Proving f(z)=e^(g(z)) on a Convex Set Omega

[michael.wes]

Homework Statement

Suppose that f is analytic on a convex set omega and that f never vanishes on omega. Prove that f(z)=e^(g(z)) for some analytic function g defined on omega.
Hint: does f'/f have a primitive on omega?

Homework Equations

$$f(z)=\sum_{k=0}^\infty a_k(z-p)^k$$

The Attempt at a Solution

I was able to prove that f'/f has a primitive on omega by the Cauchy-Goursat theorem, but I'm not sure where to go from here. Any help is appreciated!

 

[micromass]

Let F be a primitive of f'/f. Now, consider the function $$G(z)=e^{F(z)}/f(z)$$. What is it's derivative? What can you conclude from that?

 

[michael.wes]

I got that $$e^{g(z)}=cf(z)$$, for some complex constant c and some analytic function g. It's usually easy in these problems to show that the constant is 1, but this is not a concrete function, so I'm not sure how to do that.

 

[micromass]

Well, the constant is not necessairly 1, so you'll have to find something else. You'll have to modify your function g in some way such that the equation is right...