$\underset{0}{\overset{\infty }{\int }}{x}^{2\lambda -1}{\left(a+x\right)}^{-\mu -\frac{1}{2}}{e}^{-\frac{1}{2}x}{M}_{{k,\mu }_{}}^{{-\frac{1}{2}x}^{}}\left(a+x\right)\text{d}x=\frac{\mathrm{\Gamma }\left(2\lambda \right)\mathrm{\Gamma }\left(2\mu +1\right)\mathrm{\Gamma }\left(k+\mu -2\lambda +\frac{1}{2}\right)}{\mathrm{\Gamma }\left(k+\mu +\frac{1}{2}\right)\mathrm{\Gamma }\left(1-2\lambda +2\mu \right)}{a}^{\lambda -\mu -\frac{1}{2}}{M}_{k-\lambda ,\mu -\lambda }\left(a\right)$ ET II 405(20) 3.

$\underset{0}{\overset{\infty }{\int }}{x}^{2\lambda -1}{\left(a+x\right)}^{-\mu -\frac{1}{2}}{e}^{-\frac{1}{2}x}{W}_{k,\mu }\left(a+x\right)\text{d}x=\mathrm{\Gamma }\left(2\lambda \right){a}^{\lambda -\mu -\frac{1}{2}}{W}_{k-\lambda ,\mu -\lambda }\left(a\right)$ ET II 411(47) 4.

$\underset{0}{\overset{\infty }{\int }}{x}^{\lambda -1}{\left(a+x\right)}^{k-\lambda -1}{e}^{-\frac{1}{2}x}{W}_{k,\mu }\left(a+x\right)\text{d}x=\mathrm{\Gamma }\left(\lambda \right){a}^{k-1}{W}_{k-\lambda ,\mu }\left(a\right)$

ET II 411(48)

5.

$\underset{0}{\overset{\infty }{\int }}{x}^{}$

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.