"3-manifolds and their invariants"의 두 판 사이의 차이
| 40번째 줄: | 40번째 줄: | ||
Clausen function | Clausen function | ||
| + | |||
| + | <math>Cl_2(\theta)=-\int_0^{\theta} \ln |2\sin \frac{t}{2}| \,dt=\sum_{n=1}^{\infty}\frac{\sin (n\theta)}{n^2}</math> | ||
<math>\mathfrak{I}(\operatorname{Li}_2(e^{i\theta}))=\sum_{n=1}^\infty \frac{\sin n\theta}{n^2}=Cl_2(\theta)</math> | <math>\mathfrak{I}(\operatorname{Li}_2(e^{i\theta}))=\sum_{n=1}^\infty \frac{\sin n\theta}{n^2}=Cl_2(\theta)</math> | ||
| 107번째 줄: | 109번째 줄: | ||
* [http://link.aip.org/link/?JMAPAQ/49/093508/1 Evaluation of a ln tan integral arising in quantum field theory]<br> | * [http://link.aip.org/link/?JMAPAQ/49/093508/1 Evaluation of a ln tan integral arising in quantum field theory]<br> | ||
** Mark W. Coffey, J. Math. Phys. 49, 093508 (2008); doi:10.1063/1.2981311 | ** Mark W. Coffey, J. Math. Phys. 49, 093508 (2008); doi:10.1063/1.2981311 | ||
| + | * [http://arxiv.org/abs/hep-th/9811173 Determinations of rational Dedekind-zeta invariants of hyperbolic manifolds and Feynman knots and links]<br> | ||
| + | ** J.M. Borwein, D.J. Broadhurst, 1998 | ||
* [[2010년 books and articles|논문정리]] | * [[2010년 books and articles|논문정리]] | ||
2010년 3월 28일 (일) 06:27 판
introduction
- volume of knot complements
Volume of knot complement
- KnotData[]
KnotData["FigureEight", "HyperbolicVolume"]
N[%, 20]
복소이차수체의 데데킨트 제타함수
\(\zeta_{K}(2)=\frac{\pi^2}{6\sqrt{|d_K|}}\sum_{(a,d_k)=1} (\frac{d_K}{a})D(e^{2\pi ia/|d_k|})\)
\(\zeta_{\mathbb{Q}\sqrt{-3}}(2)=\frac{\pi^2}{6\sqrt{3}}(D(e^{2\pi i/3})-D(e^{4\pi i/3}))=\frac{\pi^2}{3\sqrt{3}}D(e^{2\pi i/3})\)
\(\zeta_{\mathbb{Q}\sqrt{-7}}(2)=\frac{\pi^2}{3\sqrt{7}}(D(e^{2\pi i/7})+D(e^{4\pi i/7})-D(e^{6\pi i/7}))\)
- L[x_] := Im[PolyLog[2, x]] + 1/2 Log[Abs[x]] Arg[1 - x]
N[Sum[JacobiSymbol[a, 7]*L[Exp[2 I*Pi*a/7]], {a, 1, 6}], 20]
N[L[Exp[2 I*Pi/7]] + L[Exp[4 I*Pi/7]] - L[Exp[6 I*Pi/7]], 20]
an open problem
\(\frac{24}{7\sqrt{7}}\int_{\pi/3}^{\pi/2}\ln|\frac{\tan t+\sqrt{7}}{\tan t-\sqrt{7}}|\,dt=\frac{2}{\sqrt{7}}(D(e^{2\pi i/7})+D(e^{4\pi i/7})-D(e^{6\pi i/7}))=\frac{2}{\sqrt{7}}(Cl(2\pi /7})+Cl(4\pi/7})-Cl(6\pi/7}))\)
Clausen function
\(Cl_2(\theta)=-\int_0^{\theta} \ln |2\sin \frac{t}{2}| \,dt=\sum_{n=1}^{\infty}\frac{\sin (n\theta)}{n^2}\)
\(\mathfrak{I}(\operatorname{Li}_2(e^{i\theta}))=\sum_{n=1}^\infty \frac{\sin n\theta}{n^2}=Cl_2(\theta)\)
software
history
encyclopedia
- http://ko.wikipedia.org/wiki/
- http://en.wikipedia.org/wiki/Figure-eight_knot_(mathematics)
- http://en.wikipedia.org/wiki/
- Princeton companion to mathematics(Companion_to_Mathematics.pdf)
books
- 2010년 books and articles
- http://gigapedia.info/1/
- http://gigapedia.info/1/
- http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
[[4909919|]]
articles
- A dilogarithmic integral arising in quantum field theory
- Djurdje Cvijović, J. Math. Phys. 50, 023515 (2009)JMAPAQ000050000002023515000001
- On a three-dimensional symmetric Ising tetrahedron and contributions to the theory of the dilogarithm and Clausen functions
- Mark W. Coffey, J. Math. Phys. 49, 043510 (2008); doi:10.1063/1.2902996
- Evaluation of a ln tan integral arising in quantum field theory
- Mark W. Coffey, J. Math. Phys. 49, 093508 (2008); doi:10.1063/1.2981311
- Determinations of rational Dedekind-zeta invariants of hyperbolic manifolds and Feynman knots and links
- J.M. Borwein, D.J. Broadhurst, 1998
- 논문정리
- http://www.ams.org/mathscinet
- http://www.zentralblatt-math.org/zmath/en/
- http://pythagoras0.springnote.com/
- http://math.berkeley.edu/~reb/papers/index.html[1]
- http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
- http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&s7=
- http://dx.doi.org/10.1063/1.3085764
question and answers(Math Overflow)
blogs
experts on the field