Complex Logarithm: question seems simple, must be missing something

[Mathmos6]

Homework Statement

Hi all, I'm having some trouble seeing why this question isn't trivial, maybe someone can help explain what I actually need to show - shouldn't take you long! :)

Suppose h:$\mathbb{C} \to \mathbb{C}-\{0\}$ is analytic with no zeros. Show there is an analytic function H:$\mathbb{C} \to \mathbb{C}$ such that h(z)=exp(H(z)) for all z.

Now surely H(z) is just log(h(z)), defined on (say) the principal branch? Is there some reason why the principal branch won't necessarily work, perhaps? The logarithm should be analytic on a domain with no zeros in too, right? In which case the composition with h will be analytic too. I must be missing something!

Thanks very much in advance :)

 

[Count Iblis]

You could consider the integral of h'/h.