$\frac{1}{2\pi \mathrm{mathrm}i}{\int }_{-\infty }^{(0+)}{e}^{zt+\frac{1}{2}{t}^{-1}}{t}^{\kappa }{W}_{\kappa ,\mu }\left(t^{-1}\right)\mathrm{mathrm}dt=\frac{2z^{-\kappa -\frac{1}{2}}}{\Gamma (\frac{1}{2}+\mu -\kappa )\Gamma (\frac{1}{2}-\mu -\kappa )}{K}_{2\mu }\left(2\sqrt{z}\right)$