"Appell-Lerch sums"의 두 판 사이의 차이

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(피타고라스님이 이 페이지의 이름을 Appell-Lerch sums로 바꾸었습니다.)
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<h5>introduction</h5>
  
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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
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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>
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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>
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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>
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The modified series
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: <math>\hat\mu(u,v;\tau) = \mu(u,v;\tau)-R(u-v;\tau)/2</math>
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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>
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and <em style="">y</em> = Im(τ) and
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: <math>E(z) = 2\int_0^ze^{-\pi u^2}\,du</math>
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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>
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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>
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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.
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<h5>history</h5>
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* http://www.google.com/search?hl=en&tbs=tl:1&q=
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<h5>related items</h5>
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*  
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<h5>books</h5>
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* [[4909919|찾아볼 수학책]]<br>
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* http://gigapedia.info/1/
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* http://gigapedia.info/1/
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* http://gigapedia.info/1/
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* http://gigapedia.info/1/
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* http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
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<h5>encyclopedia</h5>
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* http://ko.wikipedia.org/wiki/
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* http://en.wikipedia.org/wiki/
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* http://en.wikipedia.org/wiki/
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* http://en.wikipedia.org/wiki/
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* 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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* http://mathoverflow.net/search?q=
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* http://mathoverflow.net/search?q=
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* http://mathoverflow.net/search?q=
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<h5>blogs</h5>
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*  구글 블로그 검색<br>
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** http://blogsearch.google.com/blogsearch?q=
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** http://blogsearch.google.com/blogsearch?q=
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** http://blogsearch.google.com/blogsearch?q=
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<h5>articles</h5>
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* [[2010년 books and articles|논문정리]]
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* http://www.ams.org/mathscinet
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* http://www.zentralblatt-math.org/zmath/en/
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* http://pythagoras0.springnote.com/
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* [http://math.berkeley.edu/%7Ereb/papers/index.html http://math.berkeley.edu/~reb/papers/index.html]
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* http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
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* http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&s7=
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* http://dx.doi.org/
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<h5>experts on the field</h5>
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* http://arxiv.org/
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<h5>TeX </h5>
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* [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier]

2010년 3월 7일 (일) 15:39 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.

 

 

history

 

 

related items
  •  

 

books

 

 

 

encyclopedia

 

 

question and answers(Math Overflow)

 

 

blogs

 

 

articles

 

 

 

experts on the field

 

 

TeX 
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