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

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

7.161

1. 

$\underset{0}{\overset{1}{\int }}{x}^{\lambda -1}{(1-x^2)}^{-\frac{1}{2}\mu }\sin (ax)P_v^{\mu }(x)\text{d}x=\frac{{\pi }^{1/2}{2}^{\mu -\lambda -1}\mathrm{\Gamma }(\lambda +1)a}{\mathrm{\Gamma }(1+\frac{\lambda -\mu -v}{2})\mathrm{\Gamma }\left(\frac{3+\lambda -\mu +v}{2}\right)}\times 2F_3(\frac{1+\lambda }{2},1+\frac{\lambda }{2};\frac{3}{2},1+\frac{\lambda -\mu +v}{2},\frac{3+\lambda -\mu +v}{2};-\frac{a^2}{4})$

  ET II 314(7)

$[\mathrm{Re}\lambda >-1,\mathrm{Re}\mu <1]$

2. 

${\displaystyle {\int }_0^1{x}^{\lambda -1}{(1-x^2)}^{-\frac{1}{2}\mu }\cos (ax)P_v^{\mu }(x)\text{d}x=\frac{{\pi }^{1/2}{2}^{\mu -\lambda }\mathrm{\Gamma }(\lambda )}{\mathrm{\Gamma }\left(1+\frac{\lambda -\mu +v}{2}\right)\mathrm{\Gamma }\left(\frac{1+\lambda -\mu +v}{2}\right)}}\times 2F_3\left(\frac{\lambda }{2},\frac{\lambda +1}{2};\frac{1}{2},\frac{1+\lambda -\mu -v}{2},1+\frac{\lambda -\mu +v}{2};-\frac{a^2}{4}\right)$

  ET II 314(8)

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

3. 

${\displaystyle {\int }_0^{\infty }{(x^2-1)}^{\frac{1}{2}\mu }\sin (ax)P_v^{\mu }(x)\text{d}x=\frac{2^{\mu }{\pi }^{1/2}{a}^{-\mu -\frac{1}{2}}}{\mathrm{\Gamma }\left(\frac{1}{2}-\frac{1}{2}\mu -\frac{1}{2}v\right)}}$