Zeros of Sections of Some Power Series

Abstract

For a power series which converges in some neighborhood of the origin in the complex plane, the zeros of its partial sums often behave in a controlled manner. We give an overview of some of the major results in the study of this phenomenon in the past century, focusing on recent developments which build on the theme of asymptotic analysis. Inspired by this work, we study the asymptotic behavior of the zeros of partial sums of power series for entire functions defined by exponential integrals of a certain type. Most of the zeros of the n'th partial sum travel outwards from the origin at a rate comparable to n, so we rescale the variable by n and explicitly calculate the limit curves of these normalized zeros. We discover that the zeros' asymptotic behavior depends on the order of the critical points of the integrand in the aforementioned exponential integral.