  This paper describes an algorithm for determining the branching geometry of
algebraic functions. The graphs of these complex-valued functions have a
complicated interweaving structure that can be described by analytic branches
separated by singular points. Power expansions for the branches in discs
centered at a point can be computed using the Newton Polygon method, and
expansions around annular regions centered at the origin computed using a
version of Laurent's Theorem applied to algebraic functions. However, neither
of the methods enable a determination of the region of convergence of the power
series. In this paper, a method using analytic continuation is used to
determine the domain of analyticity for the branches, and the Root Test used to
numerically check the results.
