3rd order mock theta functions - 편집 역사

2020년 12월 28일 (월) 12:05에 Pythagoras0님의 편집

2020-12-28T12:05:47Z

2020년 11월 16일 (월) 10:56에 Pythagoras0님의 편집

2020-11-16T10:56:10Z

2020년 11월 13일 (금) 07:36에 Pythagoras0님의 편집

2020-11-13T07:36:10Z

2020년 11월 13일 (금) 07:18에 imported>Pythagoras0님의 편집

2020-11-13T07:18:12Z

imported>Pythagoras0: section 'articles' updated

2016-03-18T05:39:14Z

section 'articles' updated

imported>Pythagoras0: section 'articles' updated

2016-03-16T07:30:48Z

section 'articles' updated

2013년 12월 20일 (금) 18:09에 imported>Pythagoras0님의 편집

2013-12-20T18:09:05Z

2013년 12월 20일 (금) 18:04에 imported>Pythagoras0님의 편집

2013-12-20T18:04:08Z

2013년 10월 10일 (목) 19:29에 imported>Pythagoras0님의 편집

2013-10-10T19:29:34Z

imported>Pythagoras0: /* encyclopedia */

2013-08-08T20:32:10Z

encyclopedia

@@ 4번째 줄: / 4번째 줄: @@
 ** http://oeis.org/A000025
-−
 * the asymptotic series for coefficients of the order 3 mock theta function f(q) studied by of (Andrews 1966) and Dragonette (1952) converges to the coefficients ([http://www.springerlink.com/content/5524655155350464/ Bringmann & Ono 2006]).
 ** see [[Rank of partition and mock theta conjecture]]
@@ 24번째 줄: / 24번째 줄: @@
 </math>
 * if <math>k=2</math>, as <math>q\to i</math>, <math>f(q)-b(q)\to 4i</math>
-−
 ==harmonic weak Maass form==
-* We have a weight k=1/2, harmonic weak Maass form <math>h_3</math> under <math>\Gamma(2)</math> defined by :<math>h_3(\tau)=q^{-1/24}f(q)+R_3(q)</math> where
+* We have a weight k=1/2, harmonic weak Maass form <math>h_3</math> under <math>\Gamma(2)</math> defined by :<math>h_3(\tau)=q^{-1/24}f(q)+R_3(q)</math> where
-:<math>R_3(\tau)=\sum_{n\equiv 1\pmod 6}\operatorname{sgn}(n)\beta(n^2y/6)q^{-n^2/24}</math> where
+:<math>R_3(\tau)=\sum_{n\equiv 1\pmod 6}\operatorname{sgn}(n)\beta(n^2y/6)q^{-n^2/24}</math> where
 :<math>\displaystyle \beta(t) = \int_t^\infty u^{-1/2} e^{-\pi u} \,du=2\int_{\sqrt{x}}^{\infty} e^{-\pi t^2}\,dt</math>
 * Note that this can be rewritten as :<math>R_3(\tau)=(i/2)^{k-1} \int_{-\overline\tau}^{i\infty} (z+\tau)^{-k}\overline{g(-\overline z)}\,dz</math>
@@ 40번째 줄: / 40번째 줄: @@
 * <math>M_f(z)=q^{-1}f(q^{24})+\frac{i}{\sqrt{3}}\int_{}^{}\frac{\Theta(24z)}{}dz</math>
-−
-−
-−
 ==related items==
@@ 50번째 줄: / 50번째 줄: @@
-−
 ==computational resources==
 * https://docs.google.com/file/d/0B8XXo8Tve1cxLWNCNklCRlVXd2c/edit
-−

← 이전 판		2020년 11월 16일 (월) 10:56 판
41번째 줄:		41번째 줄:


−
−
−
−
−
−	~~==history==~~
−
−	* http://www.google.com/search?hl=en&tbs=tl:1&q=

@@ 16번째 줄: / 16번째 줄: @@
 ==asymptotic behavior at roots of unity==
-* the series converges for $|q|<1$ and the roots of unity $q$ at odd order
+* the series converges for <math>|q|<1</math> and the roots of unity <math>q</math> at odd order
-* For even order roots of unity, $f(q)$ has exponential singularities but there is a nice result to describe this behavior
+* For even order roots of unity, <math>f(q)</math> has exponential singularities but there is a nice result to describe this behavior
-* let us define $$b(q)=(1-q)(1-q^3)(1-q^5)\cdots (1-2q+2q^4-\cdots)$$, or we can write it as $$b(q)=q^{1/24}\frac{\eta(\tau)}{\eta(2\tau)}\theta(-q)$$
+* let us define :<math>b(q)=(1-q)(1-q^3)(1-q^5)\cdots (1-2q+2q^4-\cdots)</math>, or we can write it as :<math>b(q)=q^{1/24}\frac{\eta(\tau)}{\eta(2\tau)}\theta(-q)</math>
-* let $\zeta$ be even $2k$ order root of unity
+* let <math>\zeta</math> be even <math>2k</math> order root of unity
-$$
+:<math>
 \lim_{q\to \zeta} f(q)-(-1)^k b(q)=-4\sum_{n=0}^{k-1} (1+\zeta)^2(1+\zeta^2)^2\cdots (1+\zeta^n)^2\zeta^{n+1}
-$$
+</math>
-* if $k=2$, as $q\to i$, $f(q)-b(q)\to 4i$
+* if <math>k=2</math>, as <math>q\to i</math>, <math>f(q)-b(q)\to 4i</math>
 ==harmonic weak Maass form==
-* We have a weight k=1/2, harmonic weak Maass form $h_3$ under <math>\Gamma(2)</math> defined by :<math>h_3(\tau)=q^{-1/24}f(q)+R_3(q)</math> where
+* We have a weight k=1/2, harmonic weak Maass form <math>h_3</math> under <math>\Gamma(2)</math> defined by :<math>h_3(\tau)=q^{-1/24}f(q)+R_3(q)</math> where
 :<math>R_3(\tau)=\sum_{n\equiv 1\pmod 6}\operatorname{sgn}(n)\beta(n^2y/6)q^{-n^2/24}</math> where
 :<math>\displaystyle \beta(t) = \int_t^\infty u^{-1/2} e^{-\pi u} \,du=2\int_{\sqrt{x}}^{\infty} e^{-\pi t^2}\,dt</math>
 * Note that this can be rewritten as :<math>R_3(\tau)=(i/2)^{k-1} \int_{-\overline\tau}^{i\infty} (z+\tau)^{-k}\overline{g(-\overline z)}\,dz</math>
-where $g$ is the shadow
+where <math>g</math> is the shadow
 :<math>g_3(z)=\sum_{n\equiv 1\pmod 6}nq^{n^2/24}</math>
@@ 76번째 줄: / 76번째 줄: @@
 ==articles==
 * Min-Joo Jang, Byungchan Kim, On spt-crank type functions, http://arxiv.org/abs/1603.05608v1
-* George E. Andrews, Atul Dixit, Daniel Schultz, Ae Ja Yee, Overpartitions related to the mock theta function $ω(q)$, http://arxiv.org/abs/1603.04352v1
+* George E. Andrews, Atul Dixit, Daniel Schultz, Ae Ja Yee, Overpartitions related to the mock theta function <math>ω(q)</math>, http://arxiv.org/abs/1603.04352v1
 * Watson, G. N. [http://dx.doi.org/10.1112%2Fjlms%2Fs1-11.1.55 The Final Problem : An Account of the Mock Theta Functions](1936),  J. London Math. Soc. 11: 55–80
 * Dragonette, Leila A. [http://dx.doi.org/10.2307%2F1990714 Some asymptotic formulae for the mock theta series of Ramanujan](1952), Transactions of the American Mathematical Society 72: 474–500

← 이전 판		2020년 11월 13일 (금) 07:18 판
86번째 줄:		86번째 줄:
	[[분류:mock modular forms]]		[[분류:mock modular forms]]
	[[분류:math]]		[[분류:math]]
		+	[[분류:migrate]]

@@ 75번째 줄: / 75번째 줄: @@
 ==articles==
 * George E. Andrews, Atul Dixit, Daniel Schultz, Ae Ja Yee, Overpartitions related to the mock theta function $ω(q)$, http://arxiv.org/abs/1603.04352v1
 * Watson, G. N. [http://dx.doi.org/10.1112%2Fjlms%2Fs1-11.1.55 The Final Problem : An Account of the Mock Theta Functions](1936),  J. London Math. Soc. 11: 55–80

@@ 66번째 줄: / 66번째 줄: @@
 ==expositions==
 * [https://docs.google.com/file/d/0B8XXo8Tve1cxOFZTUldUc1l1a2s/edit?usp=drivesdk Rolen, Ramanujan's mock theta functions.pdf]
 * [http://www.newscientist.com/article/mg21628904.200-mathematical-proof-reveals-magic-of-ramanujans-genius.html Mathematical proof reveals magic of Ramanujan's genius] 2012-11-8
 * Andrews, George E. 2003. “Partitions: At the Interface of Q-Series and Modular Forms.” The Ramanujan Journal 7 (1-3) (March 1): 385–400. doi:10.1023/A:1026224002193.

← 이전 판		2013년 12월 20일 (금) 18:04 판
41번째 줄:		41번째 줄:


−
−	~~==expositions==~~
−	* [https://docs.google.com/file/d/0B8XXo8Tve1cxOFZTUldUc1l1a2s/edit?usp=drivesdk Rolen, Ramanujan's mock theta functions.pdf]
−	* [http://www.newscientist.com/article/mg21628904.200-mathematical-proof-reveals-magic-of-ramanujans-genius.html Mathematical proof reveals magic of Ramanujan's genius] 2012-11-8
−	* good introduction is given in Andrews article
−	** [http://link.springer.com/article/10.1023%2FA%3A1026224002193?LI=true Partitions : at the interface of q-series and modular forms]
−	** section 5
−
−
−
−	~~==articles==~~
−	* Watson, G. N. [http://dx.doi.org/10.1112%2Fjlms%2Fs1-11.1.55 The Final Problem : An Account of the Mock Theta Functions](1936), J. London Math. Soc. 11: 55–80
−	* Dragonette, Leila A. [http://dx.doi.org/10.2307%2F1990714 Some asymptotic formulae for the mock theta series of Ramanujan](1952), Transactions of the American Mathematical Society 72: 474–500
−	* Andrews, George E. [http://dx.doi.org/10.2307%2F2373202 On the theorems of Watson and Dragonette for Ramanujan's mock theta functions](1966) American Journal of Mathematics 88: 454–490


76번째 줄:		62번째 줄:
	* https://docs.google.com/file/d/0B8XXo8Tve1cxLWNCNklCRlVXd2c/edit		* https://docs.google.com/file/d/0B8XXo8Tve1cxLWNCNklCRlVXd2c/edit

		+
		+
		+	==expositions==
		+	* [https://docs.google.com/file/d/0B8XXo8Tve1cxOFZTUldUc1l1a2s/edit?usp=drivesdk Rolen, Ramanujan's mock theta functions.pdf]
		+	* [http://www.newscientist.com/article/mg21628904.200-mathematical-proof-reveals-magic-of-ramanujans-genius.html Mathematical proof reveals magic of Ramanujan's genius] 2012-11-8
		+	* Andrews, George E. 2003. “Partitions: At the Interface of Q-Series and Modular Forms.” The Ramanujan Journal 7 (1-3) (March 1): 385–400. doi:10.1023/A:1026224002193.
		+	** good introduction is given in section 5
		+
		+
		+
		+
		+	==articles==
		+	* Watson, G. N. [http://dx.doi.org/10.1112%2Fjlms%2Fs1-11.1.55 The Final Problem : An Account of the Mock Theta Functions](1936), J. London Math. Soc. 11: 55–80
		+	* Dragonette, Leila A. [http://dx.doi.org/10.2307%2F1990714 Some asymptotic formulae for the mock theta series of Ramanujan](1952), Transactions of the American Mathematical Society 72: 474–500
		+	* Andrews, George E. [http://dx.doi.org/10.2307%2F2373202 On the theorems of Watson and Dragonette for Ramanujan's mock theta functions](1966) American Journal of Mathematics 88: 454–490
		+

	[[분류:개인노트]]		[[분류:개인노트]]

← 이전 판		2013년 8월 8일 (목) 20:32 판
75번째 줄:		75번째 줄:
	* https://docs.google.com/file/d/0B8XXo8Tve1cxLWNCNklCRlVXd2c/edit		* https://docs.google.com/file/d/0B8XXo8Tve1cxLWNCNklCRlVXd2c/edit

−
−	~~==encyclopedia==~~
−
−	* http://ko.wikipedia.org/wiki/
−	* http://en.wikipedia.org/wiki/
−
−

	[[분류:개인노트]]		[[분류:개인노트]]