  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.
