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

2020년 11월 16일 (월) 04:09 기준 최신판

개요

로저스 다이로그 함수 (Rogers' 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,

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

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

http://perso.univ-rennes1.fr/luc.pirio/SELECTApirio.pdf

역사

매스매티카 파일 및 계산 리소스

https://docs.google.com/leaf?id=0B8XXo8Tve1cxMzA0M2NkMzMtYTFiNy00N2YwLTlmYzktYWI2YTYwMDMyOTQz&sort=name&layout=list&num=50

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

@@ 1번째 줄: / 1번째 줄: @@
-<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>
+==개요==
-* [[함수 다이로그 항등식(functional dilogarithm identity)]]
+* 로저스 다이로그 함수 (Rogers' dilogarithm)가 만족시키는 두 함수 항등식의 일반화
+**  2항 관계식, 반사공식(오일러) <math>0\leq x \leq 1</math> 일 때, :<math>L(x)+L(1-x)=L(1)</math>
+** [[5항 관계식 (5-term relation) |5항 관계식 (5-term relation)]] <math>0\leq x,y\leq 1</math> 일 때,
+:<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>
-<h5>개요</h5>
-* [[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>
-**  2항 관계식, 반사공식(오일러)<br><math>0\leq x \leq 1</math> 일 때, <math>L(x)+L(1-x)=L(1)</math><br>
-** [[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
-* n 변수로 구성된 <math>(n^2+3n)/2</math> 항 관계식을 찾을 수 있음
+* 가령 <math>A_n</math> 딘킨 다이어그램으로부터, n 변수로 구성된 <math>(n^2+3n)/2</math> 항 관계식을 찾을 수 있음
+* <math>2, 5, 9, 14, 20, 27, 35, 44, 54, 65,\cdots</math>
-<h5>2항 관계식</h5>
-<math>S=\left\{x,\frac{1}{x}\right\}</math>
-<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>
-<h5>5항 관계식</h5>
-<math>S=\left\{x,y,\frac{x+1}{y},\frac{y+1}{x},\frac{x+y+1}{x y}\right\}</math>
-<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>
-<h5>9항 관계식</h5>
+==2항 관계식==
+* <math>S=\left\{x,\frac{1}{x}\right\}</math>라 두면,
+:<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>
-<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>
-<math>\sum_{a\in S}L(\frac{1}{1+a})=3L(1)</math>
+==5항 관계식==
+* <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>
-<h5>14항 관계식</h5>
+==9항 관계식==
+* <math>S</math>를 다음과 같이 두자
+:<math>
+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\}</math>
+* 다이로그 함수에 대하여 다음이 성립한다
+:<math>\sum_{a\in S}L(\frac{1}{1+a})=3L(1)</math>
-<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>
-<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>
+==14항 관계식==
+:<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}, \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>
+:<math>\sum_{a\in S}L(\frac{1}{1+a})=4L(1)</math>
-<math>\sum_{a\in S}L(\frac{1}{1+a})=4L(1)</math>
+==메모==
+* http://perso.univ-rennes1.fr/luc.pirio/SELECTApirio.pdf
+==역사==
-<h5>역사</h5>
 * http://www.google.com/search?hl=en&tbs=tl:1&q=
-* [[수학사연표 (역사)|수학사연표]]
+* [[수학사 연표]]
-<h5>메모</h5>
-<h5>관련된 항목들</h5>
-<h5>매스매티카 파일 및 계산 리소스</h5>
-* [[8056064/attachments/5229616|다이로그_함수]][[8056064/attachments/5229616|_항등식(dilogarithm_functional_identity).nb]]
-* http://www.wolframalpha.com/input/?i=
-* http://functions.wolfram.com/
-* [http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]
-* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]
-* [http://numbers.computation.free.fr/Constants/constants.html Numbers, constants and computation]
-* [[매스매티카 파일 목록]]
-<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>
-*  단어사전<br>
-** http://www.google.com/dictionary?langpair=en|ko&q=
-** http://ko.wiktionary.org/wiki/
-* 발음사전 http://www.forvo.com/search/
-* [http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=&fstr= 대한수학회 수학 학술 용어집]<br>
-** http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=
-* [http://www.nktech.net/science/term/term_l.jsp?l_mode=cate&s_code_cd=MA 남·북한수학용어비교]
-* [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]
-<h5>사전 형태의 자료</h5>
-* http://ko.wikipedia.org/wiki/
-* http://en.wikipedia.org/wiki/
-* [http://eom.springer.de/default.htm The Online Encyclopaedia of Mathematics]
-* [http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]
-* [http://eqworld.ipmnet.ru/ The World of Mathematical Equations]
-<h5>리뷰논문, 에세이, 강의노트</h5>
+==매스매티카 파일 및 계산 리소스==
+* https://docs.google.com/leaf?id=0B8XXo8Tve1cxMzA0M2NkMzMtYTFiNy00N2YwLTlmYzktYWI2YTYwMDMyOTQz&sort=name&layout=list&num=50
-<h5>관련논문</h5>
-*  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.<br>
+==관련논문==
-* [http://dx.doi.org/10.1023/A:1009709927327 Algebraic Dilogarithm Identities] ,Basil Gordon  and Richard J. Mcintosh, 1997
+* 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
 * 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]
-* http://www.jstor.org/action/doBasicSearch?Query=
-* http://www.ams.org/mathscinet
-* http://dx.doi.org/10.1112/plms/s2-4.1.169
-<h5>관련도서</h5>
-*  도서내검색<br>
-** http://books.google.com/books?q=
-** http://book.daum.net/search/contentSearch.do?query=
-<h5>링크</h5>
-* [http://www.ams.org/news/math-in-the-media/mathdigest-index Summaries of Media Coverage of Math]
+[[분류:다이로그]]
-*  구글 블로그 검색<br>
-** http://blogsearch.google.com/blogsearch?q=