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oai:www.ideals.illinois.edu:2142/50667 2023-07-11 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

Berndt, Bruce C. Ahlgren, Scott Berndt, Bruce C. Stolarsky, Kenneth B. Zaharescu, Alexandru Schultz, Daniel 2014-09-16T17:25:03Z 2014-09-16T17:25:03Z 2014-08 2014-09-16 2014-08 This thesis is centered around three topics: the theory of the cubic theta functions as functions of two analytic variables, cubic modular equations, and a class of two-variable cubic modular equations. Chapter 2 is dedicated to the rst two topics, while Chapter 3 covers the last. First, the theory of cubic theta functions can be developed analogously to, but distinct from, the classical theory of elliptic functions. We will derive analogues of the Jacobian elliptic functions, and provide addition theorems, integral inversion formulas, di erential equations, and modular transformations for these functions. Second, we revisit the cubic modular equations rst derived by Ramanujan and study them in a systematic manner. The results obtained greatly extend previous work on cubic modular equations. Finally, in Chapter 3, we study modular equations for the Picard modular functions. These modular equations provide a two-variable generalization of the cubic modular equations studied in Chapter 2. Item withdrawn by Laura Spradlin (lspradl2@illinois.edu) on 2014-05-21T16:35:01Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 6 linefigure.pdf: 1498766 bytes, checksum: 7b26ce786ea66a65e51e77073835de80 (MD5) monodromy1.pdf: 14523 bytes, checksum: f212f88bdb9d7ba713428673a45f7975 (MD5) figure2.pdf: 33136 bytes, checksum: d32b2b53bd6ba73e01f4aa6a9125a90a (MD5) figure1new.pdf: 943655 bytes, checksum: 5eb23c08610c18c8f74ee73818d4da2e (MD5) ThesisDraft4DanSchultz.tex: 332403 bytes, checksum: 0b69f6f5a79e7ea29c7bd03549c8a6d3 (MD5) Schultz_Daniel.pdf: 3245463 bytes, checksum: 1ba44d467d9cd375724e122060b08cb4 (MD5) Made available in DSpace on 2014-09-16T17:25:03Z (GMT). No. of bitstreams: 7 Daniel_Schultz.pdf: 3245750 bytes, checksum: 9bec15a637889983ab89b8e712d6fe74 (MD5) ThesisDraft5DanSchultz.tex: 332404 bytes, checksum: 36070a4c151f1da31de0cee78cb1649c (MD5) linefigure.pdf: 1498766 bytes, checksum: 7b26ce786ea66a65e51e77073835de80 (MD5) monodromy1.pdf: 14523 bytes, checksum: f212f88bdb9d7ba713428673a45f7975 (MD5) figure2.pdf: 33136 bytes, checksum: d32b2b53bd6ba73e01f4aa6a9125a90a (MD5) figure1new.pdf: 943655 bytes, checksum: 5eb23c08610c18c8f74ee73818d4da2e (MD5) license.txt: 4064 bytes, checksum: 72c12b5e1f6f43fcbab596376f4e1158 (MD5) http://hdl.handle.net/2142/50667 en Copyright 2014 Daniel Schultz cubic theta functions modular equations appell hypergeometric picard modular forms Cubic theta functions and identities for Appell's F1 function text Mathematics 1257 Mathematics 0439 University of Illinois at Urbana-Champaign Dissertation Ph.D. PHD:Mathematics -UIUC 10KS0439PHD