"다이로그 항등식 (dilogarithm identities)"의 두 판 사이의 차이

수학노트
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<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">이 항목의 스프링노트 원문주소</h5>
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==개요==
  
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* [[다이로그 항등식 (dilogarithm identities)]]<br>
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* [[로저스 다이로그 함수 (Rogers' dilogarithm)]] <math>L(x)</math>
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* 다이로그 항등식이란 대수적수 <math>x_i</math>와 유리수 <math>c</math>에 대한 다음과 같은 형태의 항등식을 말함
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:<math>\sum_{i=1}^{N}L(x_i)=cL(1)</math>
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* Polylogarithm ladder라 불리기도 한다
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* 모든 다이로그 항등식을 5항 관계식을 통하여 이해할 수 있는지의 여부는 다이로그 함수에 대한 중요한 미해결 문제이다
  
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==원분다항식과 dilogarithm 항등식==
  
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<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">개요</h5>
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*  원분다항식 (단위근에 대한 [[원분다항식(cyclotomic polynomial)]] 과는 의미가 다르나 이미 용어가 굳어져 널리 사용됨):<math>\prod_{r}(1-x^r)^{e_r}=x^{e_0}</math> 여기서 <math>e_i</math> 는 정수, <math>r</math>은 자연수
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*  대응되는 dilogarithm 항등식:<math>\sum_{r}\frac{e_r}{r}L(x^r)=cL(1)</math> 여기서 c 는 유리수
  
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* [[로저스 다이로그 함수 (Rogers' dilogarithm)|로저스 dilogarithm]] <math>L(x)</math><br>
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* dilogarithm 항등식<br> 대수적수 <math>x_i</math>와 유리수 <math>c</math>에 대한 다음과 같은 형태의 항등식<br><math>\sum_{i=1}^{N}L(x_i)=cL(1)</math><br>
 
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*  Polylogarithm ladders 으로 불리기도 한다<br>
 
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*  모든 다이로그 항등식을 5항 관계식을 통하여 이해할 수 있는지의 여부는 다이로그 함수에 대한 중요한 미해결 문제이다<br>
 
  
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==유리수==
  
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<h5 style="margin: 0px; line-height: 2em;">원분다항식과 dilogarithm 항등식</h5>
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*  오일러:<math>L(1)=\frac{\pi^2}{6}</math>:<math>-2L(-1)=L(1)</math>:<math>2L(\frac{1}{2})=L(1)</math>
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*  Lewin:<math>L(\frac{1}{2^6})-2L(\frac{1}{2^3})-6L(\frac{1}{2^2})+2L(1)=0</math>:<math>L(\frac{1}{3^2})-6L(\frac{1}{3})+2L(1)=0</math>
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*  ???:<math>2 L\left(\frac{1}{p}\right)+\sum _{j=2}^{p-1} L\left(\frac{1}{j^2}\right)=L(1)</math>
  
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* 원분다항식 (단위근에 대한 [[원분다항식(cyclotomic polynomial)]] 과는 의미가 다르나 이미 용어가 굳어져 널리 사용됨)<br><math>\prod_{r}(1-x^r)^{e_r}=x^{e_0}</math><br><math>e_i</math> 는 정수, <math>r</math>은 자연수<br>
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*  대응되는 dilogarithm 항등식<br><math>\sum_{r}\frac{e_r}{r}L(x^r)=cL(1)</math><br> 여기서 c 는 유리수<br>
 
  
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==2차식==
  
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<h5 style="margin: 0px; line-height: 2em;">유리수</h5>
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* [[란덴 다이로그 항등식]]
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:<math>5L(\frac{3-\sqrt{5}}{2})=2L(1)</math>:<math>5L(\frac{-1+\sqrt{5}}{2})=3L(1)</math>
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*  콕세터(1935) <math>\rho=\tfrac{1}{2}(\sqrt{5}-1)</math>라 두자
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:<math>L(\rho^6)=4L(\rho^3)+3L(\rho^2)-6L(\rho)+\frac{7}{5}L(1)</math>
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:<math>L(\rho^{12})=2L(\rho^6)+3L(\rho^4)+4L(\rho^3)-6L(\rho^2)-\frac{21}{5}L(1)</math>
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:<math>L(\rho^{20})=2L(\rho^{10})+15L(\rho^4)-10L(\rho^2)+\frac{6}{5}L(1)</math>
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*  Lewin
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:<math>L(\rho^{24})=6L(\rho^{8})+8L(\rho^6)-6L(\rho^4)+\frac{1}{5}L(1)</math>
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* Lewin
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:<math>L(x^6)-4L(x^3)-6L(x^2)+24L(x)=7L(1)</math>
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여기서 <math>x=\frac{\sqrt{13}-3}{2}</math>
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*  Browkin
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:<math>L(x^6)-6L(x^3)+3L(x^2)+18L(x)=8L(1)</math>
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:<math>L(z^6)-3L(z^3)-6L(z^2)+9L(z)=2L(1)</math>
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여기서 <math>x=\frac{\sqrt{13}-1}{6}, z=\frac{\sqrt{13}+1}{6}</math>
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* [[르장드르 카이 함수]]
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:<math>4L(\sqrt{2}-1)-L((\sqrt{2}-1)^2)=\frac{3}{2}L(1)</math>
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:<math>4L(\frac{\sqrt{5}-1}{2})-L((\frac{\sqrt{5}-1}{2})^2)=2L(1)</math>
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:<math>4L(\sqrt{5}-2)-L((\sqrt{5}-2)^2)=L(1)</math>
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*  Loxton
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:<math>12L(\beta)+3L(\beta^{2})-2L(\beta^{3})=5L(1)</math>
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:<math>12L(\gamma)-9L(\gamma^{2})-2L(\gamma^{3})+L(\gamma^6)=3L(1)</math>
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여기서 <math>\beta=\frac{\sqrt{3}-1}{2},\gamma=\sqrt{3}-1</math>.
  
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*  오일러<br><math>L(1)=\frac{\pi^2}{6}</math><br><math>-2L(-1)=L(1)</math><br><math>2L(\frac{1}{2})=L(1)</math><br>
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==3차식==
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*  Lewin<br><math>L(\frac{1}{2^6})-2L(\frac{1}{2^3})-6L(\frac{1}{2^2})+2L(1)=0</math><br><math>L(\frac{1}{3^2})-6L(\frac{1}{3})+2L(1)=0</math><br>
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*  왓슨 <math>\alpha, -\beta, -\gamma^{-1}</math> 가 방정식<math>x^3+2x^2-x-1=0</math> 의 해라고 하자.
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*  ???<br><math>2 L\left(\frac{1}{p}\right)+\sum _{j=2}^{p-1} L\left(\frac{1}{j^2}\right)=L(1)</math><br>
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:<math>7L(\alpha^2)-7L(\alpha)+L(1)=0</math>
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:<math>7L(\beta^2)+14L(\beta)-10L(1)=0</math>
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:<math>7L(\gamma^2)+14L(\gamma)-8L(1)=0</math>
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*  Loxton & Lewin <math>x, -y, -z^{-1}</math>가 방정식 <math>x^3+3x^2-1=0</math>의 해라고 하자.
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:<math>3L(x^3)-9L(x^2)-9L(x)+7L(1)=0</math>
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:<math>3L(y^6)-6L(y^3)-27L(y^2)+18L(y)+2L(1)=0</math>
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:<math>3L(z^6)-6L(z^3)-27L(z^2)+18L(z)-2L(1)=0</math>
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*  Gordon & McIntosh <math>a, -b, -c^{-1}</math>가 방정식 <math>x^3+6x^2+3x-1=0</math>의 해라고 하자.
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:<math>2L(a^3)-2L(a^2)-11L(a)+3L(1)=0</math>
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:<math>2L(b^6)-4L(b^3)-15L(b^2)+22L(b)-6L(1)=0</math>
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:<math>2L(c^6)-4L(c^3)-15L(c^2)+22L(c)-4L(1)=0</math>
  
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==4차식==
  
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*  Gordon & McIntosh <math>\delta=(\sqrt{3+2\sqrt{5}}-1)/2</math> 는 방정식 <math>x^4+2x^3-x-1=0</math>의 해:<math>5L(\delta^3)-5L(\delta)+L(1)=0</math>:<math>L(\delta^{12})-2L(\delta^6)-6L(\delta^4)+4L(\delta^3)+3L(\delta^2)+4L(\delta)-4L(1)=0</math>
  
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<h5 style="margin: 0px; line-height: 2em;">2차식</h5>
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* 란덴<br><math>5L(\frac{3-\sqrt{5}}{2})=2L(1)</math><br><math>5L(\frac{-1+\sqrt{5}}{2})=3L(1)</math><br>
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*  콕세터(1935)<br><math>\rho=\tfrac{1}{2}(\sqrt{5}-1)</math> 는 [[황금비]]<br><math>L(\rho^6)=4L(\rho^3)+3L(\rho^2)-6L(\rho)+\frac{7\pi^2}{30}</math><br><math>L(\rho^{12})=2L(\rho^6)+3L(\rho^4)+4L(\rho^3)-6L(\rho^2)+\frac{7\pi^2}{10}</math><br><math>L(\rho^{20})=2L(\rho^{10})+15L(\rho^4)-10L(\rho^2)+\frac{\pi^2}{5}</math><br>
 
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*  Lewin<br><math>L(\rho^{24})=6L(\rho^{8})+8L(\rho^6)-6L(\rho^4)+\frac{\pi^2}{30}</math><br>
 
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*  Lewin<br><math>x=\frac{\sqrt{13}-3}{2}</math><br><math>L(x^6)-4L(x^3)-6L(x^2)+24L(x)=7L(1)</math><br>
 
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*  Browkin<br><math>x=\frac{\sqrt{13}-1}{6}</math>, <math>z=\frac{\sqrt{13}+1}{6}</math><br><math>L(x^6)-6L(x^3)+L(x^2)+18L(x)=8L(1)</math><br><math>L(z^6)-3L(z^3)-6L(z^2)+9L(z)=2L(1)</math><br>
 
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* [[르장드르 카이 함수]]<br><math>L(\sqrt{2}-1)-4L((\sqrt{2}-1)^2)=\frac{\pi^2}{4}</math><br><math>L(\frac{\sqrt{5}-1}{2})-4L((\frac{\sqrt{5}-1}{2})^2)=\frac{\pi^2}{3}</math><br><math>L(\sqrt{5}-2)-4L((\sqrt{5}-2)^2)=\frac{\pi^2}{6}</math><br>
 
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*  Loxton<br><math>x=\frac{\sqrt{3}-1}{2}</math><br><math>12L(\beta)+3L(\beta^{2})-2L(\beta^{3})=\frac{5\pi^{2}}{6}</math><br>
 
  
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==etc==
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<h5>3차식</h5>
 
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*  왓슨<br><math>\alpha, -\beta, -\gamma^{-1}</math> 가 방정식<math>x^3+2x^2-x-1=0</math> 의 해라고 하자.<br><math>7L(\alpha^2)-7L(\alpha)+L(1)=0</math><br><math>7L(\beta^2)+14L(\beta)-10L(1)=0</math><br><math>7L(\gamma^2)+14L(\gamma)-8L(1)=0</math><br>
 
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*  Loxton & Lewin<br><math>x, -y, -z^{-1}</math>가 방정식 <math>x^3+3x^2-1=0</math>의 해라고 하자.<br><math>3L(x^3)-9L(x^2)-9L(x)+7L(1)=0</math><br><math>3L(y^6)-6L(y^3)-27L(y^2)+18L(y)+2L(1)=0</math><br><math>3L(z^6)-6L(z^3)-27L(z^2)+18L(z)-2L(1)=0</math><br>
 
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*  Gordon & McIntosh<br><math>a, -b, -c^{-1}</math>가 방정식 <math>x^3+6x^2+3x-1=0</math>의 해라고 하자.<br><math>2L(a^3)-2L(a^2)-11L(a)+3L(1)=0</math><br><math>2L(b^6)-4L(b^3)-15L(b^2)+22L(b)-6L(1)=0</math><br><math>2L(c^6)-4L(c^3)-15L(c^2)+22L(c)-4L(1)=0</math><br>  <br>
 
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<h5 style="margin: 0px; line-height: 2em;">4차식</h5>
 
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*  Gordon & McIntosh<br><math>\delta=(\sqrt{3+2\sqrt{5}}-1)/2</math> 는 방정식 <math>x^4+2x^3-x-1=0</math>의 해<br><math>5L(\delta^3)-5L(\delta)+L(1)=0</math><br><math>L(\delta^{12})-2L(\delta^6)-6L(\delta^4)+4L(\delta^3)+3L(\delta^2)+4L(\delta)-4L(1)=0</math><br>
 
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<h5 style="margin: 0px; line-height: 2em;">etc</h5>
 
  
 
<math>\sum_{i=1}^{k}L(\frac{\sin^2\frac{\pi}{k+2}}{\sin^2\frac{(i+1)\pi}{k+2}})=\frac{3k}{k+2}\cdot\frac{\pi^2}{6}</math>
 
<math>\sum_{i=1}^{k}L(\frac{\sin^2\frac{\pi}{k+2}}{\sin^2\frac{(i+1)\pi}{k+2}})=\frac{3k}{k+2}\cdot\frac{\pi^2}{6}</math>
75번째 줄: 81번째 줄:
 
http://www.jstor.org/stable/2152925
 
http://www.jstor.org/stable/2152925
  
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==역사==
  
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* http://www.google.com/search?hl=en&tbs=tl:1&q=dilogarithm
 
* http://www.google.com/search?hl=en&tbs=tl:1&q=dilogarithm
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* [[수학사연표 (역사)|수학사연표]]
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* [[수학사 연표]]
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<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">메모</h5>
 
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<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">관련된 항목들</h5>
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* [[원분다항식(cyclotomic polynomial)]]<br>
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==관련된 항목들==
  
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* [[원분다항식(cyclotomic polynomial)]]
  
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<h5>매스매티카 파일 및 계산 리소스</h5>
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==매스매티카 파일 및 계산 리소스==
 
* https://docs.google.com/leaf?id=0B8XXo8Tve1cxODUxMjM0ZTctNWRlYS00NmMxLWJiZTItYjk0YTQ4YjA1YjBl&sort=name&layout=list&num=50
 
* https://docs.google.com/leaf?id=0B8XXo8Tve1cxODUxMjM0ZTctNWRlYS00NmMxLWJiZTItYjk0YTQ4YjA1YjBl&sort=name&layout=list&num=50
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* http://www.wolframalpha.com/input/?i=
 
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* http://functions.wolfram.com/
 
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* [http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]
 
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* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]
 
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* [http://numbers.computation.free.fr/Constants/constants.html Numbers, constants and computation]
 
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* [[매스매티카 파일 목록]]
 
  
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* 단어사전 http://www.google.com/dictionary?langpair=en|ko&q=
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==사전 형태의 자료==
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* 발음사전 http://www.forvo.com/search/
 
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* [http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=&fstr= 대한수학회 수학 학술 용어집]<br>
 
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** http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=
 
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* [http://www.nktech.net/science/term/term_l.jsp?l_mode=cate&s_code_cd=MA 남·북한수학용어비교]
 
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* [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]
 
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<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">사전 형태의 자료</h5>
 
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* http://ko.wikipedia.org/wiki/
 
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* http://en.wikipedia.org/wiki/
 
 
* http://en.wikipedia.org/wiki/Polylogarithm
 
* http://en.wikipedia.org/wiki/Polylogarithm
 
* http://mathworld.wolfram.com/Dilogarithm.html
 
* http://mathworld.wolfram.com/Dilogarithm.html
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* http://www.wolframalpha.com/input/?i=
 
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* [http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]
 
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* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]<br>
 
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** http://www.research.att.com/~njas/sequences/?q=
 
  
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<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">관련논문</h5>
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==관련논문==
  
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* [http://arxiv.org/abs/math.CA/9906134 A seventeenth-order polylogarithm ladder]<br>
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* [http://arxiv.org/abs/math.CA/9906134 A seventeenth-order polylogarithm ladder]
 
** David H. Bailey, David J. Broadhurst
 
** David H. Bailey, David J. Broadhurst
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* [http://dx.doi.org/10.1023/A:1009709927327 Algebraic Dilogarithm Identities]<br>
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* [http://dx.doi.org/10.1023/A:1009709927327 Algebraic Dilogarithm Identities]
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** Basil Gordon  and Richard J. Mcintosh, 1997
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** Basil Gordon and Richard J. Mcintosh, 1997
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* [http://dx.doi.org/10.1143/PTPS.118.61 Dilogarithm identities]
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* [http://dx.doi.org/10.1143/PTPS.118.61 Dilogarithm identities]<br>
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** Anatol N. Kirillov,Prog.Theor.Phys.Suppl.118:61-142, 1995
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** Anatol N. Kirillov,Prog.Theor.Phys.Suppl.118:61-142, 1995
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* [http://dx.doi.org/10.1007/BF01840426 Identities for the Rogers dilogarithm function connected with simple Lie algebras]
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* [http://dx.doi.org/10.1007/BF01840426 Identities for the Rogers dilogarithm function connected with simple Lie algebras]<br>
 
 
** A. N. Kirillov, 1989
 
** A. N. Kirillov, 1989
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* [http://dx.doi.org/10.1016/0022-314X%2884%2990077-5 The inner structure of the dilogarithm in algebraic fields]<br>
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* [http://dx.doi.org/10.1016/0022-314X%2884%2990077-5 The inner structure of the dilogarithm in algebraic fields]
 
** L. Lewin, 1984
 
** L. Lewin, 1984
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* [http://matwbn.icm.edu.pl/ksiazki/aa/aa43/aa4326.pdf Special values of the dilogarithm function]<br>
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* Loxton, J. H. 1984. “Special Values of the Dilogarithm Function.”Acta Arithmetica 43 (2): 155–166. http://matwbn.icm.edu.pl/ksiazki/aa/aa43/aa4326.pdf
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** J. H. Loxton, 1984
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* [http://dx.doi.org/10.1017/S1446788700018747 The dilogarithm in algebraic fields]
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* [http://dx.doi.org/10.1017/S1446788700018747 The dilogarithm in algebraic fields]<br>
 
 
** L. Lewin, Journal of the Australian Mathematical Society (Series A) (1982), 33 : 302-33
 
** L. Lewin, Journal of the Australian Mathematical Society (Series A) (1982), 33 : 302-33
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* [http://dx.doi.org/10.1093/qmath/os-8.1.39 A Note on Spence's Logarithmic Transcendent]<br>
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* [http://dx.doi.org/10.1093/qmath/os-8.1.39 A Note on Spence's Logarithmic Transcendent]
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** Watson, G. N., Quart. J. Math. Oxford Ser. 8, 39-42, 1937
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** Watson, G. N., Quart. J. Math. Oxford Ser. 8, 39-42, 1937
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* [http://dx.doi.org/10.1093/qmath/os-6.1.13 The functions of Schlafli and Lobatschefsky]<br>
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* [http://dx.doi.org/10.1093/qmath/os-6.1.13 The functions of Schlafli and Lobatschefsky]
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** Coxeter, H.S.M. (1935),  Quarterly Journal of Mathematics (Oxford) 6: 13–29
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** Coxeter, H.S.M. (1935), Quarterly Journal of Mathematics (Oxford) 6: 13–29
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* http://www.jstor.org/action/doBasicSearch?Query=
 
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<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">관련기사</h5>
 
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<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">블로그</h5>
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===위키데이터===
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===Spacy 패턴 목록===
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* [http://www.ams.org/mathmoments/ Mathematical Moments from the AMS]
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* [http://betterexplained.com/ BetterExplained]
 

2021년 2월 17일 (수) 05:01 기준 최신판

개요

\[\sum_{i=1}^{N}L(x_i)=cL(1)\]

  • Polylogarithm ladder라 불리기도 한다
  • 모든 다이로그 항등식을 5항 관계식을 통하여 이해할 수 있는지의 여부는 다이로그 함수에 대한 중요한 미해결 문제이다


원분다항식과 dilogarithm 항등식

  • 원분다항식 (단위근에 대한 원분다항식(cyclotomic polynomial) 과는 의미가 다르나 이미 용어가 굳어져 널리 사용됨)\[\prod_{r}(1-x^r)^{e_r}=x^{e_0}\] 여기서 \(e_i\) 는 정수, \(r\)은 자연수
  • 대응되는 dilogarithm 항등식\[\sum_{r}\frac{e_r}{r}L(x^r)=cL(1)\] 여기서 c 는 유리수



유리수

  • 오일러\[L(1)=\frac{\pi^2}{6}\]\[-2L(-1)=L(1)\]\[2L(\frac{1}{2})=L(1)\]
  • Lewin\[L(\frac{1}{2^6})-2L(\frac{1}{2^3})-6L(\frac{1}{2^2})+2L(1)=0\]\[L(\frac{1}{3^2})-6L(\frac{1}{3})+2L(1)=0\]
  • ???\[2 L\left(\frac{1}{p}\right)+\sum _{j=2}^{p-1} L\left(\frac{1}{j^2}\right)=L(1)\]



2차식

\[5L(\frac{3-\sqrt{5}}{2})=2L(1)\]\[5L(\frac{-1+\sqrt{5}}{2})=3L(1)\]

  • 콕세터(1935) \(\rho=\tfrac{1}{2}(\sqrt{5}-1)\)라 두자

\[L(\rho^6)=4L(\rho^3)+3L(\rho^2)-6L(\rho)+\frac{7}{5}L(1)\] \[L(\rho^{12})=2L(\rho^6)+3L(\rho^4)+4L(\rho^3)-6L(\rho^2)-\frac{21}{5}L(1)\] \[L(\rho^{20})=2L(\rho^{10})+15L(\rho^4)-10L(\rho^2)+\frac{6}{5}L(1)\]

  • Lewin

\[L(\rho^{24})=6L(\rho^{8})+8L(\rho^6)-6L(\rho^4)+\frac{1}{5}L(1)\]

  • Lewin

\[L(x^6)-4L(x^3)-6L(x^2)+24L(x)=7L(1)\] 여기서 \(x=\frac{\sqrt{13}-3}{2}\)

  • Browkin

\[L(x^6)-6L(x^3)+3L(x^2)+18L(x)=8L(1)\] \[L(z^6)-3L(z^3)-6L(z^2)+9L(z)=2L(1)\] 여기서 \(x=\frac{\sqrt{13}-1}{6}, z=\frac{\sqrt{13}+1}{6}\)

\[4L(\sqrt{2}-1)-L((\sqrt{2}-1)^2)=\frac{3}{2}L(1)\] \[4L(\frac{\sqrt{5}-1}{2})-L((\frac{\sqrt{5}-1}{2})^2)=2L(1)\] \[4L(\sqrt{5}-2)-L((\sqrt{5}-2)^2)=L(1)\]

  • Loxton

\[12L(\beta)+3L(\beta^{2})-2L(\beta^{3})=5L(1)\] \[12L(\gamma)-9L(\gamma^{2})-2L(\gamma^{3})+L(\gamma^6)=3L(1)\] 여기서 \(\beta=\frac{\sqrt{3}-1}{2},\gamma=\sqrt{3}-1\).

3차식

  • 왓슨 \(\alpha, -\beta, -\gamma^{-1}\) 가 방정식\(x^3+2x^2-x-1=0\) 의 해라고 하자.

\[7L(\alpha^2)-7L(\alpha)+L(1)=0\] \[7L(\beta^2)+14L(\beta)-10L(1)=0\] \[7L(\gamma^2)+14L(\gamma)-8L(1)=0\]

  • Loxton & Lewin \(x, -y, -z^{-1}\)가 방정식 \(x^3+3x^2-1=0\)의 해라고 하자.

\[3L(x^3)-9L(x^2)-9L(x)+7L(1)=0\] \[3L(y^6)-6L(y^3)-27L(y^2)+18L(y)+2L(1)=0\] \[3L(z^6)-6L(z^3)-27L(z^2)+18L(z)-2L(1)=0\]

  • Gordon & McIntosh \(a, -b, -c^{-1}\)가 방정식 \(x^3+6x^2+3x-1=0\)의 해라고 하자.

\[2L(a^3)-2L(a^2)-11L(a)+3L(1)=0\] \[2L(b^6)-4L(b^3)-15L(b^2)+22L(b)-6L(1)=0\] \[2L(c^6)-4L(c^3)-15L(c^2)+22L(c)-4L(1)=0\]

4차식

  • Gordon & McIntosh \(\delta=(\sqrt{3+2\sqrt{5}}-1)/2\) 는 방정식 \(x^4+2x^3-x-1=0\)의 해\[5L(\delta^3)-5L(\delta)+L(1)=0\]\[L(\delta^{12})-2L(\delta^6)-6L(\delta^4)+4L(\delta^3)+3L(\delta^2)+4L(\delta)-4L(1)=0\]



etc

\(\sum_{i=1}^{k}L(\frac{\sin^2\frac{\pi}{k+2}}{\sin^2\frac{(i+1)\pi}{k+2}})=\frac{3k}{k+2}\cdot\frac{\pi^2}{6}\)

http://www.jstor.org/stable/2152925




역사



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