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The purpose of this paper is to introduce the branching geometry of algebraicfunctions around singular points and to describe a simple method of determiningradii of convergence of their power expansions in terms of those singularpoints. Branching geometries are categorized into six types. Then a method ispresented to determine radii of convergence of branch power expansions usinganalytic continuation and the identification of convergence-limiting singularpoints. Test cases exhibiting a variety of branching morphologies are analyzed,and convergence results obtained through analytic continuation are checkedagainst Root Tests of the associated power series. All Root Tests agreed wellwith the results obtained by analytic continuation. Mathematica ver. 12.3 wasused to implement the numeric algorithms.

Determining radii of convergence of fractional power expansions around singular points of algebraic functions