"Quantum dilogarithm"의 두 판 사이의 차이

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*  also called the Weyl algebra<br>
 
*  also called the Weyl algebra<br>
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* <math>0<q<1</math>에 대하여 다음과 같이 정의<br><math>\int_0^a f(x) d_q x = a(1-q)\sum_{k=0}^{\infty}q^k f(aq^k )</math><br><math>\int_0^{\infty} f(x) d_q x =(1-q)\sum_{k=-\infty}^{\infty}q^k f(aq^k )</math><br>
 
* <math>0<q<1</math>에 대하여 다음과 같이 정의<br><math>\int_0^a f(x) d_q x = a(1-q)\sum_{k=0}^{\infty}q^k f(aq^k )</math><br><math>\int_0^{\infty} f(x) d_q x =(1-q)\sum_{k=-\infty}^{\infty}q^k f(aq^k )</math><br>
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<h5 style="margin: 0px; line-height: 2em;">quantum dilogarithm</h5>
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<h5 style="MARGIN: 0px; LINE-HEIGHT: 2em;">quantum dilogarithm</h5>
  
 
<math>\Psi(z)=\prod_{n=0}^{\infty}(1-zq^n)=\sum_{n\geq 0}\frac{(-1)^nq^{n(n-1)/2}}{(1-q)(1-q^2)\cdots(1-q^n)} z^n</math>
 
<math>\Psi(z)=\prod_{n=0}^{\infty}(1-zq^n)=\sum_{n\geq 0}\frac{(-1)^nq^{n(n-1)/2}}{(1-q)(1-q^2)\cdots(1-q^n)} z^n</math>
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<h5 style="MARGIN: 0px; LINE-HEIGHT: 2em;">asymptotics </h5>
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* <math>q=e^{-t}</math> and as the t goes 0 (i.e. as q goes to 1)<br>
 
* <math>q=e^{-t}</math> and as the t goes 0 (i.e. as q goes to 1)<br>
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*  In Weyl algebra, the following identity holds<br><math>(v)_{\infty}(u)_{\infty}=(u)_{\infty}(-vu)_{\infty}(v)_{\infty}</math><br>
 
*  In Weyl algebra, the following identity holds<br><math>(v)_{\infty}(u)_{\infty}=(u)_{\infty}(-vu)_{\infty}(v)_{\infty}</math><br>
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* http://www.google.com/search?hl=en&tbs=tl:1&q=
 
* http://www.google.com/search?hl=en&tbs=tl:1&q=
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* [[1 Fermion summation formula - quasi-particle interpretation|Boson and Fermion summation form]]<br>
 
* [[1 Fermion summation formula - quasi-particle interpretation|Boson and Fermion summation form]]<br>
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* [http://pythagoras0.springnote.com/pages/5012319 q-적분]
 
* [http://pythagoras0.springnote.com/pages/5012319 q-적분]
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*  Quantum dilogarithm.<br>
 
*  Quantum dilogarithm.<br>
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** V V Bazhanov and N Yu Reshetikhin, 1995 J. Phys. A: Math. Gen. 28 2217
 
** V V Bazhanov and N Yu Reshetikhin, 1995 J. Phys. A: Math. Gen. 28 2217
 
* [http://dx.doi.org/10.1142/S0217732394000447 Quantum Dilogarithm]<br>
 
* [http://dx.doi.org/10.1142/S0217732394000447 Quantum Dilogarithm]<br>
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** L.D.<em style="line-height: 2em;">Fadeev</em> and R.M.<em style="line-height: 2em;">Kashaev</em>, Mod. Phys. Lett. A. 9 (1994) p.427–434
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** L.D.<em style="LINE-HEIGHT: 2em;">Fadeev</em> and R.M.<em style="LINE-HEIGHT: 2em;">Kashaev</em>, Mod. Phys. Lett. A. 9 (1994) p.427–434
 
* http://ncatlab.org/nlab/show/quantum+dilogarithm
 
* http://ncatlab.org/nlab/show/quantum+dilogarithm
  
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* http://www.zentralblatt-math.org/zmath/en/
 
* http://www.zentralblatt-math.org/zmath/en/
 
* http://pythagoras0.springnote.com/
 
* http://pythagoras0.springnote.com/
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* [http://math.berkeley.edu/%7Ereb/papers/index.html http://math.berkeley.edu/~reb/papers/index.html][http://www.ams.org/mathscinet ]
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* http://math.berkeley.edu/~reb/papers/index.html[http://www.ams.org/mathscinet ]
 
* http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
 
* http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
 
* 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://mathoverflow.net/search?q=
 
* http://mathoverflow.net/search?q=
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*  구글 블로그 검색<br>
 
*  구글 블로그 검색<br>
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* http://arxiv.org/
 
* http://arxiv.org/
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* [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier]
 
* [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier]
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* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]
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* [http://www.research.att.com/~njas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]

2010년 6월 24일 (목) 14:36 판

introduction

 

 

quantum plane
  • also called the Weyl algebra
  • noncommutative geometry
  • \(uv=qvu\)

 

 

q-integral (Jackson integral)
  • \(0<q<1\)에 대하여 다음과 같이 정의
    \(\int_0^a f(x) d_q x = a(1-q)\sum_{k=0}^{\infty}q^k f(aq^k )\)
    \(\int_0^{\infty} f(x) d_q x =(1-q)\sum_{k=-\infty}^{\infty}q^k f(aq^k )\)
  • \(q\to 1\) 이면, \(\int_0^a f(x) d_q x \to \int_0^a f(x) dx \)

 

 

 

quantum dilogarithm

\(\Psi(z)=\prod_{n=0}^{\infty}(1-zq^n)=\sum_{n\geq 0}\frac{(-1)^nq^{n(n-1)/2}}{(1-q)(1-q^2)\cdots(1-q^n)} z^n\)

\(\Psi(z)=\exp(\frac{\operatorname{Li}_{2,q}(z)}{q-1})\)

 

\(\operatorname{Li}_{2,q}(z) = -\int_0^z{{\ln (1-t)}\over t} d_{q}t \)

\(\operatorname{Li}_2(z) = -\int_0^z{{\ln (1-t)}\over t} dt \)

 

 

asymptotics 
  • \(q=e^{-t}\) and as the t goes 0 (i.e. as q goes to 1)

\(\sum_{n=0}^{\infty}\frac{q^{\frac{A}{2}n^2+cn}}{(q)_n}\sim\exp(\frac{C}{t})\)

where C= sum of Rogers dilogarithms

 

 

quantum 5-term relation

 

 

history

 

 

related items

 

 

encyclopedia

 

 

books

 

[[4909919|]]

 

 

articles

 

 

question and answers(Math Overflow)

 

 

blogs

 

 

experts on the field

 

 

links
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