{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/86827"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/86827","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Infinite Series Identities in the Theory of Elliptic Functions and Q-Series","abstract":"We prove several infinite series identities. In Chapter 2, We extend C. L. Siegel's method of proving the Dedekind-eta function transformation by integrating some selected functions over a positively oriented polygon, generalizing Siegel's integration over a parallelogram. As consequences, we obtain a generalization of the Dedekind-eta function transformation and generalizations of other transformation formulas. In Chapter 3, we adapt B. C. Berndt and A. Zaharescu's method to establish a multi-variable theta product identity of a function of k + 1 complex variables. In Chapter 4, we give a simple new proof of the classical theta-function inversion formula. In Chapter 5, we give two general methods for proving q-series-product identities. The first method uses basic properties of roots of unity. The second method generalizes S. Bhragava's argument proving the quintuple product identity. By comparing the results from the two methods, we obtain new identities. Using these identities, we can derive certain modular equations. In Chapter 6, we evaluate certain infinite series involving hyperbolic functions by using the cubic theory of elliptic functions.","abstract_html":"We prove several infinite series identities. In Chapter 2, We extend C. L. Siegel's method of proving the Dedekind-eta function transformation by integrating some selected functions over a positively oriented polygon, generalizing Siegel's integration over a parallelogram. As consequences, we obtain a generalization of the Dedekind-eta function transformation and generalizations of other transformation formulas. In Chapter 3, we adapt B. C. Berndt and A. Zaharescu's method to establish a multi-variable theta product identity of a function of k + 1 complex variables. In Chapter 4, we give a simple new proof of the classical theta-function inversion formula. In Chapter 5, we give two general methods for proving q-series-product identities. The first method uses basic properties of roots of unity. The second method generalizes S. Bhragava's argument proving the quintuple product identity. By comparing the results from the two methods, we obtain new identities. Using these identities, we can derive certain modular equations. In Chapter 6, we evaluate certain infinite series involving hyperbolic functions by using the cubic theory of elliptic functions.","abstract_has_math":false,"creators":["Kongsiriwong, Sarachai"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Berndt, Bruce C."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-09-28T15:19:44Z","date_published":"2015-09-28T15:19:44Z","updated_at":"2026-07-22T22:26:28Z","subjects":["Mathematics"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(MiAaPQ)AAI3111562"],"render_values":[{"text":"(MiAaPQ)AAI3111562","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/86827","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Berndt, Bruce C."]},{"key":"dc:creator","label":"Author","values":["Kongsiriwong, Sarachai"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2015-09-28T15:19:44Z","10000-01-01","2003"]},{"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"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/86827","(MiAaPQ)AAI3111562"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["We prove several infinite series identities. In Chapter 2, We extend C. L. Siegel's method of proving the Dedekind-eta function transformation by integrating some selected functions over a positively oriented polygon, generalizing Siegel's integration over a parallelogram. As consequences, we obtain a generalization of the Dedekind-eta function transformation and generalizations of other transformation formulas. In Chapter 3, we adapt B. C. Berndt and A. Zaharescu's method to establish a multi-variable theta product identity of a function of k + 1 complex variables. In Chapter 4, we give a simple new proof of the classical theta-function inversion formula. In Chapter 5, we give two general methods for proving q-series-product identities. The first method uses basic properties of roots of unity. The second method generalizes S. Bhragava's argument proving the quintuple product identity. By comparing the results from the two methods, we obtain new identities. Using these identities, we can derive certain modular equations. In Chapter 6, we evaluate certain infinite series involving hyperbolic functions by using the cubic theory of elliptic functions.","Made available in DSpace on 2015-09-28T15:19:44Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 3111562.pdf: 4185658 bytes, checksum: b33c927f35f1cd4486c51c3b1b9b7d0e (MD5) Previous issue date: 2003","Embargo set by: Seth Robbins for item 88108 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","128 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2003."]},{"key":"dc:title","label":"Title","values":["Infinite Series Identities in the Theory of Elliptic Functions and Q-Series"]}]}],"canonical_facts":{"dc:contributor":["Berndt, Bruce C."],"dc:creator":["Kongsiriwong, Sarachai"],"dc:date":["2015-09-28T15:19:44Z","10000-01-01","2003"],"dc:description":["We prove several infinite series identities. In Chapter 2, We extend C. L. Siegel's method of proving the Dedekind-eta function transformation by integrating some selected functions over a positively oriented polygon, generalizing Siegel's integration over a parallelogram. As consequences, we obtain a generalization of the Dedekind-eta function transformation and generalizations of other transformation formulas. In Chapter 3, we adapt B. C. Berndt and A. Zaharescu's method to establish a multi-variable theta product identity of a function of k + 1 complex variables. In Chapter 4, we give a simple new proof of the classical theta-function inversion formula. In Chapter 5, we give two general methods for proving q-series-product identities. The first method uses basic properties of roots of unity. The second method generalizes S. Bhragava's argument proving the quintuple product identity. By comparing the results from the two methods, we obtain new identities. Using these identities, we can derive certain modular equations. In Chapter 6, we evaluate certain infinite series involving hyperbolic functions by using the cubic theory of elliptic functions.","Made available in DSpace on 2015-09-28T15:19:44Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 3111562.pdf: 4185658 bytes, checksum: b33c927f35f1cd4486c51c3b1b9b7d0e (MD5) Previous issue date: 2003","Embargo set by: Seth Robbins for item 88108 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","128 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2003."],"dc:identifier":["http://hdl.handle.net/2142/86827","(MiAaPQ)AAI3111562"],"dc:language":["eng"],"dc:subject":["Mathematics"],"dc:title":["Infinite Series Identities in the Theory of Elliptic Functions and Q-Series"],"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":"2026-07-22T22:26:28Z"}