Can the integrals of the following types be written in terms of any Special functions?
$$\int_0^\infty e^{\pm\alpha t}t^{\pm \beta t}\mathrm{d}t \quad\quad \{\alpha,\beta\}\in \mathbb{C}$$
All I have tried so far is tried to integrate numerically. I see that the plot of $e^x x^{-x}$ rapidly decreases so that the integral converges nicely. So, I am wondering if there are some functions in terms of which $\int_0^\infty e^x x^{-x}\mathrm{d}x$ can be written and whether the more general integral $\int_0^\infty e^{\pm\alpha t}t^{\pm \beta t}\mathrm{d}t \quad\quad \{\alpha,\beta\}\in \mathbb{C}$ can be written in terms of such functions for the region where integral converges.