"함수 다이로그 항등식(functional dilogarithm identity)"의 두 판 사이의 차이
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Pythagoras0 (토론 | 기여) |
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| − | + | ==개요== | |
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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> | * [[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> | ** 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> | + | ** [[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 | ||
* n 변수로 구성된 <math>(n^2+3n)/2</math> 항 관계식을 찾을 수 있음 | * n 변수로 구성된 <math>(n^2+3n)/2</math> 항 관계식을 찾을 수 있음 | ||
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| − | + | ==2항 관계식== | |
<math>S=\left\{x,\frac{1}{x}\right\}</math> | <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> | <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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| − | + | ==5항 관계식== | |
<math>S=\left\{x,y,\frac{x+1}{y},\frac{y+1}{x},\frac{x+y+1}{x y}\right\}</math> | <math>S=\left\{x,y,\frac{x+1}{y},\frac{y+1}{x},\frac{x+y+1}{x y}\right\}</math> | ||
| 35번째 줄: | 27번째 줄: | ||
<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> | <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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| − | + | ==9항 관계식== | |
<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>\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> | ||
| 45번째 줄: | 37번째 줄: | ||
<math>\sum_{a\in S}L(\frac{1}{1+a})=3L(1)</math> | <math>\sum_{a\in S}L(\frac{1}{1+a})=3L(1)</math> | ||
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| − | + | ==14항 관계식== | |
| − | <math>\left | + | <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> | ||
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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= | ||
* [[수학사연표 (역사)|수학사연표]] | * [[수학사연표 (역사)|수학사연표]] | ||
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| − | + | ==메모== | |
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| − | + | ==관련된 항목들== | |
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| − | + | ==매스매티카 파일 및 계산 리소스== | |
* https://docs.google.com/leaf?id=0B8XXo8Tve1cxMzA0M2NkMzMtYTFiNy00N2YwLTlmYzktYWI2YTYwMDMyOTQz&sort=name&layout=list&num=50 | * https://docs.google.com/leaf?id=0B8XXo8Tve1cxMzA0M2NkMzMtYTFiNy00N2YwLTlmYzktYWI2YTYwMDMyOTQz&sort=name&layout=list&num=50 | ||
| 97번째 줄: | 87번째 줄: | ||
* [[매스매티카 파일 목록]] | * [[매스매티카 파일 목록]] | ||
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| − | + | ==수학용어번역== | |
* 단어사전<br> | * 단어사전<br> | ||
** http://www.google.com/dictionary?langpair=en|ko&q= | ** http://www.google.com/dictionary?langpair=en|ko&q= | ||
** http://ko.wiktionary.org/wiki/ | ** 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=&fstr= 대한수학회 수학 학술 용어집]<br> | ||
** http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr= | ** 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://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 | + | * [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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| − | + | ==사전 형태의 자료== | |
* http://ko.wikipedia.org/wiki/ | * http://ko.wikipedia.org/wiki/ | ||
| 126번째 줄: | 116번째 줄: | ||
* [http://eqworld.ipmnet.ru/ The World of Mathematical Equations] | * [http://eqworld.ipmnet.ru/ The World of Mathematical Equations] | ||
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| − | + | ==리뷰논문, 에세이, 강의노트== | |
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| − | + | ==관련논문== | |
* 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> | * 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> | ||
| − | * [http://dx.doi.org/10.1023/A:1009709927327 Algebraic Dilogarithm Identities] ,Basil Gordon | + | * [http://dx.doi.org/10.1023/A:1009709927327 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:[http://dx.doi.org/10.1112/plms/s2-4.1.169%20 10.1112/plms/s2-4.1.169] | * 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.jstor.org/action/doBasicSearch?Query= | ||
| 145번째 줄: | 135번째 줄: | ||
* http://dx.doi.org/10.1112/plms/s2-4.1.169 | * http://dx.doi.org/10.1112/plms/s2-4.1.169 | ||
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| − | + | ==관련도서== | |
* 도서내검색<br> | * 도서내검색<br> | ||
| 155번째 줄: | 145번째 줄: | ||
** http://book.daum.net/search/contentSearch.do?query= | ** http://book.daum.net/search/contentSearch.do?query= | ||
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| − | + | ==링크== | |
* [http://www.ams.org/news/math-in-the-media/mathdigest-index Summaries of Media Coverage of Math] | * [http://www.ams.org/news/math-in-the-media/mathdigest-index Summaries of Media Coverage of Math] | ||
* 구글 블로그 검색<br> | * 구글 블로그 검색<br> | ||
** http://blogsearch.google.com/blogsearch?q= | ** http://blogsearch.google.com/blogsearch?q= | ||
2012년 10월 27일 (토) 13:10 판
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, \(L(x)+L(1-xy)+L(y)+L(\frac{1-y}{1-xy})+L\left( \frac{1-x}{1-xy} )\right)=3L(1)\)
- 2항 관계식, 반사공식(오일러)
- 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
메모
관련된 항목들
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
수학용어번역
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
리뷰논문, 에세이, 강의노트
관련논문
- 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