"Examples of mock modular forms"의 두 판 사이의 차이
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* <math>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}</math> | * <math>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}</math> | ||
| + | * good introduction is given in Andrews article <br> | ||
| + | ** Partitions : at the interface of q-series and modular forms | ||
* 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 ([http://www.springerlink.com/content/5524655155350464/ Bringmann & Ono 2006]). | * 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 ([http://www.springerlink.com/content/5524655155350464/ 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. | * 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. | ||
| + | * [http://dx.doi.org/10.1112%2Fjlms%2Fs1-11.1.55 The Final Problem : An Account of the Mock Theta Functions]<br> | ||
| + | ** Watson, G. N. (1936) | ||
| + | ** J. London Math. Soc. 11: 55–80 | ||
* [http://dx.doi.org/10.2307%2F1990714 Some asymptotic formulae for the mock theta series of Ramanujan]<br> | * [http://dx.doi.org/10.2307%2F1990714 Some asymptotic formulae for the mock theta series of Ramanujan]<br> | ||
** Dragonette, Leila A. (1952), | ** Dragonette, Leila A. (1952), | ||
** Transactions of the American Mathematical Society 72: 474–500 | ** Transactions of the American Mathematical Society 72: 474–500 | ||
| − | * | + | * [http://dx.doi.org/10.2307%2F2373202 On the theorems of Watson and Dragonette for Ramanujan's mock theta functions]<br> |
| − | ** <br> | + | ** Andrews, George E. (1966) |
| + | ** American Journal of Mathematics 88: 454–490<br> | ||
2009년 8월 9일 (일) 01:28 판
Examples
classify which examples are of which use
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]
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
- 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 (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, http://www.jstor.org/stable/2693766
- Choi, Youn-Seo (2007), "Tenth order mock theta functions in Ramanujan's lost notebook. III", Proceedings of the London Mathematical Society. Third Series94 (1): 26–52, doi:10.1112/plms/pdl006, MR2293464, ISSN0024-6115
http://mathsci.ucd.ie/~zwegers/papers/006.pdf
간단한 소개
하위주제들
하위페이지
재미있는 사실
관련된 단원
많이 나오는 질문
관련된 고교수학 또는 대학수학
관련된 다른 주제들
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=
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