"폴리로그 함수(polylogarithm)"의 두 판 사이의 차이

2013년 11월 2일 (토) 07:45 판

개요

다이로그 함수(dilogarithm) 의 일반화

정의

\[\operatorname{Li}_r(z)= \sum_{n=1}^\infty {z^n \over n^r}=\int_0^z \operatorname{Li}_{r-1}(t) \frac{dt}{t}\] \[\operatorname{Li}_3(z) =\int_0^z \operatorname{Li}_2(t) \frac{dt}{t}\]

로그함수

로그 함수

\[-\log (1-z)=z+\frac{z^2}{2}+\frac{z^3}{3}+\frac{z^4}{4}+\frac{z^5}{5}+\cdots\]

역사

메모

http://mathoverflow.net/questions/25428/what-is-special-about-polylogarithms-that-leads-to-so-many-interesting-identities

사전 형태의 자료

리뷰논문, 에세이, 강의노트

John R. Rhodes Polylogarithms ,2008
Richard Hain, Classical Polylogarithms , 1992
Some wonderful formulas ... an introduction to polylogarithms A.J. Van der Poorten, Queen's papers in Pure and Applied Mathematics, 54 (1979), 269-286 (http://www.ega-math.narod.ru/Apery2.htm )
Askey, Richard. 1982. “Book Review: Polylogarithms and Associated Functions.” American Mathematical Society. Bulletin. New Series 6 (2): 248–251. doi:10.1090/S0273-0979-1982-14998-9.

관련논문

Multiple Polylogarithms: A Brief Survey Douglas Bowman, David M. Bradley, 5 Oct 2003
Polylogarithmic ladders, hypergeometric series and the ten millionth digits of $\zeta(3)$ and $\zeta(5)$ D. J. Broadhurst, 1998
On the rapid computation of various polylogarithmic constants David Bailey; Peter Borwein; Simon Plouffe.Journal: Math. Comp. 66 (1997), 903-913.
Ramakrishnan, Analogs of the Bloch-Wigner function for higher polylogarithms, 1986
The classical polylogarithms, algebraic K-theory and $\zeta_F(n)$, Goncharov, A. Proc. of the Gelfand Seminar, Birkhauser, 113-135

@@ 1번째 줄: / 1번째 줄: @@
-==이 항목의 스프링노트 원문주소==
-* [[폴리로그 함수(polylogarithm)]]<br>
 ==개요==
-* [[다이로그 함수(dilogarithm)|다이로그 함수(dilogarithm )]] 의 일반화<br>
+* [[다이로그 함수(dilogarithm)]] 의 일반화
-* http://www.ega-math.narod.ru/Apery2.htm<br>
 ==정의==
+:<math>\operatorname{Li}_r(z)= \sum_{n=1}^\infty {z^n \over n^r}=\int_0^z \operatorname{Li}_{r-1}(t) \frac{dt}{t}</math>
+:<math>\operatorname{Li}_3(z) =\int_0^z \operatorname{Li}_2(t) \frac{dt}{t}</math>
-<math>\operatorname{Li}_r(z)= \sum_{n=1}^\infty {z^n \over n^r}=\int_0^z \operatorname{Li}_{r-1}(t) \frac{dt}{t}</math>
-<math>\operatorname{Li}_3(z) =\int_0^z \operatorname{Li}_2(t) \frac{dt}{t}</math>
 ==로그함수==
-* [[로그 함수]]:<math>-\log (1-z)=z+\frac{z^2}{2}+\frac{z^3}{3}+\frac{z^4}{4}+\frac{z^5}{5}+\cdots</math><br>
+* [[로그 함수]]
+:<math>-\log (1-z)=z+\frac{z^2}{2}+\frac{z^3}{3}+\frac{z^4}{4}+\frac{z^5}{5}+\cdots</math>
 ==역사==
 * http://www.google.com/search?hl=en&tbs=tl:1&q=
 * [[수학사 연표]]
-*
 ==메모==
-* http://mathoverflow.net/questions/25428/what-is-special-about-polylogarithms-that-leads-to-so-many-interesting-identities<br>
+* http://mathoverflow.net/questions/25428/what-is-special-about-polylogarithms-that-leads-to-so-many-interesting-identities
 * http://books.google.com/books?hl=ko&lr=&id=9G3nlZUDAhkC&oi=fnd&pg=PA391&dq=The+classical+polylogarithms,+algebraic+K-theory&ots=zst2m387di&sig=kNRuqZp_mUdFDXScW41qNbprgps#v=onepage&q=&f=false
-* [http://www.maths.dur.ac.uk/%7Edma0hg/kyoto.pdf Functional equations of polylogarithms] Herbert Gangl<br>
+* [http://www.maths.dur.ac.uk/%7Edma0hg/kyoto.pdf Functional equations of polylogarithms] Herbert Gangl
 * [http://www.maths.dur.ac.uk/%7Edma0hg/kyoto.pdf http://www.maths.dur.ac.uk/~dma0hg/kyoto.pdf]
 * [http://www.maths.dur.ac.uk/%7Ed40ppt/pdf/John_Rhodes.pdf http://www.maths.dur.ac.uk/~d40ppt/pdf/John_Rhodes.pdf]
-*  Math Overflow<br>
-** http://mathoverflow.net/search?q=
 ==관련된 항목들==
-* [[L-함수의 값 구하기 입문]]<br>
+* [[L-함수의 값 구하기 입문]]
-* [[원주율의 BBP 공식|파이에 대한 BBP 공식]]<br>
+* [[원주율의 BBP 공식|파이에 대한 BBP 공식]]
-* [[로그 사인 적분 (log sine integrals)]]<br>
+* [[로그 사인 적분 (log sine integrals)]]
-==수학용어번역==
-* 단어사전 http://www.google.com/dictionary?langpair=en|ko&q=
-* 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]
-==사전 형태의 자료==
+==사전 형태의 자료==
 * http://ko.wikipedia.org/wiki/
 * http://en.wikipedia.org/wiki/Polylogarithm
-* http://en.wikipedia.org/wiki/
-* http://www.wolframalpha.com/input/?i=
-* [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]<br>
-** http://www.research.att.com/~njas/sequences/?q=
 ==리뷰논문, 에세이, 강의노트==
@@ 104번째 줄: / 68번째 줄: @@
 * John R. Rhodes [http://www.mathematik.hu-berlin.de/%7Ekreimer/Polylogarithms.pdf Polylogarithms] ,2008
 * Richard Hain, [http://arxiv.org/abs/alg-geom/9202022 Classical Polylogarithms] , 1992
+* Some wonderful formulas ... an introduction to polylogarithms A.J. Van der Poorten, Queen's papers in Pure and Applied Mathematics, 54 (1979), 269-286 (http://www.ega-math.narod.ru/Apery2.htm )
+* Askey, Richard. 1982. “Book Review: Polylogarithms and Associated Functions.” American Mathematical Society. Bulletin. New Series 6 (2): 248–251. doi:10.1090/S0273-0979-1982-14998-9.
 ==관련논문==
+* [http://arxiv.org/abs/math/0310062 Multiple Polylogarithms: A Brief Survey] Douglas Bowman, David M. Bradley, 5 Oct 2003
+* [http://arxiv.org/abs/math.CA/9803067 Polylogarithmic ladders, hypergeometric series and the ten millionth digits of $\zeta(3)$ and $\zeta(5)$] D. J. Broadhurst, 1998
+* [http://dx.doi.org/http://dx.doi.org/10.1090%2FS0025-5718-97-00856-9 On the rapid computation of various polylogarithmic constants] David Bailey; Peter Borwein; Simon Plouffe.Journal: Math. Comp. 66 (1997), 903-913.
+*  Ramakrishnan, Analogs of the Bloch-Wigner function for higher polylogarithms, 1986
+*  The classical polylogarithms, algebraic K-theory and $\zeta_F(n)$, Goncharov, A. Proc. of the Gelfand Seminar, Birkhauser, 113-135
-*   <br>
-* [http://arxiv.org/abs/math/0310062 Multiple Polylogarithms: A Brief Survey] Douglas Bowman, David M. Bradley, 5 Oct 2003<br>
-* [http://arxiv.org/abs/math.CA/9803067 Polylogarithmic ladders, hypergeometric series and the ten millionth digits of $\zeta(3)$ and $\zeta(5)$] D. J. Broadhurst, 1998<br>
-* [http://dx.doi.org/http://dx.doi.org/10.1090%2FS0025-5718-97-00856-9 On the rapid computation of various polylogarithmic constants] David Bailey; Peter Borwein; Simon Plouffe.Journal: Math. Comp. 66 (1997), 903-913.<br>
-*  Ramakrishnan, Analogs of the Bloch-Wigner function for higher polylogarithms, 1986<br>
-*  The classical polylogarithms, algebraic K-theory and ζ. F. (n), Goncharov, A. Proc. of the Gelfand Seminar, Birkhauser, 113-135<br>
-*  Some wonderful formulas ... an introduction to polylogarithms A.J. Van der Poorten, Queen's papers in Pure and Applied Mathematics, 54 (1979), 269-286 (http://www.ega-math.narod.ru/Apery2.htm )<br>
-* http://www.jstor.org/action/doBasicSearch?Query=polylogarithm
-* http://www.jstor.org/action/doBasicSearch?Query=
-* http://www.ams.org/mathscinet
-* http://dx.doi.org/http://dx.doi.org/10.1090%2FS0025-5718-97-00856-9
 [[분류:다이로그]]