Is the local inverse of an analytical function locally analytical as well?

Question

Take for example $f(z)=e^z$, so the inverse is $z(f) = \ln(f) + n\pi i$ for an arbitrarily chosen (but fixed) branch $n\in\mathbb N$. Now if $f$ is restricted to e.g. $0<|1-f|<1$ such that the essential singularity at $f=0$ is not part of the definition region, is $z(f)$ then analytical in that region?

Answer 1

See the Lagrange inversion theorem: "$g$ is analytic at the point $b=f(a)$".