Section 10: Region of Convergence of Algebraic Power Series

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The reader is asked to review the sections in sequential order as each succeeding section builds on a previous section.

In this section we go over in detail a method of determining the branching geometry of algebraic functions and the region of convergence of their power expansions. This method is described here: On the Branching Geometry of Algebraic Functions

Background

In previous sections we described the geometry of an algebraic function: Setting the resultant, $R(f,f_w)$ equal to zero identifies the singularities of the function in the $z$-plane, and we can then sort the values in order of increasing absolute value thereby creating a set of concentric rings around the origin between which the function is analytic and splits or ramifies into branches of varying cycles. Around the origin and each singular point, the function also ramifies. Using the Newton Polygon algorithm, we can compute power expansions around the origin and each singular point for the ramified branches. And using Laurent's expansion theorem applied to algebraic functions, likewise compute power expansions of the function in the annular regions between rings. But both methods do not help us determine the region of convergence of these series. In this section we go over both a geometric method to compute the convergence domain of these series and Mathematica code to implement the algorithm.

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