Existence of Logarithms

[Tsakanikas Nickos]

Let \( \displaystyle A \) be a simply connected region that does not contain \( \displaystyle 0 \). Show that there is an analytic function \( \displaystyle F \), which is unique up to the addition of multiples of \( \displaystyle 2 \pi i \), such that \( \displaystyle {e}^{F(z)} = z , \; \forall z \in A \).

Additionally, prove the following more general version of the previous result:

Let \( \displaystyle A \) be a simply connected region and let \( \displaystyle f \) be an analytic and everywhere non-zero function on A. Show that there is an analytic function \( \displaystyle g \), which is unique up to the addition of multiples of \( \displaystyle 2 \pi i \), such that \( \displaystyle {e}^{g(z)} = f(z) , \; \forall z \in A \).