### 7.16 Combinations of associated Legendre functions, powers, and trigonometric functions

7.161

1.

$\underset{0}{\overset{1}{\int }}{x}^{\lambda -1}{\left(1-{x}^{2}\right)}^{-\frac{1}{2}\mu }\mathrm{sin}\left(ax\right){P}_{v}^{\mu }\left(x\right)\text{d}x=\frac{{\pi }^{1/2}{2}^{\mu -\lambda -1}\mathrm{\Gamma }\left(\lambda +1\right)a}{\mathrm{\Gamma }\left(1+\frac{\lambda -\mu -v}{2}\right)\mathrm{\Gamma }\left(\frac{3+\lambda -\mu +v}{2}\right)}×2{F}_{3}\left(\frac{1+\lambda }{2},1+\frac{\lambda }{2};\frac{3}{2},1+\frac{\lambda -\mu +v}{2},\frac{3+\lambda -\mu +v}{2};-\frac{{a}^{2}}{4}\right)$ ET II 314(7)

$\left[\mathrm{Re}\lambda >-1,\mathrm{Re}\mu <1\right]$ 2.

${\int }_{0}^{1}{x}^{\lambda -1}{\left(1-{x}^{2}\right)}^{-\frac{1}{2}\mu }\mathrm{cos}\left(ax\right){P}_{v}^{\mu }\left(x\right)\text{d}x=\frac{{\pi }^{1/2}{2}^{\mu -\lambda }\mathrm{\Gamma }\left(\lambda \right)}{\mathrm{\Gamma }\left(1+\frac{\lambda -\mu +v}{2}\right)\mathrm{\Gamma }\left(\frac{1+\lambda -\mu +v}{2}\right)}×2{F}_{3}\left(\frac{\lambda }{2},\frac{\lambda +1}{2};\frac{1}{2},\frac{1+\lambda -\mu -v}{2},1+\frac{\lambda -\mu +v}{2};-\frac{{a}^{2}}{4}\right)$ ET II 314(8)

$\left[\mathrm{Re}\lambda >0,\mathrm{Re}\mu <1\right]$

3.

${\int }_{0}^{\infty }{\left({x}^{2}-1\right)}^{\frac{1}{2}\mu }\mathrm{sin}\left(ax\right){P}_{v}^{\mu }\left(x\right)\text{d}x=\frac{{2}^{\mu }{\pi }^{1/2}{a}^{-\mu -\frac{1}{2}}}{\mathrm{\Gamma }\left(\frac{1}{2}-\frac{1}{2}\mu -\frac{1}{2}v\right)}$

Get Table of Integrals, Series, and Products, 8th Edition now with O’Reilly online learning.

O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.