Determining radii of convergence of fractional power expansions around singular points of algebraic functions
arXiv:2111.11883
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.
arXiv comment20 pages, 9 figures, 5 tables, Mathematica code, Ver 2 changes: Minor typos and: Correct roots of a_n(z) in sec 5.1, correct eqn 7 to z(t)=A(1-t)+Dt, changed link to website to point to section on radius of convergence, added link to web page for test 4 branch table, corrected branch types at origin for test case 4