Bounding the Radius of Convergence of Analytic Functions

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.