Show that the function $t \mapsto t^{z-1}e^{-t}$ is integrable with respect to the Lebesgue meaure on the measure space $((0,\infty),\frak{B}^1,\lambda^1)$.
Assume that a complex $c$ satisfies $\Re(z)>0$. I may use the fact that $t^z=e^{z \log(t)}$.
What I need to show is that $\int_{0}^\infty|t^{z-1}e^{-t}| d\lambda^1$ is finite. This integral is equal to $\int_0^\infty|e^{(z-1)\log(t)-t}|d\lambda^1$. Now I'm stuck at showing that this integral is indeed finite. Thank you for any hints!