함수 다이로그 항등식(functional dilogarithm identity)

개요

• 로저 다이로그 함수 (Roger's dilogarithm) 가 만족시키는 두 함수 항등식의 일반화
  
  • 2항 관계식, 반사공식(오일러)
    \(0\leq x \leq 1\) 일 때, \(L(x)+L(1-x)=L(1)\)
    
  • 5항 관계식 (5-term relation)
    \(0\leq x,y\leq 1\) 일 때, \(L(x)+L(1-xy)+L(y)+L(\frac{1-y}{1-xy})+L\left( \frac{1-x}{1-xy} )\right)=3L(1)\)
    
    
• 클러스터 대수(cluster algebra) 를 이용하여 일반화됨
• n 변수로 구성된 \((n^2+3n)/2\) 항 관계식을 찾을 수 있음

2항 관계식

\(S=\left\{x,\frac{1}{x}\right\}\)

\(\sum_{a\in S}L(\frac{1}{1+a})=L\left(\frac{1}{\frac{1}{x}+1}\right)+L\left(\frac{1}{x+1}\right)=L(1)\)

5항 관계식

\(S=\left\{x,y,\frac{x+1}{y},\frac{y+1}{x},\frac{x+y+1}{x y}\right\}\)

\(\sum_{a\in S}L(\frac{1}{1+a})=L\left(\frac{1}{\frac{x+1}{y}+1}\right)+L\left(\frac{1}{\frac{y+1}{x}+1}\right)+L\left(\frac{1}{\frac{x+y+1}{x y}+1}\right)+L\left(\frac{1}{x+1}\right)+L\left(\frac{1}{y+1}\right)=2L(1)\)

9항 관계식

\(\left\{x,y,z,\frac{x z+x+z+1}{y},\frac{x y+x z+x+y^2+y z+2 y+z+1}{x y z},\frac{x z+x+y+z+1}{x y},\frac{x z+x+y+z+1}{y z},\frac{y+1}{x},\frac{y+1}{z}\right\}\)

\(\sum_{a\in S}L(\frac{1}{1+a})=3L(1)\)

14항 관계식

\(\left\{x,z,\frac{(x+1) (z+1)}{y},\frac{z+1}{w},\frac{x z+x+y+z+1}{x y},\frac{(w+z+1) (x z+x+y+z+1)}{w y z},\frac{(y+z+1) (w (x+y+1)+x z+x+y+z+1)}{w x y z}, \frac{w (x+y+1)+x z+x+y+z+1}{y z},\frac{w y+w+y+z+1}{w z},\frac{(x+y+1) (w y+w+y+z+1)}{x y z},\frac{(w+1) (y+1)}{z},\frac{y+1}{x},w,y\right\}\)

\(\sum_{a\in S}L(\frac{1}{1+a})=4L(1)\)

역사

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

매스매티카 파일 및 계산 리소스

• https://docs.google.com/leaf?id=0B8XXo8Tve1cxMzA0M2NkMzMtYTFiNy00N2YwLTlmYzktYWI2YTYwMDMyOTQz&sort=name&layout=list&num=50
• http://www.wolframalpha.com/input/?i=
• http://functions.wolfram.com/
• NIST Digital Library of Mathematical Functions
• The On-Line Encyclopedia of Integer Sequences
• Numbers, constants and computation

• 매스매티카 파일 목록

수학용어번역

• 단어사전
  
  • http://www.google.com/dictionary?langpair=en%7Cko&q=
  • http://ko.wiktionary.org/wiki/
• 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/
• The Online Encyclopaedia of Mathematics
• NIST Digital Library of Mathematical Functions
• The World of Mathematical Equations

관련논문

• Nakanishi, Tomoki. 2011. “Dilogarithm Identities for Conformal Field Theories and Cluster Algebras: Simply Laced Case.” Nagoya Mathematical Journal 202 (June): 23–43. doi:10.1215/00277630-1260432.
• Chapoton, Frédéric. 2005. “Functional Identities for the Rogers Dilogarithm Associated to Cluster Y-Systems.” Bulletin of the London Mathematical Society 37 (5) (October 1): 755 -760. doi:10.1112/S0024609305004510.
  
• Algebraic Dilogarithm Identities ,Basil Gordon and Richard J. Mcintosh, 1997
• L.J. Rogers, On Function Sum Theorems Connected with the Series Formula Proc. London Math. Soc. (1907) s2-4(1): 169-189 doi:10.1112/plms/s2-4.1.169
• http://www.jstor.org/action/doBasicSearch?Query=
• http://www.ams.org/mathscinet
• http://dx.doi.org/10.1112/plms/s2-4.1.169

관련도서

• 도서내검색
  
  • http://books.google.com/books?q=
  • http://book.daum.net/search/contentSearch.do?query=

링크

• Summaries of Media Coverage of Math
• 구글 블로그 검색
  
  • http://blogsearch.google.com/blogsearch?q=