Examples of mock modular forms

mock theta function

• mock theta function is essentially a mock modular form of weight 1/2

 

 

mock theta conjectures

• Watson, G. N. (1936), "The Final Problem : An Account of the Mock Theta Functions", J. London Math. Soc.11: 55–80, doi:10.1112/jlms/s1-11.1.55
• Watson, G. N. (1937), "The Mock Theta Functions (2)", Proc. London Math. Soc.s2-42: 274–304, doi:10.1112/plms/s2-42.1.274
• Ramanujan's “Lost” Notebook VI: The mock theta conjectures
  
  • George E. Andrews and F. G. Garvan   
• Hickerson, Dean (1988), "A proof of the mock theta conjectures", Inventiones Mathematicae94 (3): 639–660, doi:10.1007/BF01394279, MR969247, ISSN0020-9910
  [1]

 

 

 

 

examples

• classify which examples are of which use

 

 

order 3

• \(f(q) = \sum_{n\ge 0} {q^{n^2}\over (-q;q)_n^2} = {2\over \prod_{n>0}(1-q^n)}\sum_{n\in Z}{(-1)^nq^{3n^2/2+n/2}\over 1+q^n}\)
• good introduction is given in Andrews article 
  
  • Partitions : at the interface of q-series and modular forms
  • [2]section 5
• the asymptotic series for coefficients of the order 3 mock theta function f(q) studied by of (Andrews 1966) and Dragonette (1952) converges to the coefficients (Bringmann & Ono 2006).
• In particular Mock theta functions have asymptotic expansions at cusps of the modular group, acting on the upper half-plane, that resemble those of modular forms of weight 1/2 with poles at the cusps.
• The Final Problem : An Account of the Mock Theta Functions
  
  • Watson, G. N. (1936)
  •  J. London Math. Soc. 11: 55–80
• Some asymptotic formulae for the mock theta series of Ramanujan
  
  • Dragonette, Leila A. (1952), 
  • Transactions of the American Mathematical Society 72: 474–500
• On the theorems of Watson and Dragonette for Ramanujan's mock theta functions
  
  • Andrews, George E. (1966)
  • American Journal of Mathematics 88: 454–490

 

order 5

• G.E. Andrews, The fifth and seventh order mock theta functions. Trans. Amer. Math. Soc. 293 (1986), pp. 113–134
• Andrews, George E. (1988), "Ramanujan's fifth order mock theta functions as constant terms", Ramanujan revisited (Urbana-Champaign, Ill., 1987),
• Modular Transformations of Ramanujan's Fifth and Seventh Order Mock Theta Functions
  
  • Basil Gordon and Richard J. Mcintosh

 

order 7

• Hickerson, Dean (1988), "On the seventh order mock theta functions", Inventiones Mathematicae94 (3): 661–677, doi:10.1007/BF01394280, MR969247, ISSN0020-9910

 

 

order 10

• Choi, Youn-Seo (1999), "Tenth order mock theta functions in Ramanujan's lost notebook", Inventiones Mathematicae136 (3): 497–569, doi:10.1007/s002220050318, MR1695205, ISSN0020-9910
• Choi, Youn-Seo (2000), "Tenth order mock theta functions in Ramanujan's lost notebook. II", Advances in Mathematics156 (2): 180–285, doi:10.1006/aima.2000.1948, MR1808245, ISSN0001-8708
• Choi, Youn-Seo (2007), "Tenth order mock theta functions in Ramanujan's lost notebook. III", Proceedings of the London Mathematical Society. Third Series 94 (1): 26–52, doi:10.1112/plms/pdl006, MR2293464, ISSN 0024-6115
• Choi, Youn-Seo (2002), "Tenth order mock theta functions in Ramanujan's lost notebook. IV", Transactions of the American Mathematical Society354 (2): 705–733, doi:10.1090/S0002-9947-01-02861-6, MR1862564, ISSN0002-9947,
• Zwegers' new proof  http://mathsci.ucd.ie/~zwegers/papers/006.pdf

 

 

관련된 다른 주제들

• Mock modular forms

 

표준적인 도서 및 추천도서

• Ramanujan's Lost Notebook: Part I
• Ramanujan's Lost Notebook: Part II
• 찾아볼 수학책
• http://gigapedia.info/1/Polyogarithms
• http://gigapedia.info/1/ramanujan
• http://gigapedia.info/1/
• http://gigapedia.info/1/
• http://gigapedia.info/1/
• http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=

 

 

참고할만한 자료

• http://www.zentralblatt-math.org/zmath/en/
• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/Mock_modular_form
• http://viswiki.com/en/
• http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
• http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&s7=
• 다음백과사전 http://enc.daum.net/dic100/search.do?q=