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

7.21 Integration of associated Legendre functions with respect to the order

7.211

1. 

$\begin{array}{cc}{\displaystyle {\int }_0^{\infty }{P}_{-x-\frac{1}{2}}(\cos \theta )\text{d}x=\frac{1}{2}\cos \text{ec}\left(\frac{1}{2}\theta \right)}& [0<\theta <\pi ]\end{array}$

  ET II 329(19)

2. 

$\begin{array}{cc}{\displaystyle {\int }_{-\infty }^{\infty }{P}_x(\cos \theta )\text{d}x=\cos \text{ec}\left(\frac{1}{2}\theta \right)}& [0<\theta <\pi ]\end{array}$

  ET II 329(20)

7.212

${\displaystyle {\int }_0^{\infty }{x}^{-1}\tanh (\pi x)P_{-\frac{1}{2}+ix}(\cosh a)\text{d}x=2e^{-\frac{1}{2}a}K(e^{-a})}$

  ET II 330(22)

[a > 0]

7.213

$\begin{array}{cc}{\displaystyle {\int }_0^{\infty }\frac{x\tanh (\pi x)}{a^2+x^2}{P}_{-\frac{1}{2}+ix}(\cosh b)\text{d}x=Q_{a-\frac{1}{2}}(\cosh b)}& [\mathrm{Re}a>0]\end{array}$

  ET II 387(23)