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

$\underset{-i\infty }{\overset{i\infty }{{\displaystyle \int }}}{\left[\frac{\Gamma (\frac{1}{2}-s)}{\Gamma (s)}\right]}^2{z}^s\text{d}s=2\pi i{z}^{\frac{1}{2}}[2{\pi }^{-1}{\mathit{K}}_0(4z^{\frac{1}{4}})-Y_0(4z^{\frac{1}{4}})]$

  ET II 303(33)

$\left[z>0\right]$

17. 

$\underset{-i\infty }{\overset{i\infty }{{\displaystyle \int }}}\frac{\Gamma \left(\lambda +\mu -s+\frac{1}{2}\right)\Gamma \left(\lambda +\mu -s+\frac{1}{2}\right)}{\Gamma \left(\lambda -k-s+1\right)}{z}^s\text{d}s\text{=2π}i{z}^{\lambda }{e}^{-\frac{z}{2}}{W}_{k,u}\left(z\right)$

  ET II 302(30)

$\left[Re\lambda >\left|Re\mu \right|-\frac{1}{2},\left|\arg z\right|<\frac{\pi }{2}\right]$

18. 

$\underset{-i\infty }{\overset{i\infty }{{\displaystyle \int }}}\frac{\Gamma \left(k-\lambda +s\right)\Gamma \left(\lambda -\mu -s+\frac{1}{2}\right)}{\Gamma \left(\mu -\lambda +s+\frac{1}{2}\right)}{z}^s\text{d}s=2\pi i\frac{\Gamma \left(k+\mu +\frac{1}{2}\right)}{\Gamma \left(2\mu +1\right)}{z}^{\lambda }{e}^{-\frac{z}{2}}{M}_{k,\mu }\left(z\right)$

  ET II 302(31)

$\left[Re\left(k-\lambda \right)>0,Re\left(\lambda +\mu \right)>-\frac{1}{2},\left|\arg z\right|<\frac{\pi }{2}\right]$

19¹². 

${\int }_{-i\infty }^{i\infty }\frac{\prod _{j=1}^m\Gamma \left(b_j-s\right)\prod _{j=1}^n\Gamma (}{}$