝑧
{\displaystyle{\displaystyle\sin\NVar{z}}}
m
𝑚
{\displaystyle{\displaystyle m}}
n
𝑛
{\displaystyle{\displaystyle n}}
α
n
,
m
subscript
𝛼
𝑛
𝑚
{\displaystyle{\displaystyle\alpha_{n,m}}}
Identifiers
