  We show the existence of some special coordinate systems for expressing maps
with branch points. These coordinates allow obtaining an explicit
representation formula of branch immersions and understanding the regularity of
some fundamental elements in the theory of R. D. Gulliver, R. Osserman and H.
L. Royden. For conformal maps, we prove that these coordinates exist with
certain regularity if and only if the mean curvature vector extends and is in a
particular H\"older space. In the general case, we characterize this type of
coordinate system. Also, using these coordinates, we study the regularity and
behavior of curvatures near branch points.
