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Statements
∫
-
1
1
(
𝖯
ν
(
x
)
)
2
d
x
=
π
2
-
2
sin
2
(
ν
π
)
ψ
′
(
ν
+
1
)
π
2
(
ν
+
1
2
)
,
superscript
subscript
1
1
superscript
shorthand-Ferrers-Legendre-P-first-kind
𝜈
𝑥
2
𝑥
superscript
𝜋
2
2
2
𝜈
𝜋
diffop
digamma
1
𝜈
1
superscript
𝜋
2
𝜈
1
2
{\displaystyle{\displaystyle\int_{-1}^{1}\left(\mathsf{P}_{\nu}\left(x\right)%
\right)^{2}\mathrm{d}x=\frac{\pi^{2}-2{\sin^{2}}\left(\nu\pi\right)\psi'\left(%
\nu+1\right)}{\pi^{2}\left(\nu+\frac{1}{2}\right)},}}
ν
≠
-
1
2
𝜈
1
2
{\displaystyle{\displaystyle\nu\neq-\frac{1}{2}}}
π
{\displaystyle{\displaystyle\pi}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
ψ
(
z
)
digamma
𝑧
{\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
∫
{\displaystyle{\displaystyle\int}}
𝖯
ν
(
x
)
=
𝖯
ν
0
(
x
)
shorthand-Ferrers-Legendre-P-first-kind
𝜈
𝑥
Ferrers-Legendre-P-first-kind
0
𝜈
𝑥
{\displaystyle{\displaystyle\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=%
\mathsf{P}^{0}_{\nu}\left(x\right)}}
sin
z
𝑧
{\displaystyle{\displaystyle\sin\NVar{z}}}
x
𝑥
{\displaystyle{\displaystyle x}}
ν
𝜈
{\displaystyle{\displaystyle\nu}}
