The authors investigate the following generalized incomplete gamma function \[ \Gamma(\alpha, x;b)= \int^\infty_x t^{\alpha-1} e^{-t- b/t}dt,\quad x>0,\quad \text{Re}(b)\geq 0,\quad b\neq 0, \] towards obtaining asymptotic expansions when \(x\to\infty\), and \(\alpha\) is unrestricted for negative values. For \(b= -i\omega (\omega>0)\), the asymptotic expansions of the following functions are considered: \[ C_\Gamma(\alpha, x;\omega)= \int^\infty_x t^{\alpha-1} e^{-t}\cos\Biggl({\omega\over t}\Biggr) dt\quad\text{and }S_\Gamma(\alpha, x;\omega)= \int^\infty_x t^{-\alpha- 1} e^{-t}\sin\Biggl({\omega\over t}\Biggr) dt. \] Closed form expressions are also considered for certain values of \(\alpha\).