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Statements
∫
-
1
1
𝖯
ν
(
x
)
𝖯
λ
(
x
)
d
x
=
2
(
2
sin
(
ν
π
)
sin
(
λ
π
)
(
ψ
(
ν
+
1
)
-
ψ
(
λ
+
1
)
)
+
π
sin
(
(
λ
-
ν
)
π
)
)
π
2
(
λ
-
ν
)
(
λ
+
ν
+
1
)
,
superscript
subscript
1
1
shorthand-Ferrers-Legendre-P-first-kind
𝜈
𝑥
shorthand-Ferrers-Legendre-P-first-kind
𝜆
𝑥
𝑥
2
2
𝜈
𝜋
𝜆
𝜋
digamma
𝜈
1
digamma
𝜆
1
𝜋
𝜆
𝜈
𝜋
superscript
𝜋
2
𝜆
𝜈
𝜆
𝜈
1
{\displaystyle{\displaystyle\int_{-1}^{1}\mathsf{P}_{\nu}\left(x\right)\mathsf%
{P}_{\lambda}\left(x\right)\mathrm{d}x=\frac{2\left(2\sin\left(\nu\pi\right)%
\sin\left(\lambda\pi\right)\left(\psi\left(\nu+1\right)-\psi\left(\lambda+1%
\right)\right)+\pi\sin\left((\lambda-\nu)\pi\right)\right)}{\pi^{2}(\lambda-%
\nu)(\lambda+\nu+1)},}}
-
ν
-
1
𝜈
1
{\displaystyle{\displaystyle-\nu-1}}
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}}
