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Statements
P
ν
-
μ
(
z
e
s
π
i
)
=
e
s
ν
π
i
P
ν
-
μ
(
z
)
+
2
i
sin
(
(
ν
+
1
2
)
s
π
)
e
-
s
π
i
/
2
cos
(
ν
π
)
Γ
(
μ
-
ν
)
𝑸
ν
μ
(
z
)
,
Legendre-P-first-kind
𝜇
𝜈
𝑧
superscript
𝑒
𝑠
𝜋
𝑖
superscript
𝑒
𝑠
𝜈
𝜋
𝑖
Legendre-P-first-kind
𝜇
𝜈
𝑧
2
𝑖
𝜈
1
2
𝑠
𝜋
superscript
𝑒
𝑠
𝜋
𝑖
2
𝜈
𝜋
Euler-Gamma
𝜇
𝜈
associated-Legendre-black-Q
𝜇
𝜈
𝑧
{\displaystyle{\displaystyle P^{-\mu}_{\nu}\left(ze^{s\pi i}\right)=e^{s\nu\pi
i%
}P^{-\mu}_{\nu}\left(z\right)+\frac{2i\sin\left(\left(\nu+\frac{1}{2}\right)s%
\pi\right)e^{-s\pi i/2}}{\cos\left(\nu\pi\right)\Gamma\left(\mu-\nu\right)}%
\boldsymbol{Q}^{\mu}_{\nu}\left(z\right),}}
Γ
(
z
)
Euler-Gamma
𝑧
{\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
P
ν
μ
(
z
)
Legendre-P-first-kind
𝜇
𝜈
𝑧
{\displaystyle{\displaystyle P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)}}
𝑸
ν
μ
(
z
)
associated-Legendre-black-Q
𝜇
𝜈
𝑧
{\displaystyle{\displaystyle\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(%
\NVar{z}\right)}}
π
{\displaystyle{\displaystyle\pi}}
cos
z
𝑧
{\displaystyle{\displaystyle\cos\NVar{z}}}
e
{\displaystyle{\displaystyle\mathrm{e}}}
i
imaginary-unit
{\displaystyle{\displaystyle\mathrm{i}}}
sin
z
𝑧
{\displaystyle{\displaystyle\sin\NVar{z}}}
z
𝑧
{\displaystyle{\displaystyle z}}
μ
𝜇
{\displaystyle{\displaystyle\mu}}
ν
𝜈
{\displaystyle{\displaystyle\nu}}
s
𝑠
{\displaystyle{\displaystyle s}}
