{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/20092"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/20092","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"\"On the location of zeros of solutions to w\"\" + Aw = 0 for certain entire functions A\"","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.\"","abstract_html":""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_has_math":true,"creators":["Steinbart, Enid Marguerite"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Ting, Tran Wa"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T12:28:34Z","date_published":"2011-05-07T12:28:34Z","updated_at":"2026-07-22T22:25:15Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1989 Steinbart, Enid Marguerite"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI8924947","(UMI)AAI8924947"],"render_values":[{"text":"AAI8924947","href":null,"code":true},{"text":"(UMI)AAI8924947","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/20092","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Ting, Tran Wa"]},{"key":"dc:creator","label":"Author","values":["Steinbart, Enid Marguerite"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T12:28:34Z","10000-01-01","1989"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 1989 Steinbart, Enid Marguerite"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI8924947","(UMI)AAI8924947","http://hdl.handle.net/2142/20092"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["\"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.","Made available in DSpace on 2011-05-07T12:28:34Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 8924947.pdf: 2810751 bytes, checksum: aeababa5b8bbb6c0025e8b460b84fd73 (MD5) Previous issue date: 1989","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:41:30Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:17:58-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"]},{"key":"dc:title","label":"Title","values":["\"On the location of zeros of solutions to w\"\" + Aw = 0 for certain entire functions A\""]}]}],"canonical_facts":{"dc:contributor":["Ting, Tran Wa"],"dc:creator":["Steinbart, Enid Marguerite"],"dc:date":["2011-05-07T12:28:34Z","10000-01-01","1989"],"dc: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.\"","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.","Made available in DSpace on 2011-05-07T12:28:34Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 8924947.pdf: 2810751 bytes, checksum: aeababa5b8bbb6c0025e8b460b84fd73 (MD5) Previous issue date: 1989","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:41:30Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:17:58-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"],"dc:identifier":["AAI8924947","(UMI)AAI8924947","http://hdl.handle.net/2142/20092"],"dc:language":["eng"],"dc:rights":["Copyright 1989 Steinbart, Enid Marguerite"],"dc:subject":["Mathematics"],"dc:title":["\"On the location of zeros of solutions to w\"\" + Aw = 0 for certain entire functions A\""],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:15Z"}