5항 관계식 (5-term relation)

이 항목의 스프링노트 원문주소

• 5항 관계식 (5-term relation)

 

 

개요

• 다이로그함수(dilogarithm) 가 만족시키는 중요한 항등식
• 함수 다이로그 항등식(functional dilogarithm identity) 으로 일반화됨

 

 

 로저스 다이로그 함수

• 로저스 다이로그 함수 (Roger's dilogarithm) 의 정의
  \(x\in (0,1)\)에서 로저스 dilogarithm을 다음과 같이 정의
  \(L(x)=\operatorname{Li}_2(x)+\frac{1}{2}\log x\log (1-x)=-\frac{1}{2}\int_{0}^{x}\frac{\log(1-y)}{y}+\frac{\log(1-y)}{1-y}dy\)
  

 

 

 5항 관계식

• 로저스 다이로그 함수 \(L(x)\)에 대하여 다음이 성립한다
  \(0\leq x,y\leq 1\) 일 때,
  \(L(x)+L(1-xy)+L(y)+L(\frac{1-y}{1-xy})+L\Left( \frac{1-x}{1-xy} )\right)=\frac{\pi^2}{2}\)
  
• \(1-x_{i}=x_{i-1}x_{i+1}\), \(x_0=x\), \(x_2=y\)로 정의되는 점화식은 주기가 5인 수열이 된다
  \(x_0=x, x_1=1-xy, x_2=y, x_3=\frac{1-y}{1-xy}, x_4=\frac{1-x}{1-xy}\)
  
• 집합 \(\{0,1,\infty,y,xy\}\) 에서 4개의 원소를 뽑아 얻어지는 교차비(cross ratio)

 

 

 q-이항정리를 통한 증명

• [GM1997]참고
• q-이항정리
  \(\sum_{n=0}^{\infty} \frac{(a;q)_n}{(q;q)_n}b^n=\frac{(ab;q)_{\infty}}{(b;q)_{\infty}}\)
  
• z를 \((1-az)b=1-z\) 의 해로 정의, 즉
  \(z=\frac{1-b}{1-ab}\)
  
• \(q=e^{-t}\)이고 t가 0으로 갈 때, 양변의 근사식은 다음과 같다
  좌변  \(\frac{\operatorname{Li}_2(az)-\operatorname{Li}_2(a)-\operatorname{Li}_2(z)+\operatorname{Li}_2(1)-\log z\log b}{t}\)
  우변 \(\frac{\operatorname{Li}_2(b)-\operatorname{Li}_2(ab)}{t}\)
  
• 양변의 근사식을 비교하여 5항 관계식을 얻는다
  \(\operatorname{Li}_2(az)-\operatorname{Li}_2(a)-\operatorname{Li}_2(z)+\operatorname{Li}_2(1)-\log z\log b}=\operatorname{Li}_2(b)-\operatorname{Li}_2(ab)}\)
  

 

 

재미있는 사실

" I like explicit, hands-on formulas. To me they have a beauty of their own. They can be deep or not. As an example, imagine you have a series of numbers such that if you add 1 to any number you will get the product of its left and right neighbors. Then this series will repeat itself at every fifth step! For instance, if you start with 3, 4 then the sequence continues: 3, 4, 5/3, 2/3, 1, 3, 4, 5/3, etc. The difference between a mathematician and a nonmathematician is not just being able to discover something like this, but to care about it and to be curious about why it's true, what it means, and what other things in mathematics it might be connected with. In this particular case, the statement itself turns out to be connected with a myriad of deep topics in advanced mathematics: hyperbolic geometry, algebraic K-theory, the Schrodinger equation of quantum mechanics, and certain models of quantum field theory. I find this kind of connection between very elementary and very deep mathematics overwhelmingly beautiful." Don Zagier (Mathematicians: An Outer View of the Inner World):

http://math.stackexchange.com/questions/11650/what-is-the-connection-of-the-sequence-3-4-5-3-2-3-1-with-deep-topics

• Math Overflow http://mathoverflow.net/search?q=
• 네이버 지식인 http://kin.search.naver.com/search.naver?where=kin_qna&query=

 

 

역사

 

• http://www.google.com/search?hl=en&tbs=tl:1&q=
• 수학사연표
•  

 

 

메모

• http://golem.ph.utexas.edu/wiki/gst/show/The+Dilogarithm+Function

 

 

 

관련된 항목들

• q-이항정리
• 함수 다이로그 항등식(functional dilogarithm identity)

 

 

수학용어번역

• 단어사전 http://www.google.com/dictionary?langpair=en%7Cko&q=
• 발음사전 http://www.forvo.com/search/
• 대한수학회 수학 학술 용어집
  
  • http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=
• 남·북한수학용어비교
• 대한수학회 수학용어한글화 게시판

 

 

사전 형태의 자료

• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/
• http://www.wolframalpha.com/input/?i=
• NIST Digital Library of Mathematical Functions
• The On-Line Encyclopedia of Integer Sequences
  
  • http://www.research.att.com/~njas/sequences/?q=

 

 

관련논문

[1]

• Kashaev, R. M. 2004. “The q-binomial formula and the Rogers dilogarithm identity”. math/0407078 (7월 6). http://arxiv.org/abs/math/0407078
  
• [GM1997]Algebraic Dilogarithm Identities ,Basil Gordon  and Richard J. Mcintosh, 1997
• Zdzisław Wojtkowiak, Functional equations of iterated integrals with regular singularities, Nagoya Math. J. Volume 142 (1996), 145-159. http://projecteuclid.org/euclid.nmj/1118772047
• Moak, Daniel S. 1984. “The $q$-analogue of Stirling’s formula”. The Rocky Mountain Journal of Mathematics 14 (2): 403–413. doi:10.1216/RMJ-1984-14-2-403.

• http://www.jstor.org/action/doBasicSearch?Query=
• http://www.ams.org/mathscinet
• http://dx.doi.org/