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

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

  ET II 405(20)

$[\text{Re }\lambda >0,\mathrm{Re}(k+\mu -2\lambda )>-\frac{1}{2}]$

3. 

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

  ET II 411(47)

$[|\arg a|<\pi ,\text{ Re }\lambda >0]|$

4. 

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

  ET II 411(48)

$[|\arg a|<\pi ,\text{ Re }\lambda >0]|$

5. 

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