{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/45365"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/45365","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Identities involving theta functions and analogues of theta functions","abstract":"My dissertation is mainly about various identities involving theta functions and analogues of theta functions. In Chapter 1, we give a completely elementary proof of Ramanujan's circular summation formula of theta functions and its generalizations given by S. H. Chan and Z. -G. Liu, and J. M. Zhu, who used the theory of elliptic functions. In contrast to all other proofs, our proofs are elementary. An application of this summation formula is given. In Chapter 2, we analyze various generalized two-dimensional lattice sums, one of which arose from the solution to a certain Poisson equation. We evaluate certain lattice sums in closed form using results from Ramanujan's theory of theta functions, continued fractions and class invariants. Many nice explicit examples are given. In Chapter 3, we study one page in Ramanujan's lost notebook that is devoted to claims about a certain integral with two parameters. One claim gives an inversion formula for the integral that is similar to the transformation formula for theta functions. Other claims are remindful of Gauss sums. In this chapter, we prove all the claims made by Ramanujan about this integral.","abstract_html":"My dissertation is mainly about various identities involving theta functions and analogues of theta functions. In Chapter 1, we give a completely elementary proof of Ramanujan's circular summation formula of theta functions and its generalizations given by S. H. Chan and Z. -G. Liu, and J. M. Zhu, who used the theory of elliptic functions. In contrast to all other proofs, our proofs are elementary. An application of this summation formula is given. In Chapter 2, we analyze various generalized two-dimensional lattice sums, one of which arose from the solution to a certain Poisson equation. We evaluate certain lattice sums in closed form using results from Ramanujan's theory of theta functions, continued fractions and class invariants. Many nice explicit examples are given. In Chapter 3, we study one page in Ramanujan's lost notebook that is devoted to claims about a certain integral with two parameters. One claim gives an inversion formula for the integral that is similar to the transformation formula for theta functions. Other claims are remindful of Gauss sums. In this chapter, we prove all the claims made by Ramanujan about this integral.","abstract_has_math":false,"creators":["Xu, Ping"],"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.","Ahlgren, Scott","Stolarsky, Kenneth B.","Zaharescu, Alexandru"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2013,"date_issued":"2013-08-22T16:37:57Z","date_published":"2013-08-22T16:37:57Z","updated_at":"2026-07-22T22:25:34Z","subjects":["Circular summation formula","elementary proof","two-dimensional lattice sums","Poisson equation","theta functions","analogue of theta functions","analogue of Gauss sums"],"languages":["en"],"rights":["Copyright 2013 Ping Xu"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/45365","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Berndt, Bruce C.","Ahlgren, Scott","Stolarsky, Kenneth B.","Zaharescu, Alexandru"]},{"key":"dc:creator","label":"Author","values":["Xu, Ping"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2013-08-22T16:37:57Z","2013-08"]},{"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":["Circular summation formula","elementary proof","two-dimensional lattice sums","Poisson equation","theta functions","analogue of theta functions","analogue of Gauss sums"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2013 Ping Xu"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/45365"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["My dissertation is mainly about various identities involving theta functions and analogues of theta functions. In Chapter 1, we give a completely elementary proof of Ramanujan's circular summation formula of theta functions and its generalizations given by S. H. Chan and Z. -G. Liu, and J. M. Zhu, who used the theory of elliptic functions. In contrast to all other proofs, our proofs are elementary. An application of this summation formula is given. In Chapter 2, we analyze various generalized two-dimensional lattice sums, one of which arose from the solution to a certain Poisson equation. We evaluate certain lattice sums in closed form using results from Ramanujan's theory of theta functions, continued fractions and class invariants. Many nice explicit examples are given. In Chapter 3, we study one page in Ramanujan's lost notebook that is devoted to claims about a certain integral with two parameters. One claim gives an inversion formula for the integral that is similar to the transformation formula for theta functions. Other claims are remindful of Gauss sums. In this chapter, we prove all the claims made by Ramanujan about this integral.","Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2013-07-09T16:07:48Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 1 Xu_Ping.pdf: 488152 bytes, checksum: 7316f588bedf939190e2b3e500be8c68 (MD5)","Made available in DSpace on 2013-08-22T16:37:57Z (GMT). No. of bitstreams: 2 Ping_Xu.pdf: 488152 bytes, checksum: 7316f588bedf939190e2b3e500be8c68 (MD5) license.txt: 4056 bytes, checksum: 585245f3c9eb952638524554e4742239 (MD5)"]},{"key":"dc:title","label":"Title","values":["Identities involving theta functions and analogues of theta functions"]}]}],"canonical_facts":{"dc:contributor":["Berndt, Bruce C.","Ahlgren, Scott","Stolarsky, Kenneth B.","Zaharescu, Alexandru"],"dc:creator":["Xu, Ping"],"dc:date":["2013-08-22T16:37:57Z","2013-08"],"dc:description":["My dissertation is mainly about various identities involving theta functions and analogues of theta functions. In Chapter 1, we give a completely elementary proof of Ramanujan's circular summation formula of theta functions and its generalizations given by S. H. Chan and Z. -G. Liu, and J. M. Zhu, who used the theory of elliptic functions. In contrast to all other proofs, our proofs are elementary. An application of this summation formula is given. In Chapter 2, we analyze various generalized two-dimensional lattice sums, one of which arose from the solution to a certain Poisson equation. We evaluate certain lattice sums in closed form using results from Ramanujan's theory of theta functions, continued fractions and class invariants. Many nice explicit examples are given. In Chapter 3, we study one page in Ramanujan's lost notebook that is devoted to claims about a certain integral with two parameters. One claim gives an inversion formula for the integral that is similar to the transformation formula for theta functions. Other claims are remindful of Gauss sums. In this chapter, we prove all the claims made by Ramanujan about this integral.","Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2013-07-09T16:07:48Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 1 Xu_Ping.pdf: 488152 bytes, checksum: 7316f588bedf939190e2b3e500be8c68 (MD5)","Made available in DSpace on 2013-08-22T16:37:57Z (GMT). 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