{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/86780"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/86780","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"The Construction and Applications of Modular Equations","abstract":"We introduce also two parametrizations hk,n and h'k,n of the theta-function 4q . For q=e2piz , Im z > 0, define 4q : =fq,q=n =-infinityinfinityqn2 =:q30,2z , where q3 is classical theta function. For any positive real numbers n and k, define hk,n=4e-p n/k k1/44e-p nk=q 30,-n/k k1/4q30, -nk and h'k,n=4 -e-2pn/k k1/44-e-2p nk =q30,1+2-n/ kk1/4q 30,1+2-nk . We show how they are connected with rk,n, r'k,n and Ramanujan-Weber class invariants and give some new theta-function identities. In addition, we establish some formulas for hk,n and h'k,n by using those theta-function identities and find some values of hk,n and h'k,n . Finally, we offer explicit formulas for 4-e-np and 4e-np for any positive real numbers n. We give some examples.","abstract_html":"We introduce also two parametrizations hk,n and h'k,n of the theta-function 4q . For q=e2piz , Im z > 0, define 4q : =fq,q=n =-infinityinfinityqn2 =:q30,2z , where q3 is classical theta function. For any positive real numbers n and k, define hk,n=4e-p n/k k1/44e-p nk=q 30,-n/k k1/4q30, -nk and h'k,n=4 -e-2pn/k k1/44-e-2p nk =q30,1+2-n/ kk1/4q 30,1+2-nk . We show how they are connected with rk,n, r'k,n and Ramanujan-Weber class invariants and give some new theta-function identities. In addition, we establish some formulas for hk,n and h'k,n by using those theta-function identities and find some values of hk,n and h'k,n . Finally, we offer explicit formulas for 4-e-np and 4e-np for any positive real numbers n. We give some examples.","abstract_has_math":false,"creators":["Yi, Jinhee"],"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":"2015-09-28T15:19:30Z","date_published":"2015-09-28T15:19:30Z","updated_at":"2026-07-22T22:26:27Z","subjects":["Mathematics"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(MiAaPQ)AAI3017261"],"render_values":[{"text":"(MiAaPQ)AAI3017261","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/86780","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":["Yi, Jinhee"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2015-09-28T15:19:30Z","10000-01-01","2001"]},{"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/2142/86780","(MiAaPQ)AAI3017261"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["We introduce also two parametrizations hk,n and h'k,n of the theta-function 4q . For q=e2piz , Im z > 0, define 4q : =fq,q=n =-infinityinfinityqn2 =:q30,2z , where q3 is classical theta function. For any positive real numbers n and k, define hk,n=4e-p n/k k1/44e-p nk=q 30,-n/k k1/4q30, -nk and h'k,n=4 -e-2pn/k k1/44-e-2p nk =q30,1+2-n/ kk1/4q 30,1+2-nk . We show how they are connected with rk,n, r'k,n and Ramanujan-Weber class invariants and give some new theta-function identities. In addition, we establish some formulas for hk,n and h'k,n by using those theta-function identities and find some values of hk,n and h'k,n . Finally, we offer explicit formulas for 4-e-np and 4e-np for any positive real numbers n. We give some examples.","Made available in DSpace on 2015-09-28T15:19:30Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 3017261.pdf: 4334866 bytes, checksum: 3563146fd088258751ea49489bb21e14 (MD5) Previous issue date: 2001","Embargo set by: Seth Robbins for item 88061 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","157 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2001."]},{"key":"dc:title","label":"Title","values":["The Construction and Applications of Modular Equations"]}]}],"canonical_facts":{"dc:contributor":["Berndt, Bruce C."],"dc:creator":["Yi, Jinhee"],"dc:date":["2015-09-28T15:19:30Z","10000-01-01","2001"],"dc:description":["We introduce also two parametrizations hk,n and h'k,n of the theta-function 4q . For q=e2piz , Im z > 0, define 4q : =fq,q=n =-infinityinfinityqn2 =:q30,2z , where q3 is classical theta function. For any positive real numbers n and k, define hk,n=4e-p n/k k1/44e-p nk=q 30,-n/k k1/4q30, -nk and h'k,n=4 -e-2pn/k k1/44-e-2p nk =q30,1+2-n/ kk1/4q 30,1+2-nk . We show how they are connected with rk,n, r'k,n and Ramanujan-Weber class invariants and give some new theta-function identities. In addition, we establish some formulas for hk,n and h'k,n by using those theta-function identities and find some values of hk,n and h'k,n . Finally, we offer explicit formulas for 4-e-np and 4e-np for any positive real numbers n. We give some examples.","Made available in DSpace on 2015-09-28T15:19:30Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 3017261.pdf: 4334866 bytes, checksum: 3563146fd088258751ea49489bb21e14 (MD5) Previous issue date: 2001","Embargo set by: Seth Robbins for item 88061 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","157 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2001."],"dc:identifier":["http://hdl.handle.net/2142/86780","(MiAaPQ)AAI3017261"],"dc:language":["eng"],"dc:subject":["Mathematics"],"dc:title":["The Construction and Applications of Modular Equations"],"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":"2026-07-22T22:26:27Z"}