"함수 다이로그 항등식(functional dilogarithm identity)"의 두 판 사이의 차이

수학노트
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(사용자 2명의 중간 판 31개는 보이지 않습니다)
1번째 줄: 1번째 줄:
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<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">이 항목의 수학노트 원문주소</h5>
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==개요==
  
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* 로저스 다이로그 함수 (Rogers' dilogarithm)가 만족시키는 두 함수 항등식의 일반화
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**  2항 관계식, 반사공식(오일러) <math>0\leq x \leq 1</math> 일 때, :<math>L(x)+L(1-x)=L(1)</math>
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** [[5항 관계식 (5-term relation) |5항 관계식 (5-term relation)]] <math>0\leq x,y\leq 1</math> 일 때,  
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:<math>L(x)+L(1-xy)+L(y)+L\left(\frac{1-y}{1-xy}\right)+L\left(\frac{1-x}{1-xy} \right)=3L(1)</math>
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<h5>개요</h5>
 
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* [[search?q=%EB%A1%9C%EC%A0%80%20%EB%8B%A4%EC%9D%B4%EB%A1%9C%EA%B7%B8%20%ED%95%A8%EC%88%98%20%28Roger%27s%20dilogarithm%29&parent id=8056064|로저 다이로그 함수 (Roger's dilogarithm)]] 가 만족시키는 두 함수 항등식의 일반화<br>
 
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**  2항 관계식, 반사공식(오일러)<br><math>0\leq x \leq 1</math> 일 때, <math>L(x)+L(1-x)=L(1)</math><br>
 
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** [[5항 관계식 (5-term relation) |5항 관계식 (5-term relation)]]<br><math>0\leq x,y\leq 1</math> 일 때, <math>L(x)+L(1-xy)+L(y)+L(\frac{1-y}{1-xy})+L\Left( \frac{1-x}{1-xy} )\right)=3L(1)</math><br>  <br>
 
 
* 클러스터 대수(cluster algebra) 를 이용하여 일반화됨
 
* 클러스터 대수(cluster algebra) 를 이용하여 일반화됨
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* n 변수로 구성된 <math>(n^2+3n)/2</math> 항 관계식을 찾을 수 있음
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* 가령 <math>A_n</math> 딘킨 다이어그램으로부터, n 변수로 구성된 <math>(n^2+3n)/2</math> 항 관계식을 찾을 수 있음
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* <math>2, 5, 9, 14, 20, 27, 35, 44, 54, 65,\cdots</math>
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<h5>2항 관계식</h5>
 
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<math>S=\left\{x,\frac{1}{x}\right\}</math>
 
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<math>\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)</math>
 
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<h5>5항 관계식</h5>
 
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<math>S=\left\{x,y,\frac{x+1}{y},\frac{y+1}{x},\frac{x+y+1}{x y}\right\}</math>
 
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<math>\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)</math>
 
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<h5>9항 관계식</h5>
 
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<math>\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\}</math>
 
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<math>\sum_{a\in S}L(\frac{1}{1+a})=3L(1)</math>
 
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<h5>14항 관계식</h5>
 
  
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<math>\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},</math>
 
  
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<math>\left.\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\}</math>
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==2항 관계식==
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* <math>S=\left\{x,\frac{1}{x}\right\}</math>라 두면,
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:<math>\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)</math>
  
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<math>\sum_{a\in S}L(\frac{1}{1+a})=4L(1)</math>
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==5항 관계식==
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* <math>S=\left\{x,y,\frac{x+1}{y},\frac{y+1}{x},\frac{x+y+1}{x y}\right\}</math> 이면,
 +
:<math>\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)</math>
  
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<h5>재미있는 사실</h5>
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==9항 관계식==
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* <math>S</math>를 다음과 같이 두자
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:<math>
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S=\left\{x,y,z,\frac{(x+1) (z+1)}{y},\frac{(x+y+1) (y+z+1)}{x y z},\\
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\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\}</math>
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* 다이로그 함수에 대하여 다음이 성립한다
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:<math>\sum_{a\in S}L(\frac{1}{1+a})=3L(1)</math>
  
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* Math Overflow http://mathoverflow.net/search?q=
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==14항 관계식==
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* 네이버 지식인 http://kin.search.naver.com/search.naver?where=kin_qna&query=
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:<math>
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\left\{x,z,\frac{(x+1) (z+1)}{y},\frac{z+1}{w},\frac{x z+x+y+z+1}{x y},
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\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},\\
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\frac{w y+w+y+z+1}{w z},\frac{(x+y+1) (w y+w+y+z+1)}{x y z},
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\frac{(w+1) (y+1)}{z},\frac{y+1}{x},w,y\right\}
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</math>
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:<math>\sum_{a\in S}L(\frac{1}{1+a})=4L(1)</math>
  
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==메모==
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* http://perso.univ-rennes1.fr/luc.pirio/SELECTApirio.pdf
  
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<h5>역사</h5>
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==역사==
  
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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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* [[수학사연표 (역사)|수학사연표]]
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* [[수학사 연표]]
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<h5>메모</h5>
 
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<h5>관련된 항목들</h5>
 
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<h5>매스매티카 파일 및 계산 리소스</h5>
 
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* [[8056064/attachments/5229616|다이로그_함수]][[8056064/attachments/5229616|_항등식(dilogarithm_functional_identity).nb]]
 
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* http://www.wolframalpha.com/input/?i=
 
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* http://functions.wolfram.com/
 
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* [http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]
 
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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://numbers.computation.free.fr/Constants/constants.html Numbers, constants and computation]
 
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* [[매스매티카 파일 목록]]
 
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<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">수학용어번역</h5>
 
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*  단어사전<br>
 
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** http://www.google.com/dictionary?langpair=en|ko&q=
 
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** http://ko.wiktionary.org/wiki/
 
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* 발음사전 http://www.forvo.com/search/
 
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* [http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=&fstr= 대한수학회 수학 학술 용어집]<br>
 
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** http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=
 
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* [http://www.nktech.net/science/term/term_l.jsp?l_mode=cate&s_code_cd=MA 남·북한수학용어비교]
 
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* [http://kms.or.kr/home/kor/board/bulletin_list_subject.asp?bulletinid=%7BD6048897-56F9-43D7-8BB6-50B362D1243A%7D&boardname=%BC%F6%C7%D0%BF%EB%BE%EE%C5%E4%B7%D0%B9%E6&globalmenu=7&localmenu=4 대한수학회 수학용어한글화 게시판]
 
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<h5>사전 형태의 자료</h5>
 
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* http://ko.wikipedia.org/wiki/
 
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* http://en.wikipedia.org/wiki/
 
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* [http://eom.springer.de/default.htm The Online Encyclopaedia of Mathematics]
 
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* [http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]
 
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* [http://eqworld.ipmnet.ru/ The World of Mathematical Equations]
 
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<h5>리뷰논문, 에세이, 강의노트</h5>
 
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<h5>관련논문</h5>
 
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* [http://dx.doi.org/10.1023/A:1009709927327 Algebraic Dilogarithm Identities] ,Basil Gordon  and Richard J. Mcintosh, 1997
 
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* http://www.jstor.org/action/doBasicSearch?Query=
 
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* http://www.ams.org/mathscinet
 
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* http://dx.doi.org/
 
  
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<h5>관련도서</h5>
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==매스매티카 파일 및 계산 리소스==
  
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* 도서내검색<br>
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* https://docs.google.com/leaf?id=0B8XXo8Tve1cxMzA0M2NkMzMtYTFiNy00N2YwLTlmYzktYWI2YTYwMDMyOTQz&sort=name&layout=list&num=50
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** http://books.google.com/books?q=
 
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** http://book.daum.net/search/contentSearch.do?query=
 
  
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<h5>링크</h5>
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==관련논문==
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* Tomoki Nakanishi, Rogers dilogarithms of higher degree and generalized cluster algebras, arXiv:1605.04777 [math.QA], May 16 2016, http://arxiv.org/abs/1605.04777
 +
* Soudères, Ismaël. “Functional Equations for Rogers Dilogarithm.” arXiv:1509.02869 [math], September 9, 2015. http://arxiv.org/abs/1509.02869.
 +
* Kerr, Matt, James D. Lewis, and Patrick Lopatto. “Simplicial Abel-Jacobi Maps and Reciprocity Laws.” arXiv:1502.05459 [math], February 18, 2015. http://arxiv.org/abs/1502.05459.
 +
* Herbert Gangl, Functional equations and ladders for polylogarithms http://www.maths.dur.ac.uk/~dma0hg/ladders2_ams.pdf
 +
* 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.” <em>Bulletin of the London Mathematical Society</em> 37 (5) (October 1): 755 -760. doi:10.1112/S0024609305004510.
 +
* Herbert Gangl, Functional equations of polylogarithms http://www.mathematik.hu-berlin.de/~maphy/kyoto.pdf
 +
* Zagier, D. "Special Values and Functional Equations of Polylogarithms." Appendix A in Structural Properties of Polylogarithms (Ed. L. Lewin) http://people.mpim-bonn.mpg.de/zagier/files/tex/LewinPolylogarithms/fulltext.pdf
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* L.J. Rogers, On Function Sum Theorems Connected with the Series Formula Proc. London Math. Soc. (1907) s2-4(1): 169-189 doi:[http://dx.doi.org/10.1112/plms/s2-4.1.169%20 10.1112/plms/s2-4.1.169]
  
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* [http://www.ams.org/news/math-in-the-media/mathdigest-index Summaries of Media Coverage of Math]
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[[분류:다이로그]]
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*  구글 블로그 검색<br>
 
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** http://blogsearch.google.com/blogsearch?q=
 

2020년 11월 16일 (월) 04:09 기준 최신판

개요

  • 로저스 다이로그 함수 (Rogers' dilogarithm)가 만족시키는 두 함수 항등식의 일반화

\[L(x)+L(1-xy)+L(y)+L\left(\frac{1-y}{1-xy}\right)+L\left(\frac{1-x}{1-xy} \right)=3L(1)\]

  • 클러스터 대수(cluster algebra) 를 이용하여 일반화됨
  • 가령 \(A_n\) 딘킨 다이어그램으로부터, n 변수로 구성된 \((n^2+3n)/2\) 항 관계식을 찾을 수 있음
  • \(2, 5, 9, 14, 20, 27, 35, 44, 54, 65,\cdots\)


2항 관계식

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

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



5항 관계식

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

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


9항 관계식

  • \(S\)를 다음과 같이 두자

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

  • 다이로그 함수에 대하여 다음이 성립한다

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


14항 관계식

\[ \left\{x,z,\frac{(x+1) (z+1)}{y},\frac{z+1}{w},\frac{x z+x+y+z+1}{x y}, \frac{(w+z+1) (x z+x+y+z+1)}{w y z}\\,\frac{(y+z+1) (w (x+y+1)+x z+x+y+z+1)}{w x y z}, \frac{w (x+y+1)+x z+x+y+z+1}{y z},\\ \frac{w y+w+y+z+1}{w z},\frac{(x+y+1) (w y+w+y+z+1)}{x y z}, \frac{(w+1) (y+1)}{z},\frac{y+1}{x},w,y\right\} \] \[\sum_{a\in S}L(\frac{1}{1+a})=4L(1)\]


메모

역사



매스매티카 파일 및 계산 리소스



관련논문

  • Tomoki Nakanishi, Rogers dilogarithms of higher degree and generalized cluster algebras, arXiv:1605.04777 [math.QA], May 16 2016, http://arxiv.org/abs/1605.04777
  • Soudères, Ismaël. “Functional Equations for Rogers Dilogarithm.” arXiv:1509.02869 [math], September 9, 2015. http://arxiv.org/abs/1509.02869.
  • Kerr, Matt, James D. Lewis, and Patrick Lopatto. “Simplicial Abel-Jacobi Maps and Reciprocity Laws.” arXiv:1502.05459 [math], February 18, 2015. http://arxiv.org/abs/1502.05459.
  • Herbert Gangl, Functional equations and ladders for polylogarithms http://www.maths.dur.ac.uk/~dma0hg/ladders2_ams.pdf
  • 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.
  • Herbert Gangl, Functional equations of polylogarithms http://www.mathematik.hu-berlin.de/~maphy/kyoto.pdf
  • Zagier, D. "Special Values and Functional Equations of Polylogarithms." Appendix A in Structural Properties of Polylogarithms (Ed. L. Lewin) http://people.mpim-bonn.mpg.de/zagier/files/tex/LewinPolylogarithms/fulltext.pdf
  • 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
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