"Appell-Lerch sums"의 두 판 사이의 차이
| 81번째 줄: | 81번째 줄: | ||
<h5>related items</h5> | <h5>related items</h5> | ||
| − | * | + | * [[Kac-Wakimoto modules]] |
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| 135번째 줄: | 137번째 줄: | ||
<h5>articles</h5> | <h5>articles</h5> | ||
| − | + | * [http://dx.doi.org/10.1007/s002200000315 Integrable highest weight modules over affine superalgebras and Appell’s function]<br> | |
| + | ** Kac V.G., Wakimoto M, Commun. Math. Phys. '''215'''(3), 631–682 (2001)<br> | ||
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| + | * [http://front.math.ucdavis.edu/0807.4834 Mock Theta Functions]<br> | ||
| + | ** Sander [http://front.math.ucdavis.edu/author/S.Zwegers Zwegers], 2002 | ||
* [[2010년 books and articles|논문정리]] | * [[2010년 books and articles|논문정리]] | ||
2010년 3월 7일 (일) 15:47 판
introduction
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.
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- 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)
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- Sander Zwegers, 2002
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