{"id":{"repo_id":"calpoly","oai_identifier":"oai:digitalcommons.calpoly.edu:theses-4097"},"canonical_url":"https://search.dev.ndltd.org/etd/calpoly/oai:digitalcommons.calpoly.edu:theses-4097","repository":{"repo_id":"calpoly","name":"Cal Poly","base_url":"https://digitalcommons.calpoly.edu/do/oai/"},"display":{"title":"A Synthesis of Classical Boundary Theorems","abstract":"Bounded analytic functions on the open unit disk D = {z ∈ C | |z| < 1} are a fre-<br />quent area of study in complex function theory. While it is easy to understand the<br />behavior of analytic functions on sequences with limit points inside D, the theory<br />becomes much more complicated as sequences converge to the boundary, ∂D. In this<br />thesis, we will explore boundary theorems, which can guarantee specific desired be-<br />havior of these analytic functions. The thesis describes an elementary approach to<br />proving Fatou’s Non-Tangential Limit Theorem, as well as proofs and discussion of<br />the subsequent classical boundary theorems for specific points, Julia’s Theorem and<br />the Julia-Carathéodory Theorem. This thesis serves as a synthesis of these boundary<br />theorems in order to fill a gap in the overarching literature.<br /><br />","abstract_html":"Bounded analytic functions on the open unit disk D = {z ∈ C | |z| &lt; 1} are a fre-&lt;br /&gt;quent area of study in complex function theory. While it is easy to understand the&lt;br /&gt;behavior of analytic functions on sequences with limit points inside D, the theory&lt;br /&gt;becomes much more complicated as sequences converge to the boundary, ∂D. In this&lt;br /&gt;thesis, we will explore boundary theorems, which can guarantee specific desired be-&lt;br /&gt;havior of these analytic functions. The thesis describes an elementary approach to&lt;br /&gt;proving Fatou’s Non-Tangential Limit Theorem, as well as proofs and discussion of&lt;br /&gt;the subsequent classical boundary theorems for specific points, Julia’s Theorem and&lt;br /&gt;the Julia-Carathéodory Theorem. This thesis serves as a synthesis of these boundary&lt;br /&gt;theorems in order to fill a gap in the overarching literature.&lt;br /&gt;&lt;br /&gt;","abstract_has_math":false,"creators":["Dakhlia, Lukas A"],"institution":null,"degree_name":"MS in Mathematics","degree_level":null,"degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Ryan Tully-Doyle","Mathematics","College of Science and Mathematics"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2022,"date_issued":"2022-06-01T07:00:00Z","date_published":"2022-06-01T07:00:00Z","updated_at":"2026-07-24T01:32:29Z","subjects":["Fatou","Julia","Carathéodory","Boundary","Complex","Analysis"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["10.15368/theses.2022.74"],"render_values":[{"text":"10.15368/theses.2022.74","href":"https://doi.org/10.15368/theses.2022.74","code":true}]}]},"links":{"outbound_url":"https://digitalcommons.calpoly.edu/theses/2496","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Ryan Tully-Doyle","Mathematics","College of Science and Mathematics"]},{"key":"dc:creator","label":"Author","values":["Dakhlia, Lukas A"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2022-06-10T07:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_name","label":"Degree Name","values":["MS in Mathematics"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Fatou","Julia","Carathéodory","Boundary","Complex","Analysis"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://digitalcommons.calpoly.edu/theses/2496","10.15368/theses.2022.74"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Bounded analytic functions on the open unit disk D = {z ∈ C | |z| < 1} are a fre-<br />quent area of study in complex function theory. While it is easy to understand the<br />behavior of analytic functions on sequences with limit points inside D, the theory<br />becomes much more complicated as sequences converge to the boundary, ∂D. In this<br />thesis, we will explore boundary theorems, which can guarantee specific desired be-<br />havior of these analytic functions. The thesis describes an elementary approach to<br />proving Fatou’s Non-Tangential Limit Theorem, as well as proofs and discussion of<br />the subsequent classical boundary theorems for specific points, Julia’s Theorem and<br />the Julia-Carathéodory Theorem. This thesis serves as a synthesis of these boundary<br />theorems in order to fill a gap in the overarching literature.<br /><br />"]},{"key":"dc:title","label":"Title","values":["A Synthesis of Classical Boundary Theorems"]}]}],"canonical_facts":{"dc:contributor":["Ryan Tully-Doyle","Mathematics","College of Science and Mathematics"],"dc:creator":["Dakhlia, Lukas A"],"dc:date.available":["2022-06-10T07:00:00Z"],"dc:description.abstract":["Bounded analytic functions on the open unit disk D = {z ∈ C | |z| < 1} are a fre-<br />quent area of study in complex function theory. While it is easy to understand the<br />behavior of analytic functions on sequences with limit points inside D, the theory<br />becomes much more complicated as sequences converge to the boundary, ∂D. In this<br />thesis, we will explore boundary theorems, which can guarantee specific desired be-<br />havior of these analytic functions. The thesis describes an elementary approach to<br />proving Fatou’s Non-Tangential Limit Theorem, as well as proofs and discussion of<br />the subsequent classical boundary theorems for specific points, Julia’s Theorem and<br />the Julia-Carathéodory Theorem. This thesis serves as a synthesis of these boundary<br />theorems in order to fill a gap in the overarching literature.<br /><br />"],"dc:identifier":["https://digitalcommons.calpoly.edu/theses/2496","10.15368/theses.2022.74"],"dc:subject":["Fatou","Julia","Carathéodory","Boundary","Complex","Analysis"],"dc:title":["A Synthesis of Classical Boundary Theorems"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_name":["MS in Mathematics"]},"updated_at":"2026-07-24T01:32:29Z"}