Table of Integrals, Series, and Products, 8th Edition

7

Definite Integrals of Special Functions

7.1-7.2 Associated Legendre Functions

7.11 Associated Legendre functions

7.111

${\displaystyle \underset{\cos \phi }{\overset{1}{\int }}{P}_v(x)\text{d}x=\sin \phi P_v^{-1}(\cos \phi )}$

  MO 90

7.112

1. 

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

  SM III 185, WH

2. 

${\displaystyle \underset{-1}{\overset{1}{\int }}{Q}_n^m(x)P_n^m(x)\text{d}x={(-1)}^m\frac{1-{(-1)}^{n+k}(n+m)!}{(k-n)(k+n+1)(n-m)!}}$

  EH I 171(18)

3. 

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

  EH I 170(7) ...