"3-manifolds and their invariants"의 두 판 사이의 차이

수학노트
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2번째 줄: 2번째 줄:
  
 
* volume of knot complements
 
* volume of knot complements
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*  volume of figure 8 knot complement<br>
 
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**  2.02988321281930725<br><math>V=\frac{9\sqrt{3}}{\pi^2}\zeta_{\mathbb{Q}(\sqrt{-3})}(2)=3D(e^{\frac{2i\pi}{3}})=2D(e^{\frac{i\pi}{3}})=2.029883212819\cdots</math><br>
 
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*  what is <math>\zeta_{\mathbb{Q}(\sqrt{-3})}(2)</math>? numrically 1.285190955484149<br>
 
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#  L[x_] := Im[PolyLog[2, x]] + 1/2 Log[Abs[x]] Arg[1 - x]<br> f[x_, y_] :=<br>  L[x] + L[1 - x*y] + L[y] + L[(1 - y)/(1 - x*y)] + L[(1 - x)/(1 - x*y)]<br> Print["five term relation"]<br> Table[f[i, j], {i, 0.1, 0.9, 0.1}, {j, 0.1, 0.9, 0.1}] // TableForm<br> N[3 L[Exp[2 I*Pi/3]], 20]<br> N[2 L[Exp[I*Pi/3]], 20]<br> N[3 (L[Exp[2 I*Pi/3]] - L[Exp[4 I*Pi/3]])/2, 20]<br> N[Pi^2*L[Exp[2 I*Pi/3]]/(3 Sqrt[3]), 20]<br>
 
  
 
 
 
 
90번째 줄: 85번째 줄:
 
<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">articles</h5>
 
<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">articles</h5>
  
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* [http://link.aip.org/link/?JMAPAQ/49/043510/1 On a three-dimensional symmetric Ising tetrahedron and contributions to the theory of the dilogarithm and Clausen functions]<br>
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** Mark W. Coffey, J. Math. Phys. 49, 043510 (2008); doi:10.1063/1.2902996
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* [http://link.aip.org/link/?JMAPAQ/49/093508/1 Evaluation of a ln tan integral arising in quantum field theory]<br>
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** Mark W. Coffey, J. Math. Phys. 49, 093508 (2008); doi:10.1063/1.2981311
  
 
* [[2010년 books and articles|논문정리]]
 
* [[2010년 books and articles|논문정리]]

2010년 3월 28일 (일) 05:46 판

introduction
  • volume of knot complements

 

 

Volume of knot complement
  1. KnotData[]
    KnotData["FigureEight", "HyperbolicVolume"]
    N[%, 20]

 

 

 

복소이차수체의 데데킨트 제타함수

\(\zeta_{K}(2)=\frac{\pi^2}{6\sqrt{|d_K|}}\sum_{(a,d_k)=1} (\frac{d_K}{a})D(e^{2\pi ia/|d_k|})\)

\(\zeta_{\mathbb{Q}\sqrt{-3}}(2)=\frac{\pi^2}{6\sqrt{3}}(D(e^{2\pi i/3})-D(e^{4\pi i/3}))=\frac{\pi^2}{3\sqrt{3}}D(e^{2\pi i/3})\)

\(\zeta_{\mathbb{Q}\sqrt{-7}}(2)=\frac{\pi^2}{3\sqrt{7}}(D(e^{2\pi i/7})+D(e^{4\pi i/7})-D(e^{6\pi i/7}))\)

  1. 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]

 

 

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