## With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

No credit card required

### 7.21 Integration of associated Legendre functions with respect to the order

7.211

1.

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

ET II 329(19)

2.

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

ET II 329(20)

7.212

${\int }_{0}^{\infty }{x}^{-1}\mathrm{tanh}\left(\pi x\right){P}_{-\frac{1}{2}+ix}\left(\mathrm{cosh}a\right)\text{d}x=2{e}^{-\frac{1}{2}a}K\left({e}^{-a}\right)$

ET II 330(22)

[a > 0]

7.213

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

ET II 387(23)

## With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

No credit card required