Find an analytic function that satisfies the condition

Question

If $f=u+iv$ is analytic in the region $D$ and continous in the closure of $D$ then find an analytic function $g$, continous in the closure of $D$ such that

$\ln|g|=u$

My idea is to take the function $g(z)=e^{f(z)}$
Since $f$ is analytic and continous it implies that $g$ satisfies the required conditions and also

$|g(z)|=|e^{f(z)}|=|e^{u(x,y)+iv(x,y)}|=e^{u(x,y)}|e^{iv(x,y)}|=e^{u(x,y)}$

so

$\ln|g|=\ln(e^{u(x,y)})=u(x,y)$

I am not sure if this is the right solution. Thank you very much!

Answer 1

Yes, $g(z)=e^{f(z)}$ is a function such that $ \ln |g|=u.$. If $k \in \mathbb Z$, then $g_k(z)=e^{f(z)+2k \pi i}$ will also do the job.