نمایش مبدأ برای فرم ماک ماژولار

در [[ریاضیات]]، '''فرم ماک ماژولار''' {{انگلیسی|Mock Modular Form}}، بخش [[هولومورف|هولومورفیک]] از یک فرم ماس (Maass Form) ضعیف هارمونیک بوده و یک '''تابع تتای ماک''' اساساً فرم ماژولار ماکی با وزن {{sfrac|1|2}} است. اولین مثال‌ها از توابع تتای ماک توسط [[سرینیواسا رامانوجان]] در آخرین نامه‌اش در سال ۱۹۲۰ به [[گادفری هارولد هاردی|گ.هـ. هاردی]] و در دفترچهٔ گم‌شده‌اش توصیف شده‌است. سندر زووگرز (Sander Zwegers) کشف نمود که افزودن برخی از توابع غیر-هولومورفیک بهشان، آن‌ها را تبدیل به فرم‌های ماس ضعیف می‌کند.{{sfn|Zwegers|2001}}{{sfn|Zwegers|2002}}

== تاریخچه ==
{{quote box
 | align=left
 | width=33%
 | quote= «فرض کنید تابعی به فرم اویلری وجود داشته و فرض کنید که تمام نقاط یا بی‌نهایت از آن‌ها، نقاط تکین نمایی باشند، و همچنین فرض کنید که در این نقاط، فرم مجانبی با همان ترتیب و سادگی موارد (A) و (B) نزدیک شود. حال سؤال این است که: آیا این تابع به صورت جمع دو تابعی است که یکی از آن‌ها تابع <math>\theta</math> و دیگری تابع (بدیهی) <math>O(1)</math> در تمام نقاط <math>e^{2m\pi i/n}</math> می‌باشد؟ … وقتی چنین نباشد، من به این تابع، تابع <math>\theta</math>ی ماک می‌گویم.»
 | source=تعریف اصلی رامانوجان از تابع تتای ماک{{sfn|Ramanujan|2000|loc=Appendix II}}
}}
نامه ۱۲ ژانویه ۱۹۲۰ میلادی رامانوجان به هاردی،{{sfn|Ramanujan|2000|loc=Appendix II}} ١٧ مثال از توابعی را که او به نام توابع تتای ماک می‌نامید، فهرست کرده، و دفترچه گم‌شده اش{{sfn|Ramanujan|1988}} شامل چندین مثال دیگر نیز می‌باشد (منظور رامانوجان از اصطلاح «تابع تتا»، چیزیست که امروزه به یک فرم ماژولار معروف است). رامانوجان اشاره کرد که این توابع دارای [[بسط مجانبی]] در کاسپ‌ها بوده، که با فرم‌های ماژولاری با وزن {{sfn|1|2}} شباهت داشته، به طوری که احتمالاً دارای قطب‌هایی در کاسپ‌ها بوده اما نمی‌توان آن‌ها را برحسب توابع تتای «معمولی» نوشت. او توابعی با خواص مشابه را «توابع تتای ماک» نامید. زووگرز (Zwegers)، بعدها ارتباط تابع تتای ماک با فرم‌های ماس ضعیف کشف نمود.

رامانوجان به توابع تتای ماکش، '''مرتبه''' (order) نسبت داد، که تعریف واضح و مشخصی نداشت. قبل از کار زووگرز، مراتب توابع تتای ماک شناخته شده شامل این موارد بودند:

{{وسط‌چین}}
<math>3, 5, 6, 7, 8, 10.</math>
{{پایان وسط‌چین}}

مفهوم رامانوجان از مرتبه، بعدها تبدیل به کونداکتورِ [[کاراکتر دیریکله|کاراکتر]] نبنتیپوس (Nebentypus character) با فرم‌های ماس هارمونیک از وزن {{sfn|1|2}} شد که توابع تتای ماک رامانوجان را به عنوان تصویرهای هولومورفیک می‌پذیرند.

== ارجاعات ==
{{پانویس|چپ‌چین=بله}}

== منابع ==
{{چپ‌چین}}
{{refbegin|30em}}
* {{Citation| title = On the theorems of Watson and Dragonette for Ramanujan's mock theta functions
 | last = Andrews | first = George E. | year = 1966
 | journal = [[American Journal of Mathematics]]
 | volume = 88 | issue = 2 | pages = 454–490
 | doi = 10.2307/2373202 | issn = 0002-9327 | jstor = 2373202 | mr = 0200258
}}
* {{Citation| title = The fifth and seventh order mock theta functions
 | last = Andrews | first = George E. | year = 1986
 | journal = [[Transactions of the American Mathematical Society]]
 | volume = 293 | issue = 1 | pages = 113–134
 | doi = 10.2307/2000275 | issn = 0002-9947 | jstor = 2000275 | mr = 814916
 | doi-access = free
}}
* {{Citation| chapter = Ramanujan's fifth order mock theta functions as constant terms
 | last = Andrews | first = George E. | year = 1988
 | title = Ramanujan revisited (Urbana-Champaign, Ill. , 1987)
 | publisher = [[Academic Press]] | location = Boston, MA
 | pages = 47–56
 | mr = 938959
}}
* {{Citation| chapter = Mock theta functions
 | last = Andrews | first = George E. | year = 1989
 | title = Theta functions—Bowdoin 1987, Part 2 (Brunswick, ME, 1987)
 | publisher = [[American Mathematical Society]] | location = Providence, R.I.
 | volume = 49 | series = Proc. Sympos. Pure Math.
 | pages = 283–298
 | mr = 1013178
}}
* {{Citation| title = Ramanujan's lost notebook. VI. The mock theta conjectures
 | last1 = Andrews | first1 = George E.
 | last2 = Garvan | first2 = F. G.
 | journal = [[Advances in Mathematics]]
 | year = 1989 | volume = 73 | issue = 2 | pages = 242–255
 | doi = 10.1016/0001-8708(89)90070-4 | issn = 0001-8708 | mr = 987276
 | doi-access = free
}}
* {{Citation| title = Ramanujan's lost notebook. VII. The sixth order mock theta functions
 | last1 = Andrews | first1 = George E.
 | last2 = Hickerson | first2 = Dean
 | journal = [[Advances in Mathematics]]
 | year = 1991 | volume = 89 | issue = 1 | pages = 60–105
 | doi = 10.1016/0001-8708(91)90083-J | issn = 0001-8708 | mr = 1123099
 | doi-access = free
}}
* {{citation| title = Sur les fonctions doublement périodiques de troisième espèce
 | last = Appell | first = P. | year = 1884
 | journal = Annales Scientifiques de l'École Normale Supérieure
 | volume = 1 | pages = 135–164
 | doi = 10.24033/asens.236
 | doi-access = free
}}
* {{Citation| title = Sixth order mock theta functions
 | last1 = Berndt | first1 = Bruce C.
 | last2 = Chan | first2 = Song Heng
 | journal = [[Advances in Mathematics]]
 | year = 2007 | volume = 216 | issue = 2 | pages = 771–786
 | doi = 10.1016/j.aim.2007.06.004 | issn = 0001-8708 | mr = 2351377
 | doi-access = free
}}
* {{Cite journal | title = q-series and weight 3/2 Maass forms
 | last1 = Bringmann | first1 = Kathrin
 | last2 = Folsom | first2 = Amanda
 | last3 = Ono | first3 = Ken
 | author1-link = Kathrin Bringmann
 | author2-link = Amanda Folsom
 | author3-link = Ken Ono
 | journal = Compositio Mathematica
 | year = 2009 | volume = 145 | issue = 3 | pages = 541–552
 | url = https://afolsom.people.amherst.edu/BringmannFolsomOno-Compositio.pdf
 | doi = 10.1112/S0010437X09004072
| s2cid = 7688222 | doi-access = free
 }}
* {{Citation| title = The f(q) mock theta function conjecture and partition ranks
 | last1 = Bringmann | first1 = Kathrin
 | last2 = Ono | first2 = Ken
 | journal = [[Inventiones Mathematicae]]
 | year = 2006 | volume = 165 | issue = 2 | pages = 243–266
 | url = http://www.math.wisc.edu/~ono/reprints/098.pdf | url-status = dead
 | archive-url = https://web.archive.org/web/20060829065534/http://www.math.wisc.edu/~ono/reprints/098.pdf
 | archive-date = 2006-08-29
 | bibcode = 2006InMat.165..243B | doi = 10.1007/s00222-005-0493-5 | issn = 0020-9910 | mr = 2231957
| s2cid = 120388256 }}
* {{Citation| title = Lifting cusp forms to Maass forms with an application to partitions
 | last1 = Bringmann | first1 = Kathrin
 | last2 = Ono | first2 = Ken
 | journal = [[Proceedings of the National Academy of Sciences|Proceedings of the National Academy of Sciences of the United States of America]]
 | year = 2007 | volume = 104 | issue = 10 | pages = 3725–3731
 | url = http://www.math.wisc.edu/~ono/reprints/106.pdf | url-status = dead
 | archive-url = https://web.archive.org/web/20080828193744/http://www.math.wisc.edu/~ono/reprints/106.pdf
 | archive-date = 2008-08-28
 | bibcode = 2007PNAS..104.3725B | doi = 10.1073/pnas.0611414104 | issn = 0027-8424 | mr = 2301875 | pmc = 1820651 | pmid = 17360420
}}
* {{Citation| title = Dyson's ranks and Maass forms
 | last1 = Bringmann | first1 = Kathrin
 | last2 = Ono | first2 = Ken
 | journal = [[Annals of Mathematics]]
 | year = 2010 | volume = 171 | page = 419
 | url = http://www.math.wisc.edu/~ono/reprints/098.pdf | url-status = dead
 | archive-url = https://web.archive.org/web/20060829065534/http://www.math.wisc.edu/~ono/reprints/098.pdf
 | archive-date = 2006-08-29
 | doi = 10.4007/annals.2010.171.419
| doi-access = free
 }}
* {{Citation| title = On two geometric theta lifts
 | last1 = Bruinier | first1 = Jan Hendrik
 | last2 = Funke | first2 = Jens
 | journal = [[Duke Mathematical Journal]]
 | year = 2004 | volume = 125 | issue = 1 | pages = 45–90
 | arxiv = math/0212286 | doi = 10.1215/S0012-7094-04-12513-8 | issn = 0012-7094 | mr = 2097357
| s2cid = 2078210 }}
* {{Citation| title = Tenth order mock theta functions in Ramanujan's lost notebook
 | last = Choi | first = Youn-Seo | year = 1999
 | journal = Inventiones Mathematicae
 | volume = 136 | issue = 3 | pages = 497–569
 | bibcode = 1999InMat.136..497C | doi = 10.1007/s002220050318 | issn = 0020-9910 | mr = 1695205
| s2cid = 125193659 }}
* {{Citation| title = Tenth order mock theta functions in Ramanujan's lost notebook. II
 | last = Choi | first = Youn-Seo | year = 2000
 | journal = [[Advances in Mathematics]]
 | volume = 156 | issue = 2 | pages = 180–285
 | doi = 10.1006/aima.2000.1948 | issn = 0001-8708 | mr = 1808245
 | doi-access = free
}}
* {{Citation| title = Tenth order mock theta functions in Ramanujan's lost notebook. IV
 | last = Choi | first = Youn-Seo | year = 2002
 | journal = Transactions of the American Mathematical Society
 | volume = 354 | issue = 2 | pages = 705–733
 | doi = 10.1090/S0002-9947-01-02861-6 | issn = 0002-9947 | jstor = 2693766 | mr = 1862564
 | doi-access = free
}}
* {{Citation| title = Tenth order mock theta functions in Ramanujan's lost notebook. III
 | last = Choi | first = Youn-Seo | year = 2007
 | journal = Proceedings of the London Mathematical Society
 | volume = 94 | issue = 1 | pages = 26–52
 | doi = 10.1112/plms/pdl006 | issn = 0024-6115 | mr = 2293464
}}
* {{cite arxiv| mode = cs2| title = Quantum Black Holes, Wall Crossing, and Mock Modular Forms
 | last1 = Dabholkar | first1 = Atish
 | last2 = Murthy | first2 = Sameer
 | last3 = Zagier | first3 = Don
 | year = 2012
 | class = hep-th
 | eprint = 1208.4074
}}
* {{Citation| title = Some asymptotic formulae for the mock theta series of Ramanujan
 | last = Dragonette | first = Leila A. | year = 1952
 | journal = Transactions of the American Mathematical Society
 | volume = 72 | issue = 3 | pages = 474–500
 | doi = 10.2307/1990714 | issn = 0002-9947 | jstor = 1990714 | mr = 0049927
 | doi-access = free
}}
* {{Citation| title = Basic hypergeometric series and applications
 | last = Fine | first = Nathan J. | year = 1988
 | publisher = American Mathematical Society | location = Providence, R.I.
 | volume = 27 | series = Mathematical Surveys and Monographs
 | isbn = 978-0-8218-1524-3 | mr = 956465
}}
* {{Citation| title = The coefficients of the ω(q) mock theta function
 | last = Garthwaite | first = Sharon Anne | year = 2008
 | journal = International Journal of Number Theory
 | volume = 4 | issue = 6 | pages = 1027–1042
 | doi = 10.1142/S1793042108001869 | issn = 1793-0421 | mr = 2483310
}}
* {{Citation| title = Some eighth order mock theta functions
 | last1 = Gordon | first1 = Basil
 | last2 = McIntosh | first2 = Richard J.
 | author1-link = Basil Gordon
 | journal = Journal of the London Mathematical Society
 | year = 2000 | volume = 62 | issue = 2 | pages = 321–335
 | url = http://journals.cambridge.org/action/displayAbstract?aid=63263
 | doi = 10.1112/S0024610700008735 | issn = 0024-6107 | mr = 1783627
}}
* {{Citation| title = A proof of the mock theta conjectures
 | last = Hickerson | first = Dean | year = 1988a
 | journal = Inventiones Mathematicae
 | volume = 94 | issue = 3 | pages = 639–660
 | bibcode = 1988InMat..94..639H | doi = 10.1007/BF01394279 | issn = 0020-9910 | mr = 969247
| s2cid = 122492320 }}
* {{Citation| title = On the seventh order mock theta functions
 | last = Hickerson | first = Dean | year = 1988b
 | journal = Inventiones Mathematicae
 | volume = 94 | issue = 3 | pages = 661–677
 | bibcode = 1988InMat..94..661H | doi = 10.1007/BF01394280 | issn = 0020-9910 | mr = 969247
| s2cid = 121384412 }}
* {{Citation| title = Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus
 | last1 = Hirzebruch | first1 = Friedrich
 | last2 = Zagier | first2 = Don
 | author1-link = Friedrich Hirzebruch
 | journal = Inventiones Mathematicae
 | year = 1976 | volume = 36 | pages = 57–113
 | bibcode = 1976InMat..36...57H | doi = 10.1007/BF01390005 | hdl = 21.11116/0000-0004-399B-E | issn = 0020-9910 | mr = 0453649
 | s2cid = 56568473 | hdl-access = free
}}
* {{Citation| title = Modular forms and quantum invariants of 3-manifolds
 | last1 = Lawrence | first1 = Ruth
 | last2 = Zagier | first2 = Don
 | journal = The Asian Journal of Mathematics
 | year = 1999 | volume = 3 | issue = 1 | pages = 93–107
 | doi = 10.4310/AJM.1999.v3.n1.a5 | issn = 1093-6106 | mr = 1701924
 | doi-access = free
}}
* {{citation| title = Bemerkungen zur Theorie der elliptischen Funktionen
 | last = Lerch | first = M. | year = 1892
 | journal = Jahrbuch uber die Fortschritte der Mathematik
 | volume = 24 | pages = 442–445
}}
* {{Citation| title = Second order mock theta functions
 | last = McIntosh | first = Richard J. | year = 2007
 | journal = Canadian Mathematical Bulletin
 | volume = 50 | issue = 2 | pages = 284–290
 | url = http://journals.cms.math.ca/cgi-bin/vault/view/mcintosh8634 | url-status = dead
 | archive-url = https://archive.today/20121209210921/http://journals.cms.math.ca/cgi-bin/vault/view/mcintosh8634
 | archive-date = 2012-12-09
 | doi = 10.4153/CMB-2007-028-9 | issn = 0008-4395 | mr = 2317449
| s2cid = 119499438 }}
* {{Citation| title = The lost notebook and other unpublished papers
 | last = Ramanujan | first = Srinivasa | year = 1988
 | publisher = [[Springer-Verlag]] | location = Berlin, New York
 | isbn = 978-3-540-18726-4 | mr = 947735
}}
* {{Citation| title = Collected papers of Srinivasa Ramanujan
 | last = Ramanujan | first = Srinivasa | year = 2000
 | location = Providence, R.I.
 | volume = 123 | issue = 3104 | page = 631
 | journal = Nature
 | bibcode = 1929Natur.123..631L | doi = 10.1038/123631a0 | isbn = 978-0-8218-2076-6 | mr = 2280843
| s2cid = 44812911 }}
* {{citation| title = Über die Mock-Thetafunktionen siebenter Ordnung. (On the mock theta functions of seventh order)
 | last = Selberg | first = A. | year = 1938
 | journal = Archiv for Mathematik og Naturvidenskab
 | volume = 41 | pages = 3–15
}} Reprinted in volume I of his collected works.
* {{Citation| title = Higher-level Appell functions, modular transformations, and characters
 | last1 = Semikhatov | first1 = A. M.
 | last2 = Taormina | first2 = A.
 | last3 = Tipunin | first3 = I. Yu
 | author2-link = Anne Taormina
 | journal = Communications in Mathematical Physics
 | year = 2005 | volume = 255 | issue = 2 | pages = 469–512
 | arxiv = math/0311314 | bibcode = 2005CMaPh.255..469S | doi = 10.1007/s00220-004-1280-7 | issn = 0010-3616 | mr = 2129953
| s2cid = 14466569 }}
* {{Citation| chapter = Modular forms of weight 1/2
 | last1 = Serre | first1 = Jean-Pierre
 | last2 = Stark | first2 = H. M.
 | author1-link = Jean-Pierre Serre
 | year = 1977
 | title = Modular functions of one variable, VI (Proc. Second Internat. Conf. , Univ. Bonn, Bonn, 1976)
 | publisher = [[Springer-Verlag]] | location = Berlin, New York
 | volume = 627 | series = Lecture Notes in Mathematics
 | pages = 27–67
 | doi = 10.1007/BFb0065296 | isbn = 978-3-540-08530-0 | mr = 0472707
}}
* {{Citation| title = The Non-Compact Elliptic Genus: Mock or Modular
 | last = Troost | first = J. | year = 2010
 | journal = Journal of High Energy Physics
 | volume = 2010 | issue = 6 | page = 104
 | arxiv = 1004.3649 | bibcode = 2010JHEP...06..104T | doi = 10.1007/JHEP06(2010)104
| s2cid = 84838021 }}
* {{citation| title = The Final Problem: An Account of the Mock Theta Functions
 | last = Watson | first = G. N. | year = 1936
 | journal = Journal of the London Mathematical Society
 | volume = 11 | pages = 55–80
 | doi = 10.1112/jlms/s1-11.1.55
}}
* {{citation| title = The Mock Theta Functions (2)
 | last = Watson | first = G. N. | year = 1937
 | journal = Proceedings of the London Mathematical Society
 | volume = s2-42 | pages = 274–304
 | doi = 10.1112/plms/s2-42.1.274
}}
* {{Citation| title = Nombres de classes et formes modulaires de poids 3/2
 | last = Zagier | first = Don | year = 1975
 | journal = Comptes Rendus de l'Académie des Sciences, Série A et B
 | volume = 281 | issue = 21 | pages = Ai, A883–A886
 | issn = 0151-0509 | mr = 0429750
}}
* {{Citation| title = Ramanujan's mock theta functions and their applications (after Zwegers and Ono-Bringmann)
 | last = Zagier | first = Don | year = 2009
 | orig-year = First published 2007
 | volume = 326 | pages = 143–164
 | department = Séminaire Bourbaki. Exp. 986
 | journal = Astérisque
 | url = http://people.mpim-bonn.mpg.de/zagier/files/aster/326/fulltext.pdf
 | issn = 0303-1179 | mr = 2605321
 | ref = {{harvid|Zagier|2007}}
}}
* {{Citation| chapter = Mock θ-functions and real analytic modular forms
 | last = Zwegers | first = S. P. | year = 2001
 | title = q-series with applications to combinatorics, number theory, and physics (Urbana, IL, 2000)
 | publisher = American Mathematical Society | location = Providence, R.I.
 | volume = 291 | series = Contemp. Math.
 | pages = 269–277
 | chapter-url = http://mathsci.ucd.ie/~zwegers/papers/001.pdf
 | isbn = 978-0-8218-2746-8 | mr = 1874536
}}{{dead link|date=February 2018 |bot=InternetArchiveBot |fix-attempted=yes}}
* {{Citation| title = Mock Theta Functions
 | last = Zwegers | first = S. P. | year = 2002
 | publisher = Utrecht PhD thesis
 | url = http://igitur-archive.library.uu.nl/dissertations/2003-0127-094324/inhoud.htm
 | isbn = 90-393-3155-3
}}
* {{citation| title = Appell–Lerch sums as mock modular forms
 | last = Zwegers | first = S. P. | year = 2008
 | url = http://mathsci.ucd.ie/~zwegers/presentations/002.pdf
}}{{dead link|date=February 2018 |bot=InternetArchiveBot |fix-attempted=yes}}
{{refend}}
{{پایان چپ‌چین}}

== برای مطالعه بیشتر ==
{{چپ‌چین}}
{{refbegin}}
* {{Citation| chapter = Mock theta functions, ranks and Maass forms
 | last = Ono | first = Ken | year = 2008
 | author-link = Ken Ono
 | title = Surveys in Number Theory
 | editor-last = Alladi | editor-first = Krishnaswami | editor-link = Krishnaswami Alladi
 | publisher = [[Springer-Verlag]]
 | volume = 17 | series = Developments in Mathematics
 | pages = 119–141
 | isbn = 978-0-387-78509-7 | zbl = 1183.11064
 | ref = none
}}
{{refend}}
{{پایان چپ‌چین}}

== پیوند به بیرون ==
{{چپ‌چین}}
* [https://web.archive.org/web/20081221143605/http://www.mpim-bonn.mpg.de/Events/This+Year+and+Prospect/Mock+theta+functions/ International Conference: Mock theta functions and applications 2009]
* [https://web.archive.org/web/20081204111537/http://www.math.psu.edu/andrews/biblio.html Papers on mock theta functions] by [[George Andrews (mathematician)|George Andrews]]
* [http://www.mi.uni-koeln.de/~kbringma/papers.html Papers on mock theta functions] by [[Kathrin Bringmann]]
* [https://web.archive.org/web/20100620082612/http://www.math.wisc.edu/~ono/reprints/index.html Papers on mock theta functions] by [[Ken Ono]]
* [https://web.archive.org/web/20081023234404/http://mathsci.ucd.ie/~zwegers/ Papers on mock theta functions] by [[Sander Zwegers]]
* {{MathWorld|urlname=MockThetaFunction|title=Mock Theta Function}}
{{پایان چپ‌چین}}

[[رده:فرم‌های مدولار]]
[[رده:Q-آنالوگ‌ها]]
[[رده:سرینیواسا رامانوجان]]