"Appell-Lerch sums"의 두 판 사이의 차이

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<h5>introduction</h5>
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==introduction==
  
 
* one way to construct mock theta functions
 
* one way to construct mock theta functions
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* [[3rd order mock theta functions]]
 
* [[3rd order mock theta functions]]
  
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<h5>Appell-Lerch sum</h5>
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==Appell-Lerch sum==
  
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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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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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The Appell–Lerch series is
 
The Appell–Lerch series is
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<math>\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}</math>
 
<math>\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}</math>
  
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where
 
where
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<math>\displaystyle q= e^{2\pi i \tau},\quad a= e^{2\pi i u},\quad b= e^{2\pi i v}</math>
 
<math>\displaystyle q= e^{2\pi i \tau},\quad a= e^{2\pi i u},\quad b= e^{2\pi i v}</math>
  
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and
 
and
  
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<math>\theta(v,\tau) = \sum_{n\in Z}(-1)^n b^{n+1/2}q^{(n+1/2)^2/2}</math>
 
<math>\theta(v,\tau) = \sum_{n\in Z}(-1)^n b^{n+1/2}q^{(n+1/2)^2/2}</math>
  
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The modified series
 
The modified series
  
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: <math>\hat\mu(u,v;\tau) = \mu(u,v;\tau)-R(u-v;\tau)/2</math>
 
: <math>\hat\mu(u,v;\tau) = \mu(u,v;\tau)-R(u-v;\tau)/2</math>
  
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where
 
where
  
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: <math>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}</math>
 
: <math>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}</math>
  
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and <em style="">y</em> = Im(τ) and
 
and <em style="">y</em> = Im(τ) and
  
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: <math>E(z) = 2\int_0^ze^{-\pi u^2}\,du</math>
 
: <math>E(z) = 2\int_0^ze^{-\pi u^2}\,du</math>
  
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satisfies the following transformation properties
 
satisfies the following transformation properties
  
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: <math>\hat\mu(u+1,v;\tau) = a^{-1}bq^{-1/2}\hat\mu(u+\tau,v;\tau) = -\hat\mu(u,v;\tau),</math>
 
: <math>\hat\mu(u+1,v;\tau) = a^{-1}bq^{-1/2}\hat\mu(u+\tau,v;\tau) = -\hat\mu(u,v;\tau),</math>
  
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: <math>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).</math>
 
: <math>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).</math>
  
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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.
 
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.
  
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<h5>higher level Appell function</h5>
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==higher level Appell function==
  
 
*  higher-level Appell functions<br>
 
*  higher-level Appell functions<br>
 
** a particular instance of indefinite theta series
 
** a particular instance of indefinite theta series
  
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<h5>history</h5>
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==history==
  
 
* http://www.google.com/search?hl=en&tbs=tl:1&q=
 
* http://www.google.com/search?hl=en&tbs=tl:1&q=
  
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<h5>related items</h5>
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==related items==
  
 
* [[Kac-Wakimoto modules]]
 
* [[Kac-Wakimoto modules]]
 
* [[indefinite theta functions]]
 
* [[indefinite theta functions]]
  
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<h5>encyclopedia</h5>
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==encyclopedia==
  
 
* http://ko.wikipedia.org/wiki/
 
* http://ko.wikipedia.org/wiki/
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* Princeton companion to mathematics([[2910610/attachments/2250873|Companion_to_Mathematics.pdf]])
 
* Princeton companion to mathematics([[2910610/attachments/2250873|Companion_to_Mathematics.pdf]])
  
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<h5>question and answers(Math Overflow)</h5>
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==question and answers(Math Overflow)==
  
 
* http://mathoverflow.net/search?q=
 
* http://mathoverflow.net/search?q=
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* http://mathoverflow.net/search?q=
 
* http://mathoverflow.net/search?q=
  
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<h5>articles</h5>
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==articles==
  
 
* [http://dx.doi.org/10.1088/1751-8113/42/30/304010 Superconformal Algebras and Mock Theta Functions Tohru Eguchi]<br>
 
* [http://dx.doi.org/10.1088/1751-8113/42/30/304010 Superconformal Algebras and Mock Theta Functions Tohru Eguchi]<br>
 
** Kazuhiro Hikami, 2009
 
** Kazuhiro Hikami, 2009
 
* [http://www.math.wisc.edu/~ono/reprints/122.pdf Some characters of Kac and Wakimoto and nonholomorphic modular functions.]<br>
 
* [http://www.math.wisc.edu/~ono/reprints/122.pdf Some characters of Kac and Wakimoto and nonholomorphic modular functions.]<br>
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** K. Bringmann and K. Ono, Math. Annalen 345, pages 547-558 (2009)
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** K. Bringmann and K. Ono, Math. Annalen 345, pages 547-558 (2009)
 
* [http://mathsci.ucd.ie/~zwegers/presentations/002.pdf Appell-Lerch sums as mock modular forms]<br>
 
* [http://mathsci.ucd.ie/~zwegers/presentations/002.pdf Appell-Lerch sums as mock modular forms]<br>
 
** Zwegers, 2008
 
** Zwegers, 2008
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* [http://front.math.ucdavis.edu/0807.4834 Mock Theta Functions]<br>
 
* [http://front.math.ucdavis.edu/0807.4834 Mock Theta Functions]<br>
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** Sander [http://front.math.ucdavis.edu/author/S.Zwegers Zwegers], 2002
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** Sander [http://front.math.ucdavis.edu/author/S.Zwegers Zwegers], 2002
 
* [http://dx.doi.org/10.1007/s002200000315 Integrable highest weight modules over affine superalgebras and Appell’s function]<br>
 
* [http://dx.doi.org/10.1007/s002200000315 Integrable highest weight modules over affine superalgebras and Appell’s function]<br>
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** Kac V.G., Wakimoto M, Commun. Math. Phys. '''215'''(3), 631–682 (2001)<br>
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** Kac V.G., Wakimoto M, Commun. Math. Phys. '''215'''(3), 631–682 (2001)<br>
 
*  N = 2 superconformal minimal models<br>
 
*  N = 2 superconformal minimal models<br>
 
* [http://ptp.ipap.jp/link?PTP/77/793/ Character Formula of C<1 Unitary representation of N=2 Superconformal Algebra]<br>
 
* [http://ptp.ipap.jp/link?PTP/77/793/ Character Formula of C<1 Unitary representation of N=2 Superconformal Algebra]<br>
 
**  1987<br>
 
**  1987<br>
 
* [http://www.jstor.org/stable/1969153 On a Function Which Occurs in the Theory of the Structure of Polymers]<br>
 
* [http://www.jstor.org/stable/1969153 On a Function Which Occurs in the Theory of the Structure of Polymers]<br>
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**  C. Truesdell, The Annals of Mathematics, Second Series, Vol. 46, No. 1 (Jan., 1945), pp. 144-157<br>
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**  C. Truesdell, The Annals of Mathematics, Second Series, Vol. 46, No. 1 (Jan., 1945), pp. 144-157<br>
  
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*  <br>
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*   <br>
 
* http://dx.doi.org/10.1007/s00220-004-1280-7
 
* http://dx.doi.org/10.1007/s00220-004-1280-7

2012년 10월 25일 (목) 11:09 판

introduction

  • one way to construct mock theta functions
  • characters of representations in (nonrational) conformal field theory models based on Lie superalgebras\
  • 3rd order mock theta functions



Appell-Lerch sum

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

  • higher-level Appell functions
    • a particular instance of indefinite theta series



history



related items



encyclopedia



question and answers(Math Overflow)



articles

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