Given $$f_1(z)=\int_0^\infty t^{z-1}e^{-t}dt$$ where $\operatorname{Re}(z)>0$ $$f_2(z)=\sum_{n=0}^\infty\frac{(-1)^n}{(n+z)(n!)}+\int_1^\infty t^{z-1}e^{-t}dt$$ except for the values $z=0,-1,\ldots$.
How to prove that $f_2$ is an analytic continuation of $f_1$?
I have been simply trying to represent function $f_2$ as a Taylor series of the $f_1$'s to show that $f_1$ is equal to $f_2$ in the intersecting domains. But dont know how to do this.