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Title:On the location of zeros of solutions to w" + Aw = 0 for certain entire functions A
Author(s):Steinbart, Enid Marguerite
Doctoral Committee Chair(s):Ting, Tran Wa
Department / Program:Mathematics
Discipline:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Degree:Ph.D.
Genre:Dissertation
Subject(s):Mathematics
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.
We also study the asymptotic behavior of the zeros of nontrivial solutions in zero-rich regions. We show that if $w$ is a nontrivial solution of $(\*)$ with infinitely many zeros in a zero-rich region, then the zeros of $w$ in this region must approach a curve of the form Im$P(z)$ = $K$ where $K$ is a constant and $P$ is a polynomial determined by $A$. We establish a bound for the rate at which the zeros approach this curve.
In addition to the classical techniques of ordinary differential equations, we employ methods developed by Walter Strodt. Strodt's general theory examines the existence and asymptotic behavior of solutions to certain first order differential equations in the complex domain.
Issue Date:1989
Type:Text
Language:English
URI:http://hdl.handle.net/2142/20092
Rights Information:Copyright 1989 Steinbart, Enid Marguerite
Date Available in IDEALS:2011-05-07
Identifier in Online Catalog:AAI8924947
OCLC Identifier:(UMI)AAI8924947


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