{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/23595"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/23595","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Zeros of partial sums of power series","abstract":"A classical result of Jentzsch states that, for a power series with radius of convergence one, every point on the unit circle is a limit point of zeros of partial sums of the power series. In this thesis, the relationship between the properties of the function and the behavior of the zeros of its partial sums is investigated. We prove that the set of limit points of xeros of partial sums of rational functions can contain an open disk outside the unit circle, while that of a power series with the unit circle as its natural boundary can be empty outside the unit circle. We also prove that, for any closed set in the complement of the open unit disk and any finite set of points inside the unit circle, there is a power series such that the set of the limit points of zeros of partial sums is the union of the two given sets and the unit circle. A condition is given on the coefficients of power series which ensures that the limit points of zeros of partial sums are contained in the closed unit disk. We obtain Jentzsch-type theorems about power series in several complex variables. Examples illustrate that Jentzsch-type theorems in several complex variables depend on the way in which the partial sums are defined and on the convergence domain of the power series.","abstract_html":"A classical result of Jentzsch states that, for a power series with radius of convergence one, every point on the unit circle is a limit point of zeros of partial sums of the power series. In this thesis, the relationship between the properties of the function and the behavior of the zeros of its partial sums is investigated. We prove that the set of limit points of xeros of partial sums of rational functions can contain an open disk outside the unit circle, while that of a power series with the unit circle as its natural boundary can be empty outside the unit circle. We also prove that, for any closed set in the complement of the open unit disk and any finite set of points inside the unit circle, there is a power series such that the set of the limit points of zeros of partial sums is the union of the two given sets and the unit circle. A condition is given on the coefficients of power series which ensures that the limit points of zeros of partial sums are contained in the closed unit disk. We obtain Jentzsch-type theorems about power series in several complex variables. Examples illustrate that Jentzsch-type theorems in several complex variables depend on the way in which the partial sums are defined and on the convergence domain of the power series.","abstract_has_math":false,"creators":["Qian, Xiaoling"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Rubel, Lee A.","Miles, Joseph B."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T14:20:01Z","date_published":"2011-05-07T14:20:01Z","updated_at":"2026-07-22T22:25:22Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1995 Qian, Xiaoling"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9624465","(UMI)AAI9624465"],"render_values":[{"text":"AAI9624465","href":null,"code":true},{"text":"(UMI)AAI9624465","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/23595","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Rubel, Lee A.","Miles, Joseph B."]},{"key":"dc:creator","label":"Author","values":["Qian, Xiaoling"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T14:20:01Z","10000-01-01","1995"]},{"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 1995 Qian, Xiaoling"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9624465","(UMI)AAI9624465","http://hdl.handle.net/2142/23595"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["A classical result of Jentzsch states that, for a power series with radius of convergence one, every point on the unit circle is a limit point of zeros of partial sums of the power series. In this thesis, the relationship between the properties of the function and the behavior of the zeros of its partial sums is investigated. We prove that the set of limit points of xeros of partial sums of rational functions can contain an open disk outside the unit circle, while that of a power series with the unit circle as its natural boundary can be empty outside the unit circle. We also prove that, for any closed set in the complement of the open unit disk and any finite set of points inside the unit circle, there is a power series such that the set of the limit points of zeros of partial sums is the union of the two given sets and the unit circle. A condition is given on the coefficients of power series which ensures that the limit points of zeros of partial sums are contained in the closed unit disk. We obtain Jentzsch-type theorems about power series in several complex variables. Examples illustrate that Jentzsch-type theorems in several complex variables depend on the way in which the partial sums are defined and on the convergence domain of the power series.","Made available in DSpace on 2011-05-07T14:20:01Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9624465.pdf: 1563224 bytes, checksum: 37fdc3542ee65d0a21e8edd70ba52bfc (MD5) Previous issue date: 1995","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T15:05:32Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:31:24-05:00 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":["Zeros of partial sums of power series"]}]}],"canonical_facts":{"dc:contributor":["Rubel, Lee A.","Miles, Joseph B."],"dc:creator":["Qian, Xiaoling"],"dc:date":["2011-05-07T14:20:01Z","10000-01-01","1995"],"dc:description":["A classical result of Jentzsch states that, for a power series with radius of convergence one, every point on the unit circle is a limit point of zeros of partial sums of the power series. In this thesis, the relationship between the properties of the function and the behavior of the zeros of its partial sums is investigated. We prove that the set of limit points of xeros of partial sums of rational functions can contain an open disk outside the unit circle, while that of a power series with the unit circle as its natural boundary can be empty outside the unit circle. We also prove that, for any closed set in the complement of the open unit disk and any finite set of points inside the unit circle, there is a power series such that the set of the limit points of zeros of partial sums is the union of the two given sets and the unit circle. A condition is given on the coefficients of power series which ensures that the limit points of zeros of partial sums are contained in the closed unit disk. We obtain Jentzsch-type theorems about power series in several complex variables. Examples illustrate that Jentzsch-type theorems in several complex variables depend on the way in which the partial sums are defined and on the convergence domain of the power series.","Made available in DSpace on 2011-05-07T14:20:01Z (GMT). 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