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Statements
P
ν
(
cosh
ξ
1
cosh
ξ
2
-
sinh
ξ
1
sinh
ξ
2
cos
ϕ
)
=
P
ν
(
cosh
ξ
1
)
P
ν
(
cosh
ξ
2
)
+
2
∑
m
=
1
∞
(
-
1
)
m
P
ν
-
m
(
cosh
ξ
1
)
P
ν
m
(
cosh
ξ
2
)
cos
(
m
ϕ
)
,
shorthand-Legendre-P-first-kind
𝜈
subscript
𝜉
1
subscript
𝜉
2
subscript
𝜉
1
subscript
𝜉
2
italic-ϕ
shorthand-Legendre-P-first-kind
𝜈
subscript
𝜉
1
shorthand-Legendre-P-first-kind
𝜈
subscript
𝜉
2
2
superscript
subscript
𝑚
1
superscript
1
𝑚
Legendre-P-first-kind
𝑚
𝜈
subscript
𝜉
1
Legendre-P-first-kind
𝑚
𝜈
subscript
𝜉
2
𝑚
italic-ϕ
{\displaystyle{\displaystyle P_{\nu}\left(\cosh\xi_{1}\cosh\xi_{2}-\sinh\xi_{1%
}\sinh\xi_{2}\cos\phi\right)=P_{\nu}\left(\cosh\xi_{1}\right)P_{\nu}\left(%
\cosh\xi_{2}\right)+2\sum_{m=1}^{\infty}(-1)^{m}P^{-m}_{\nu}\left(\cosh\xi_{1}%
\right)P^{m}_{\nu}\left(\cosh\xi_{2}\right)\cos\left(m\phi\right),}}
P
ν
μ
(
z
)
Legendre-P-first-kind
𝜇
𝜈
𝑧
{\displaystyle{\displaystyle P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)}}
cos
z
𝑧
{\displaystyle{\displaystyle\cos\NVar{z}}}
cosh
z
𝑧
{\displaystyle{\displaystyle\cosh\NVar{z}}}
sinh
z
𝑧
{\displaystyle{\displaystyle\sinh\NVar{z}}}
P
ν
(
z
)
=
P
ν
0
(
z
)
shorthand-Legendre-P-first-kind
𝜈
𝑧
Legendre-P-first-kind
0
𝜈
𝑧
{\displaystyle{\displaystyle P_{\NVar{\nu}}\left(\NVar{z}\right)=P^{0}_{\nu}%
\left(z\right)}}
m
𝑚
{\displaystyle{\displaystyle m}}
ν
𝜈
{\displaystyle{\displaystyle\nu}}
ϕ
italic-ϕ
{\displaystyle{\displaystyle\phi}}
ξ
1
subscript
𝜉
1
{\displaystyle{\displaystyle\xi_{1}}}
