Appell-Lerch sums - 편집 역사

2020년 11월 13일 (금) 16:49에 Pythagoras0님의 편집

2020-11-13T16:49:45Z

2020년 11월 13일 (금) 15:02에 imported>Pythagoras0님의 편집

2020-11-13T15:02:25Z

2014년 7월 15일 (화) 14:21에 imported>Pythagoras0님의 편집

2014-07-15T14:21:33Z

2013년 8월 5일 (월) 19:07에 imported>Pythagoras0님의 편집

2013-08-05T19:07:29Z

2013년 7월 26일 (금) 16:23에 imported>Pythagoras0님의 편집

2013-07-26T16:23:33Z

2013년 7월 25일 (목) 14:20에 imported>Pythagoras0님의 편집

2013-07-25T14:20:18Z

imported>Pythagoras0: /* articles */

2013-07-25T13:43:29Z

articles

imported>Pythagoras0: /* articles */

2013-07-25T13:42:43Z

articles

imported>Pythagoras0: /* articles */

2013-07-25T13:40:36Z

articles

2013년 2월 8일 (금) 22:09에 imported>Pythagoras0님의 편집

2013-02-08T22:09:03Z

@@ 14번째 줄: / 14번째 줄: @@
 * Zwegers used them to show that mock theta functions are essentially mock modular forms.
 * The Appell–Lerch series is
-$$
+:<math>
 \mu(u,v;\tau) = \frac{i a^{1/2}}{\theta_{11}(v;\tau)}\sum_{n\in Z}\frac{(-b)^nq^{n(n+1)/2}}{1-aq^n}
-$$
+</math>
 where
 :<math>\displaystyle q= e^{2\pi i \tau},\quad a= e^{2\pi i u},\quad b= e^{2\pi i v}</math>
@@ 22번째 줄: / 22번째 줄: @@
 :<math>\theta_{11}(v,\tau) = \sum_{n\in Z}(-1)^n b^{n+1/2}q^{(n+1/2)^2/2}</math>
 ===completion by adding a non-holomorphic part===
-* definte $\hat\mu(u,v;\tau)$ by
+* definte <math>\hat\mu(u,v;\tau)</math> by
 :<math>\hat\mu(u,v;\tau) = \mu(u,v;\tau)-R(u-v;\tau)/2</math>
 where
 :<math>R(z;\tau) = \sum_{\nu\in \mathbb{Z}+1/2}(-1)^{\nu-1/2}[{\rm sign}(\nu)-E\left((\nu+\frac{\Im(z)}{y})\sqrt{2y}\right)]e^{-2\pi i \nu z}q^{-\nu^2/2}</math>
-and $y=\Im(\tau)$ and
+and <math>y=\Im(\tau)</math> and
 :<math>E(z) = 2\int_0^ze^{-\pi u^2}\,du</math>
@@ 32번째 줄: / 32번째 줄: @@
 ===Mordell integral===
 * [[Mordell integrals]]
-$$
+:<math>
 \mu(u,v;\tau)+\sqrt{\frac{i}{\tau}}e^{\pi i \frac{(u-v)^2}{\tau}}\mu(\frac{u}{\tau},\frac{v}{\tau};-\frac{1}{\tau})=\frac{1}{2}M(u-v;\tau)
-$$
+</math>
 where
-$$M(v;\tau)=\int_{-\infty}^{\infty} \frac{e^{\pi i \tau x^2-2\pi v x}}{\cosh (\pi x)} dx$$
+:<math>M(v;\tau)=\int_{-\infty}^{\infty} \frac{e^{\pi i \tau x^2-2\pi v x}}{\cosh (\pi x)} dx</math>
-* non-holomorpic part is an incomplete period integral of the modular form $\eta^3$
+* non-holomorpic part is an incomplete period integral of the modular form <math>\eta^3</math>
-$$
+:<math>
 iR(0;\tau)=\int_{-\bar{\tau}}^{i\infty}\frac{\eta(z)^3}{\sqrt{\frac{z+\tau}{i}}}\,dz
-$$
+</math>
 * property
-$$
+:<math>
 R(z;\tau)+\sqrt{\frac{i}{\tau}}R(\frac{z}{\tau};-\frac{1}{\tau})=M(z;\tau)
-$$
+</math>
@@ 57번째 줄: / 57번째 줄: @@
 ===special case===
 * mock theta function
-$$
+:<math>
 \mu(z;\tau):= \mu(z,z;\tau)= \frac{i e^{\pi i z}}{\theta_{11}(z;\tau)}\sum_{n\in Z}\frac{(-1)^nq^{n(n+1)/2}e^{2\pi i n z}}{1-q^ne^{2\pi i z}}
-$$
+</math>
 * Mordell integrals
-$$
+:<math>
 \mu(z;\tau)+\sqrt{\frac{i}{\tau}}\mu(\frac{z}{\tau};-\frac{1}{\tau})=\frac{1}{2}M(0;\tau)=\frac{1}{2}\int_{-\infty}^{\infty} \frac{e^{\pi i \tau x^2}}{\cosh (\pi x)} dx
-$$
+</math>
 * completion
-$$
+:<math>
 \hat{\mu}(z;\tau)=\mu(z;\tau)-R(0;\tau)/2
-$$
+</math>
 * modularity
-$$
+:<math>
 \hat{\mu}(\frac{z}{\tau};\frac{-1}{\tau})=-\sqrt{\frac{\tau}{i}}\hat{\mu}(z;\tau)
-$$
+</math>
 ==higher level Appell function==

← 이전 판		2020년 11월 13일 (금) 15:02 판
123번째 줄:		123번째 줄:
	[[분류:mock modular forms]]		[[분류:mock modular forms]]
	[[분류:math]]		[[분류:math]]
		+	[[분류:migrate]]

@@ 106번째 줄: / 106번째 줄: @@
 ==articles==
 * Tohru Eguchi and Kazuhiro Hikami [http://dx.doi.org/10.1088/1751-8113/42/30/304010 Superconformal Algebras and Mock Theta Functions], 2009
 * [http://www.math.wisc.edu/~ono/reprints/122.pdf Some characters of Kac and Wakimoto and nonholomorphic modular functions.]

@@ 14번째 줄: / 14번째 줄: @@
 * Zwegers used them to show that mock theta functions are essentially mock modular forms.
 * The Appell–Lerch series is
-:<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>
+$$
 where
 :<math>\displaystyle q= e^{2\pi i \tau},\quad a= e^{2\pi i u},\quad b= e^{2\pi i v}</math>
 and
-:<math>\theta(v,\tau) = \sum_{n\in Z}(-1)^n b^{n+1/2}q^{(n+1/2)^2/2}</math>
+:<math>\theta_{11}(v,\tau) = \sum_{n\in Z}(-1)^n b^{n+1/2}q^{(n+1/2)^2/2}</math>
-* The modified series
+===completion by adding a non-holomorphic part===
 :<math>\hat\mu(u,v;\tau) = \mu(u,v;\tau)-R(u-v;\tau)/2</math>
 where
-:<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 \mathbb{Z}+1/2}(-1)^{\nu-1/2}[{\rm sign}(\nu)-E\left((\nu+\frac{\Im(z)}{y})\sqrt{2y}\right)]e^{-2\pi i \nu z}q^{-\nu^2/2}</math>
 and $y=\Im(\tau)$ and
 :<math>E(z) = 2\int_0^ze^{-\pi u^2}\,du</math>
-satisfies the following transformation properties
 :<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>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>
@@ 32번째 줄: / 55번째 줄: @@
 ==higher level Appell function==
-*  higher-level Appell functions<br>
+*  higher-level Appell functions
 ** a particular instance of indefinite theta series
@@ 68번째 줄: / 109번째 줄: @@
 * [http://www.math.wisc.edu/~ono/reprints/122.pdf Some characters of Kac and Wakimoto and nonholomorphic modular functions.]
 ** 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]
+* Zwegers [http://mathsci.ucd.ie/~zwegers/presentations/002.pdf Appell-Lerch sums as mock modular forms], 2008
-** Zwegers, 2008
+* A. M. Semikhatov [http://dx.doi.org/10.1007/s00220-008-0677-0 Higher String Functions, Higher-Level Appell Functions, and the Logarithmic sℓ︿2k/u(1) &nbsp;CFT Model]
-* [http://dx.doi.org/10.1007/s00220-008-0677-0 Higher String Functions, Higher-Level Appell Functions, and the Logarithmic sℓ︿2k/u(1) &nbsp;CFT Model]
+* A.M. Semikhatov [http://dx.doi.org/10.1007/s00220-004-1280-7 Higher-Level Appell Functions, Modular Transformations, and Characters]
-** A. M. Semikhatov
+* Sander [http://front.math.ucdavis.edu/author/S.Zwegers Zwegers], [http://front.math.ucdavis.edu/0807.4834 Mock Theta Functions], 2002
 * [http://dx.doi.org/10.1007/s002200000315 Integrable highest weight modules over affine superalgebras and Appell’s function]
 ** Kac V.G., Wakimoto M, Commun. Math. Phys. '''215'''(3), 631–682 (2001)

← 이전 판		2013년 7월 26일 (금) 16:23 판
54번째 줄:		54번째 줄:
	* [[Kac-Wakimoto modules]]		* [[Kac-Wakimoto modules]]
	* [[indefinite theta functions]]		* [[indefinite theta functions]]
		+	* [[Mathieu moonshine]]

@@ 126번째 줄: / 126번째 줄: @@
 ==articles==
-* [http://dx.doi.org/10.1088/1751-8113/42/30/304010 Superconformal Algebras and Mock Theta Functions Tohru Eguchi]
+* Tohru Eguchi and Kazuhiro Hikami [http://dx.doi.org/10.1088/1751-8113/42/30/304010 Superconformal Algebras and Mock Theta Functions], 2009
 * [http://www.math.wisc.edu/~ono/reprints/122.pdf Some characters of Kac and Wakimoto and nonholomorphic modular functions.]
 ** K. Bringmann and K. Ono, Math. Annalen 345, pages 547-558 (2009)
@@ 143번째 줄: / 142번째 줄: @@
 * Yutaka Matsuo [http://ptp.ipap.jp/link?PTP/77/793/ Character Formula of C<1 Unitary representation of N=2 Superconformal Algebra] , Prog. Theor. Phys. Vol. 77 No. 4 (1987) pp. 793-797
 * C. Truesdell [http://www.jstor.org/stable/1969153 On a Function Which Occurs in the Theory of the Structure of Polymers], The Annals of Mathematics, Second Series, Vol. 46, No. 1 (Jan., 1945), pp. 144-157
 [[분류:개인노트]]
 [[분류:math and physics]]
 [[분류:mock modular forms]]
 [[분류:math]]

@@ 143번째 줄: / 143번째 줄: @@
 ** Kac V.G., Wakimoto M, Commun. Math. Phys. '''215'''(3), 631–682 (2001)<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>
+* Yutaka Matsuo [http://ptp.ipap.jp/link?PTP/77/793/ Character Formula of C<1 Unitary representation of N=2 Superconformal Algebra] , Prog. Theor. Phys. Vol. 77 No. 4 (1987) pp. 793-797
 * [http://www.jstor.org/stable/1969153 On a Function Which Occurs in the Theory of the Structure of Polymers]<br>
 **  C. Truesdell, The Annals of Mathematics, Second Series, Vol. 46, No. 1 (Jan., 1945), pp. 144-157<br>

← 이전 판		2013년 2월 8일 (금) 22:09 판
154번째 줄:		154번째 줄:
	[[분류:math and physics]]		[[분류:math and physics]]
	[[분류:mock modular forms]]		[[분류:mock modular forms]]
		+	[[분류:math]]