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7

# Definite Integrals of Special Functions

## 7.1-7.2 Associated Legendre Functions

### 7.11 Associated Legendre functions

7.111

$\underset{\mathrm{cos}\phi }{\overset{1}{\int }}{P}_{v}\left(x\right)\text{d}x=\mathrm{sin}\phi {P}_{v}^{-1}\left(\mathrm{cos}\phi \right)$

MO 90

7.112

1.

$\begin{array}{cccc}\underset{-1}{\overset{1}{\int }}{P}_{n}^{m}\left(x\right){P}_{k}^{m}\left(x\right)\text{d}x& =\hfill & 0\hfill & \left[n\ne k\right]\\ =\hfill & \frac{2}{2n+1}\frac{\left(n+m\right)!}{\left(n-m\right)!}\hfill & \left[n=k\right]\end{array}$

SM III 185, WH

2.

$\underset{-1}{\overset{1}{\int }}{Q}_{n}^{m}\left(x\right){P}_{n}^{m}\left(x\right)\text{d}x={\left(-1\right)}^{m}\frac{1-{\left(-1\right)}^{n+k}\left(n+m\right)!}{\left(k-n\right)\left(k+n+1\right)\left(n-m\right)!}$

EH I 171(18)

3.

$\begin{array}{lll}\underset{-1}{\overset{1}{\int }}{P}_{v}\left(x\right){P}_{\sigma }\left(x\right)\text{d}x\hfill & =\frac{2\pi \mathrm{sin}\pi \left(\sigma -v\right)+4\mathrm{sin}\left(\pi v\right)\mathrm{sin}\left(\pi \sigma \right)\left[\psi \left(v+1\right)-\psi \left(\sigma +1\right)\right]}{{\pi }^{2}\left(\sigma -v\right)\left(\sigma +v+1\right)}\hfill & \left[\sigma +v+1\ne 0\right]\hfill \\ =\frac{{\pi }^{2}-2{\left(\mathrm{sin}\pi v\right)}^{2}{\psi }^{\prime }\left(v+1\right)}{{\pi }^{2}\left(v+\frac{1}{2}\right)}\hfill & \left[\sigma =v\right]\hfill \end{array}$

EH I 170(7) ...

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