I need to prove that if $f$ is an entire function (holomorphic in all the complex space) and $f = f'$ then there is a complex constant $K$ such that $f(z) = Ke^z$.
Moreover, I need to show that $e^z = \sum\limits_{n \ge 0} \frac{z^n}{n!}$.
Well I am not so sure what to do here. I see that $f-f' = $ constant, but I don't think it helps. I can clearly show that if $f(z) = Ke^z$ it satisfies the conditions, but I am not sure how to prove that $f$ is necessarily such a function.
Help would be appreciated.