Existence of Logarithms
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Existence of Logarithms
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 \( \displaystyle {e}^{g(z)} = f(z) , \; \forall z \in A \).
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 \( \displaystyle {e}^{g(z)} = f(z) , \; \forall z \in A \).
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