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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 UrbanaChampaign 
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 "zeroscarce" regions and "zerorich" regions. We call an unbounded region in the complex plane "zeroscarce" if every nontrivial solution of $(\*)$ has only finitely many zeros in the region. In contrast, a region is called "zerorich" 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 zerorich regions. We show that if $w$ is a nontrivial solution of $(\*)$ with infinitely many zeros in a zerorich 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:  20110507 
Identifier in Online Catalog:  AAI8924947 
OCLC Identifier:  (UMI)AAI8924947 
This item appears in the following Collection(s)

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois 
Dissertations and Theses  Mathematics