폴리로그 함수(polylogarithm)

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

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

메모

• Scheider, René. “The de Rham Realization of the Elliptic Polylogarithm in Families.” arXiv:1408.3819 [math], August 17, 2014. http://arxiv.org/abs/1408.3819.
• Jameson, Polylogarithms, multiple zeta values, and the series of Hjortnaes and Comtet
• 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
• Functional equations of polylogarithms Herbert Gangl
• http://www.maths.dur.ac.uk/~dma0hg/kyoto.pdf
• http://www.maths.dur.ac.uk/~d40ppt/pdf/John_Rhodes.pdf

관련된 항목들

• 원주율의 BBP 공식
• 로그 사인 적분 (log sine integrals)

사전 형태의 자료

• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/Polylogarithm

리뷰논문, 에세이, 강의노트

• Vergu, C. “Polylogarithm Identities, Cluster Algebras and the N=4 Supersymmetric Theory.” arXiv:1512.08113 [hep-Th], December 26, 2015. http://arxiv.org/abs/1512.08113.
• John R. Rhodes Polylogarithms ,2008
• Bowman, Douglas, and David M. Bradley. “Multiple Polylogarithms: A Brief Survey.” arXiv:math/0310062, October 5, 2003. http://arxiv.org/abs/math/0310062.
• Hain, Richard. “Classical Polylogarithms.” arXiv:alg-geom/9202022, February 20, 1992. http://arxiv.org/abs/alg-geom/9202022.
• 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.
• 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 )

관련논문

• Henn, Johannes M., Alexander V. Smirnov, and Vladimir A. Smirnov. “Evaluating Multiple Polylogarithm Values at Sixth Roots of Unity up to Weight Six.” arXiv:1512.08389 [hep-Ph, Physics:hep-Th, Physics:math-Ph], December 28, 2015. http://arxiv.org/abs/1512.08389.
• Rudenko, Daniil. “On the Functional Equations for Polylogarithms in One Variable.” arXiv:1511.09110 [math], November 2, 2015. http://arxiv.org/abs/1511.09110.
• Sakugawa, Kenji, and Shin-ichiro Seki. “On Functional Equations of Finite Multiple Polylogarithms.” arXiv:1509.07653 [math], September 25, 2015. http://arxiv.org/abs/1509.07653.
• 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