Scaling Limits for Partial Sums of Power Series

TL;DR

This thesis investigates the scaling limits of partial sums of power series for certain entire functions, providing insights into their zeros and verifying aspects of the Saff-Varga Width Conjecture.

Contribution

It introduces new results on the asymptotic behavior and zero distribution of partial sums for a class of entire functions, partially confirming the Saff-Varga Width Conjecture.

Findings

Scaling limits of partial sums in various directions

Verification of the Saff-Varga Width Conjecture for certain functions

Illustrative figures for classical special functions

Abstract

In this thesis we show that the partial sums of the Maclaurin series for a certain class of entire functions possess scaling limits in various directions in the complex plane. In doing so we obtain information about the zeros of the partial sums. We will only assume that these entire functions have a certain asymptotic behavior at infinity. With this information we will partially verify for this class of functions a conjecture on the location of the zeros of their partial sums known as the Saff-Varga Width Conjecture. The thesis begins by introducing the basics (and the not-so-basics) of the topic and includes a detailed survey of the relevant literature. The body of the thesis is then divided into chapters which look at different asymptotic regimes. Following that a chapter is dedicated to applying the results obtained to several well-known functions including the sine and cosine functions, Bessel functions, confluent hypergeometric functions, Airy functions, and parabolic cylinder functions. This chapter includes many figures which clearly illustrate the behavior we're interested in. The thesis closes with an appendix which aims to serve as an instructive introduction to the Laplace method of asymptotic analysis of integrals.