Picard's great theorem says that any analytic function of one complex variable defined on a punctured neighborhood of an essential singularity takes on every complex value (with at most one possible exception) infinitely often. Is there a similarly general statement that can be made about functions defined in the vicinity of a branch point? (Or a specific type of branch point, e.g. an algebraic or transcendental or logarithmic branch point? For example, is it the case that the image of a function defined near a logarithmic branch point is always unbounded?)
Clearly we need to be a bit more careful about exactly what we mean by "in the vicinity" than in the case of an isolated singularity. For example, we could consider an arbitrary open subset of the domain of the corresponding global analytic function that contains the branch point, or we could consider a single branch of the function and take an open subset of $\mathbb{C} \setminus (\text{branch point}) \setminus (\text{branch})$. I'm curious whether any general statements can be made in either case.
To give a few examples (in the "global analytic function" case):
- The image of $z^{1/n}$ for natural $n$, which has an algebraic branch point at $0$, is a neighborhood of $0$.
- The image of the function $\ln(z)$, which has an (obviously) logarithmic singularity near $0$, includes the entire half-plane $\mathrm{Re}(z) < x$ for some real $x$.