"Appell-Lerch sums"의 두 판 사이의 차이
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<h5>introduction</h5> | <h5>introduction</h5> | ||
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| + | <h5>Appell-Ler</h5> | ||
Appell–Lerch sums were first studied by [http://en.wikipedia.org/wiki/Paul_%C3%89mile_Appell Paul Émile Appell] ([http://en.wikipedia.org/wiki/Mock_theta_function#CITEREFAppell1884 1884]) and [http://en.wikipedia.org/wiki/Mathias_Lerch Mathias Lerch] ([http://en.wikipedia.org/wiki/Mock_theta_function#CITEREFLerch1892 1892]). Watson studied the order 3 mock theta functions by expressing them in terms of Appell–Lerch sums, and Zwegers used them to show that mock theta functions are essentially mock modular forms. | Appell–Lerch sums were first studied by [http://en.wikipedia.org/wiki/Paul_%C3%89mile_Appell Paul Émile Appell] ([http://en.wikipedia.org/wiki/Mock_theta_function#CITEREFAppell1884 1884]) and [http://en.wikipedia.org/wiki/Mathias_Lerch Mathias Lerch] ([http://en.wikipedia.org/wiki/Mock_theta_function#CITEREFLerch1892 1892]). Watson studied the order 3 mock theta functions by expressing them in terms of Appell–Lerch sums, and Zwegers used them to show that mock theta functions are essentially mock modular forms. | ||
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<h5>articles</h5> | <h5>articles</h5> | ||
| − | * | + | * [http://www.math.wisc.edu/~ono/reprints/122.pdf Some characters of Kac and Wakimoto and nonholomorphic modular functions.]<br> |
| − | ** K. Bringmann and K. Ono, | + | ** K. Bringmann and K. Ono, Math. Annalen 345, pages 547-558 (2009) |
| − | + | * [http://dx.doi.org/10.1007/s00220-008-0677-0 Higher String Functions, Higher-Level Appell Functions, and the Logarithmic sℓ︿2k/u(1) CFT Model]<br> | |
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** A. M. Semikhatov | ** A. M. Semikhatov | ||
| − | * | + | * [http://dx.doi.org/10.1007/s00220-004-1280-7 Higher-Level Appell Functions, Modular Transformations, and Characters]<br> |
** A.M. Semikhatov | ** A.M. Semikhatov | ||
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* http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q= | * 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= | * 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= | ||
| − | * http://dx.doi.org/ | + | * http://dx.doi.org/10.1007/s00220-004-1280-7 |
2010년 4월 7일 (수) 07:39 판
introduction
Appell-Ler
Appell–Lerch sums were first studied by Paul Émile Appell (1884) and Mathias Lerch (1892). Watson studied the order 3 mock theta functions by expressing them in terms of Appell–Lerch sums, and Zwegers used them to show that mock theta functions are essentially mock modular forms.
The Appell–Lerch series is
\(\mu(u,v;\tau) = \frac{a^{1/2}}{\theta(v;\tau)}\sum_{n\in Z}\frac{(-b)^nq^{n(n+1)/2}}{1-aq^n}\)
where
\(\displaystyle q= e^{2\pi i \tau},\quad a= e^{2\pi i u},\quad b= e^{2\pi i v}\)
and
\(\theta(v,\tau) = \sum_{n\in Z}(-1)^n b^{n+1/2}q^{(n+1/2)^2/2}\)
The modified series
\[\hat\mu(u,v;\tau) = \mu(u,v;\tau)-R(u-v;\tau)/2\]
where
\[R(z;\tau) = \sum_{\nu\in Z+1/2}(-1)^{\nu-1/2}({\rm sign}(\nu)-E((\nu+\Im(z)/y)\sqrt{2y}))e^{-2\pi i \nu z}q^{-\nu^2/2}\]
and y = Im(τ) and
\[E(z) = 2\int_0^ze^{-\pi u^2}\,du\]
satisfies the following transformation properties
\[\hat\mu(u+1,v;\tau) = a^{-1}bq^{-1/2}\hat\mu(u+\tau,v;\tau) = -\hat\mu(u,v;\tau),\]
\[e^{2\pi i/8}\hat\mu(u,v;\tau+1) = \hat\mu(u,v;\tau) = -(\tau/i)^{-1/2}e^{\pi i (u-v)^2/\tau}\hat\mu(u/\tau,v/\tau;-1/\tau).\]
In other words the modified Appell–Lerch series transforms like a modular form with respect to τ. Since mock theta functions can be expressed in terms of Appell–Lerch series this means that mock theta functions transform like modular forms if they have a certain non-analytic series added to them.
higher level Appell function
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articles
- Some characters of Kac and Wakimoto and nonholomorphic modular functions.
- K. Bringmann and K. Ono, Math. Annalen 345, pages 547-558 (2009)
- Higher String Functions, Higher-Level Appell Functions, and the Logarithmic sℓ︿2k/u(1) CFT Model
- A. M. Semikhatov
- Higher-Level Appell Functions, Modular Transformations, and Characters
- A.M. Semikhatov
- Mock Theta Functions
- Sander Zwegers, 2002
- Integrable highest weight modules over affine superalgebras and Appell’s function
- Kac V.G., Wakimoto M, Commun. Math. Phys. 215(3), 631–682 (2001)
- Kac V.G., Wakimoto M, Commun. Math. Phys. 215(3), 631–682 (2001)
- 논문정리
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- 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=
- http://dx.doi.org/10.1007/s00220-004-1280-7
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