Bounding the Radius of Convergence of Analytic Functions

TL;DR

This paper presents a robust method for determining whether a circular contour contains poles and for bounding the radius of convergence of complex-analytic functions, aiding in contour integration tasks.

Contribution

It introduces a new technique to identify poles within a contour and to estimate the radius of convergence for complex functions, enhancing numerical analysis methods.

Findings

Method effectively detects poles within contours.

Provides reliable bounds on the radius of convergence.

Applicable to complex functions in physics and engineering.

Abstract

Contour integration is a crucial technique in many numeric methods of interest in physics ranging from differentiation to evaluating functions of matrices. It is often important to determine whether a given contour contains any poles or branch cuts, either to make use of these features or to avoid them. A special case of this problem is that of determining or bounding the radius of convergence of a function, as this provides a known circle around a point in which a function remains analytic. We describe a method for determining whether or not a circular contour of a complex-analytic function contains any poles. We then build on this to produce a robust method for bounding the radius of convergence of a complex-analytic function.