I have a Complex Analysis assessment question about holomorphic functions:
Let f be a function on a plane and satisfies $f'(z) = f(z)$ and $f(0) = 1$
i) Give an example of a function with this property.
ii) By considering $g(z) = f(z)f(-z)$, show that $f(-z) = 1/f(z)$ for all z. (You may assume that a function which is holomorphic on the plane with zero derivative is constant).
I have an answer for part i): $e^z$. (in fact, I can't actually think of any other functions that satisfy these conditions, do any exist?)
Part ii), however, I have no idea how to answer. I can obviously show it using $f(z) = e^z$ and substituting it into the equation, but I'm not sure that's the right way of going about it. Could anyone offer some help?
Thanks a lot.