A branched covering map 
is a continuous map
that restricts to a covering map away from a subset
of 
called the branch locus. Points of 
where this restriction fails to be locally a homeomorphism
are ramification points, and their images are branch
points.
For a nonconstant holomorphic map between Riemann surfaces, local coordinates can be chosen near a point 
and its image so that the map has the form 
. The positive
integer 
is the ramification index; the map is unbranched
at 
when 
and branched there when 
.
