"On the location of zeros of solutions to w"" + Aw = 0 for certain entire functions A"
type: CreativeWork
description: "Consider the differential equation $(\*)$ $w\sp{\prime\prime}$ + $Aw$ = 0 where $A$ is of the form $A$($z$) = $\sum\sbsp{j=1}{m}Q\sb{j}(z)$exp$P\sb{j}(z)$ with $Q\sb{j}$ and $P\sb{j}$ polynomials in $z$. We study the location in the complex plane of the zeros of solutions to $(\*)$. Under mild hypotheses on the $P\sb{j}$'s and $Q\sb{j}$'s, we show that there exist both ""zero-scarce"" regions and ""zero-rich"" regions. We call an unbounded region in the complex plane ""zero-scarce"" if every nontrivial solution of $(\*)$ has only finitely many zeros in the region. In contrast, a region is called ""zero-rich"" if there exists a nontrivial solution of $(\*)$ with infinitely many zeros in the region."
abstract: "Consider the differential equation $(\*)$ $w\sp{\prime\prime}$ + $Aw$ = 0 where $A$ is of the form $A$($z$) = $\sum\sbsp{j=1}{m}Q\sb{j}(z)$exp$P\sb{j}(z)$ with $Q\sb{j}$ and $P\sb{j}$ polynomials in $z$. We study the location in the complex plane of the zeros of solutions to $(\*)$. Under mild hypotheses on the $P\sb{j}$'s and $Q\sb{j}$'s, we show that there exist both ""zero-scarce"" regions and ""zero-rich"" regions. We call an unbounded region in the complex plane ""zero-scarce"" if every nontrivial solution of $(\*)$ has only finitely many zeros in the region. In contrast, a region is called ""zero-rich"" if there exists a nontrivial solution of $(\*)$ with infinitely many zeros in the region."
author: Steinbart, Enid Marguerite
language: eng
keywords: Mathematics
date published: 2011-05-07T12:28:34Z
date modified: 2026-07-22T22:25:15Z
provider: University of Illinois - Urbana-Champaign
publisher: Networked Digital Library of Theses and Dissertations
type: WebPage
source organization: University of Illinois at Urbana-Champaign

PropertyValue
property ID: Handle
value: http://hdl.handle.net/2142/20092