"3rd order mock theta functions"의 두 판 사이의 차이
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imported>Pythagoras0 |
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==introduction== | ==introduction== | ||
− | * Ramanujan's 3rd order mock theta function is defined by :<math>f(q) = \sum_{n\ge 0} {q^{n^2}\over (-q;q)_n^2} =1+\sum_{n\ge 1} \frac{q^{n^2}}{(1+q)^2(1+q^2)^2\cdots{(1+q^{n})^2}}={2\over \prod_{n>0}(1-q^n)}\sum_{n\in Z}{(-1)^nq^{3n^2/2+n/2}\over 1+q^n}</math> | + | * Ramanujan's 3rd order mock theta function is defined by |
− | ** | + | :<math>f(q) = \sum_{n\ge 0} {q^{n^2}\over (-q;q)_n^2} =1+\sum_{n\ge 1} \frac{q^{n^2}}{(1+q)^2(1+q^2)^2\cdots{(1+q^{n})^2}}={2\over \prod_{n>0}(1-q^n)}\sum_{n\in Z}{(-1)^nq^{3n^2/2+n/2}\over 1+q^n}</math> |
− | ** | + | ** http://www.research.att.com/~njas/sequences/A000025 |
+ | ** http://www.research.att.com/~njas/sequences/b000025 | ||
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** see [[Rank of partition and mock theta conjecture]] | ** see [[Rank of partition and mock theta conjecture]] | ||
* In particular Mock theta functions have asymptotic expansions at cusps of the modular group, acting on the upper half-plane, that resemble those of modular forms of weight 1/2 with poles at the cusps. | * In particular Mock theta functions have asymptotic expansions at cusps of the modular group, acting on the upper half-plane, that resemble those of modular forms of weight 1/2 with poles at the cusps. | ||
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==asymptotics at 1== | ==asymptotics at 1== | ||
* If <math>q=e^{-t}</math>, around <math>t\sim 0</math>, the asymptotic behavior is given by :<math>f(q) = 1+\sum_{n\ge 1} \frac{q^{n^2}}{(1+q)^2(1+q^2)^2\cdots{(1+q^{n})^2}}\sim 4/3</math> | * If <math>q=e^{-t}</math>, around <math>t\sim 0</math>, the asymptotic behavior is given by :<math>f(q) = 1+\sum_{n\ge 1} \frac{q^{n^2}}{(1+q)^2(1+q^2)^2\cdots{(1+q^{n})^2}}\sim 4/3</math> | ||
* see also [[Asymptotic analysis of basic hypergeometric series]] | * see also [[Asymptotic analysis of basic hypergeometric series]] | ||
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==asymptotic behavior at roots of unity== | ==asymptotic behavior at roots of unity== | ||
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==shadow== | ==shadow== | ||
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* <math>\Theta(24z)=q-5q^{25}+7q^{49}-11q^{121}+13q^{169}-\cdots</math> | * <math>\Theta(24z)=q-5q^{25}+7q^{49}-11q^{121}+13q^{169}-\cdots</math> | ||
* <math>M_f(z)=q^{-1}f(q^{24})+\frac{i}{\sqrt{3}}\int_{}^{}\frac{\Theta(24z)}{}dz</math> | * <math>M_f(z)=q^{-1}f(q^{24})+\frac{i}{\sqrt{3}}\int_{}^{}\frac{\Theta(24z)}{}dz</math> | ||
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==expositions== | ==expositions== | ||
* [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 | * [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== | ==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 | |
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− | * [http://dx.doi.org/10.2307%2F1990714 Some asymptotic formulae for the mock theta series of Ramanujan] | ||
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− | * [http://dx.doi.org/10.2307%2F2373202 On the theorems of Watson and Dragonette for Ramanujan's mock theta functions] | ||
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− | + | ==computational resources== | |
+ | * https://docs.google.com/file/d/0B8XXo8Tve1cxLWNCNklCRlVXd2c/edit | ||
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[[분류:개인노트]] | [[분류:개인노트]] | ||
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[[분류:math and physics]] | [[분류:math and physics]] | ||
[[분류:mock modular forms]] | [[분류:mock modular forms]] | ||
[[분류:math]] | [[분류:math]] |
2013년 3월 17일 (일) 15:35 판
introduction
- Ramanujan's 3rd order mock theta function is defined by
\[f(q) = \sum_{n\ge 0} {q^{n^2}\over (-q;q)_n^2} =1+\sum_{n\ge 1} \frac{q^{n^2}}{(1+q)^2(1+q^2)^2\cdots{(1+q^{n})^2}}={2\over \prod_{n>0}(1-q^n)}\sum_{n\in Z}{(-1)^nq^{3n^2/2+n/2}\over 1+q^n}\]
- 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 (Bringmann & Ono 2006).
- In particular Mock theta functions have asymptotic expansions at cusps of the modular group, acting on the upper half-plane, that resemble those of modular forms of weight 1/2 with poles at the cusps.
asymptotics at 1
- If \(q=e^{-t}\), around \(t\sim 0\), the asymptotic behavior is given by \[f(q) = 1+\sum_{n\ge 1} \frac{q^{n^2}}{(1+q)^2(1+q^2)^2\cdots{(1+q^{n})^2}}\sim 4/3\]
- see also Asymptotic analysis of basic hypergeometric series
asymptotic behavior at roots of unity
- the series converges for $|q|<1$ and the roots of unity $q$ at odd order
- For even order roots of unity, $f(q)$ 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 $$b(q)=q^{1/24}\frac{\eta(\tau)}{\eta(2\tau)}\theta(-q)$$
harmonic weak Maass form
- We have a weight k=1/2, harmonic weak Maass form $h_3$ under \(\Gamma(2)\) defined by \[h_3(\tau)=q^{-1/24}f(q)+R_3(q)\] where
\(R_3(\tau)=\sum_{n\equiv 1\pmod 6}\operatorname{sgn}(n)\beta_{1/2}(n^2y/6)q^{-n^2/24}\) where \(\displaystyle \beta(t) = \int_t^\infty u^{-1/2} e^{-\pi u} \,du=2\int_{\sqrt{x}}^{\infty} e^{-\pi t^2}\,dt\)
- Note that this can be rewritten as \[R_3(\tau)=(i/2)^{k-1} \int_{-\overline\tau}^{i\infty} (z+\tau)^{-k}\overline{g(-\overline z)}\,dz\]
- shadow = weight 3/2 theta function \[g_3(z)=\sum_{n\equiv 1\pmod 6}nq^{n^2/24}\]
shadow
- \(\Theta(24z)=q-5q^{25}+7q^{49}-11q^{121}+13q^{169}-\cdots\)
- \(M_f(z)=q^{-1}f(q^{24})+\frac{i}{\sqrt{3}}\int_{}^{}\frac{\Theta(24z)}{}dz\)
expositions
- Mathematical proof reveals magic of Ramanujan's genius 2012-11-8
- good introduction is given in Andrews article
articles
- Watson, G. N. The Final Problem : An Account of the Mock Theta Functions(1936), J. London Math. Soc. 11: 55–80
- Dragonette, Leila A. Some asymptotic formulae for the mock theta series of Ramanujan(1952), Transactions of the American Mathematical Society 72: 474–500
- Andrews, George E. On the theorems of Watson and Dragonette for Ramanujan's mock theta functions(1966) American Journal of Mathematics 88: 454–490
history
computational resources