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

TL;DR

This paper introduces a geometric framework for algebraic functions around singular points and presents a simple method to determine the radii of convergence of their power series expansions, validated by numerical tests.

Contribution

It categorizes branching geometries of algebraic functions and proposes a straightforward approach to find convergence radii using analytic continuation and singular point analysis.

Findings

Root Tests confirmed the convergence radii obtained by analytic continuation.

Six types of branching geometries are classified.

The method is validated with diverse test cases.

Abstract

The purpose of this paper is to introduce the branching geometry of algebraic functions around singular points and to describe a simple method of determining radii of convergence of their power expansions in terms of those singular points. Branching geometries are categorized into six types. Then a method is presented to determine radii of convergence of branch power expansions using analytic continuation and the identification of convergence-limiting singular points. Test cases exhibiting a variety of branching morphologies are analyzed, and convergence results obtained through analytic continuation are checked against Root Tests of the associated power series. All Root Tests agreed well with the results obtained by analytic continuation. Mathematica ver. 12.3 was used to implement the numeric algorithms.