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Statements
𝖰
ν
μ
(
cos
θ
)
=
π
1
/
2
Γ
(
ν
+
μ
+
1
)
(
sin
θ
)
μ
2
μ
+
1
Γ
(
μ
+
1
2
)
Γ
(
ν
-
μ
+
1
)
(
∫
0
∞
(
sinh
t
)
2
μ
(
cos
θ
+
i
sin
θ
cosh
t
)
ν
+
μ
+
1
d
t
+
∫
0
∞
(
sinh
t
)
2
μ
(
cos
θ
-
i
sin
θ
cosh
t
)
ν
+
μ
+
1
d
t
)
,
Ferrers-Legendre-Q-first-kind
𝜇
𝜈
𝜃
superscript
𝜋
1
2
Euler-Gamma
𝜈
𝜇
1
superscript
𝜃
𝜇
superscript
2
𝜇
1
Euler-Gamma
𝜇
1
2
Euler-Gamma
𝜈
𝜇
1
superscript
subscript
0
superscript
𝑡
2
𝜇
superscript
𝜃
𝑖
𝜃
𝑡
𝜈
𝜇
1
𝑡
superscript
subscript
0
superscript
𝑡
2
𝜇
superscript
𝜃
𝑖
𝜃
𝑡
𝜈
𝜇
1
𝑡
{\displaystyle{\displaystyle\mathsf{Q}^{\mu}_{\nu}\left(\cos\theta\right)=%
\frac{\pi^{1/2}\Gamma\left(\nu+\mu+1\right)(\sin\theta)^{\mu}}{2^{\mu+1}\Gamma%
\left(\mu+\frac{1}{2}\right)\Gamma\left(\nu-\mu+1\right)}\*\left(\int_{0}^{%
\infty}\frac{(\sinh t)^{2\mu}}{(\cos\theta+i\sin\theta\cosh t)^{\nu+\mu+1}}%
\mathrm{d}t+\int_{0}^{\infty}\frac{(\sinh t)^{2\mu}}{(\cos\theta-i\sin\theta%
\cosh t)^{\nu+\mu+1}}\mathrm{d}t\right),}}
ℜ
μ
>
-
1
2
𝜇
1
2
{\displaystyle{\displaystyle\Re\mu>-\tfrac{1}{2}}}
ℜ
(
ν
±
μ
)
>
-
1
plus-or-minus
𝜈
𝜇
1
{\displaystyle{\displaystyle\Re(\nu\pm\mu)>-1}}
0
<
θ
<
π
0
𝜃
𝜋
{\displaystyle{\displaystyle 0<\theta<\pi}}
ℜ
μ
>
-
1
2
𝜇
1
2
{\displaystyle{\displaystyle\Re\mu>-\tfrac{1}{2}}}
ℜ
(
ν
±
μ
)
>
-
1
plus-or-minus
𝜈
𝜇
1
{\displaystyle{\displaystyle\Re(\nu\pm\mu)>-1}}
Γ
(
z
)
Euler-Gamma
𝑧
{\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
𝖰
ν
μ
(
x
)
Ferrers-Legendre-Q-first-kind
𝜇
𝜈
𝑥
{\displaystyle{\displaystyle\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}%
\right)}}
π
{\displaystyle{\displaystyle\pi}}
cos
z
𝑧
{\displaystyle{\displaystyle\cos\NVar{z}}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
cosh
z
𝑧
{\displaystyle{\displaystyle\cosh\NVar{z}}}
sinh
z
𝑧
{\displaystyle{\displaystyle\sinh\NVar{z}}}
i
imaginary-unit
{\displaystyle{\displaystyle\mathrm{i}}}
∫
{\displaystyle{\displaystyle\int}}
ℜ
absent
{\displaystyle{\displaystyle\Re}}
sin
z
𝑧
{\displaystyle{\displaystyle\sin\NVar{z}}}
μ
𝜇
{\displaystyle{\displaystyle\mu}}
ν
𝜈
{\displaystyle{\displaystyle\nu}}
