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Statements
P
ν
μ
(
x
)
=
1
Γ
(
1
-
μ
)
(
x
+
1
x
-
1
)
μ
/
2
ϕ
-
i
(
2
ν
+
1
)
(
-
μ
,
μ
)
(
arcsinh
(
(
1
2
x
-
1
2
)
1
/
2
)
)
.
Legendre-P-first-kind
𝜇
𝜈
𝑥
1
Euler-Gamma
1
𝜇
superscript
𝑥
1
𝑥
1
𝜇
2
Jacobi-hypergeometric-phi
𝜇
𝜇
imaginary-unit
2
𝜈
1
hyperbolic-inverse-sine
superscript
1
2
𝑥
1
2
1
2
{\displaystyle{\displaystyle P^{\mu}_{\nu}\left(x\right)=\frac{1}{\Gamma\left(%
1-\mu\right)}\left(\frac{x+1}{x-1}\right)^{\mu/2}\phi^{(-\mu,\mu)}_{-\mathrm{i%
}(2\nu+1)}\left(\operatorname{arcsinh}\left((\tfrac{1}{2}x-\tfrac{1}{2})^{%
\ifrac{1}{2}}\right)\right).}}
Γ
(
z
)
Euler-Gamma
𝑧
{\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
ϕ
λ
(
α
,
β
)
(
t
)
Jacobi-hypergeometric-phi
𝛼
𝛽
𝜆
𝑡
{\displaystyle{\displaystyle\phi^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{\lambda%
}}\left(\NVar{t}\right)}}
P
ν
μ
(
z
)
Legendre-P-first-kind
𝜇
𝜈
𝑧
{\displaystyle{\displaystyle P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)}}
arcsinh
z
hyperbolic-inverse-sine
𝑧
{\displaystyle{\displaystyle\operatorname{arcsinh}\NVar{z}}}
i
imaginary-unit
{\displaystyle{\displaystyle\mathrm{i}}}
x
𝑥
{\displaystyle{\displaystyle x}}
μ
𝜇
{\displaystyle{\displaystyle\mu}}
ν
𝜈
{\displaystyle{\displaystyle\nu}}
Identifiers
