"폴리로그 함수(polylogarithm)"의 두 판 사이의 차이

2020년 11월 13일 (금) 03:10 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

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

사전 형태의 자료

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

Ngoc Hoang, Gérard Duchamp, Hoang Ngoc Minh, The algebra of Kleene stars of the plane and polylogarithms, arXiv:1602.02801[math.CO], February 05 2016, http://arxiv.org/abs/1602.02801v2, 10.1145/1235, http://dx.doi.org/10.1145/1235
Kenji Sakugawa, Shin-ichiro Seki, Finite and étale polylogarithms, http://arxiv.org/abs/1603.05811v1
Frellesvig, Hjalte, Damiano Tommasini, and Christopher Wever. “On the Reduction of Generalized Polylogarithms to \(\text{Li}_n\) and \(\text{Li}_{2,2}\) and on the Evaluation Thereof.” arXiv:1601.02649 [hep-Ph], January 11, 2016. http://arxiv.org/abs/1601.02649.
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

@@ 71번째 줄: / 71번째 줄: @@
 * Ngoc Hoang, Gérard Duchamp, Hoang Ngoc Minh, The algebra of Kleene stars of the plane and polylogarithms, arXiv:1602.02801[math.CO], February 05 2016, http://arxiv.org/abs/1602.02801v2, 10.1145/1235, http://dx.doi.org/10.1145/1235
 * Kenji Sakugawa, Shin-ichiro Seki, Finite and étale polylogarithms, http://arxiv.org/abs/1603.05811v1
-* Frellesvig, Hjalte, Damiano Tommasini, and Christopher Wever. “On the Reduction of Generalized Polylogarithms to $\text{Li}_n$ and $\text{Li}_{2,2}$ and on the Evaluation Thereof.” arXiv:1601.02649 [hep-Ph], January 11, 2016. http://arxiv.org/abs/1601.02649.
+* Frellesvig, Hjalte, Damiano Tommasini, and Christopher Wever. “On the Reduction of Generalized Polylogarithms to <math>\text{Li}_n</math> and <math>\text{Li}_{2,2}</math> and on the Evaluation Thereof.” arXiv:1601.02649 [hep-Ph], January 11, 2016. http://arxiv.org/abs/1601.02649.
 * 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.
-* [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://arxiv.org/abs/math.CA/9803067 Polylogarithmic ladders, hypergeometric series and the ten millionth digits of <math>\zeta(3)</math> and <math>\zeta(5)</math>] 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
+*  The classical polylogarithms, algebraic K-theory and <math>\zeta_F(n)</math>, Goncharov, A. Proc. of the Gelfand Seminar, Birkhauser, 113-135
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