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

2010년 2월 9일 (화) 13:50 판

로저스 dilogarithm \(L(x)\)
dilogarithm 항등식
대수적수 \(x_i\)와 유리수 \(c\)에 대한 다음과 같은 형태의 항등식
\(\sum_{i=1}^{N}L(x_i)=cL(1)\)
Polylogarithm ladders 으로 불리기도 한다다

\(L(1)=\frac{\pi^2}{6}\)

\(-2L(-1)=L(1)\)

\(2L(\frac{1}{2})=L(1)\)

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

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

\(\rho=\tfrac{1}{2}(\sqrt{5}-1)\) 는 황금비

\(L(\rho^6)=4L(\rho^3)+3L(\rho^2)-6L(\rho)+\frac{7\pi^2}{30}\)

\(L(\rho^{12})=2L(\rho^6)+3L(\rho^4)+4L(\rho^3)-6L(\rho^2)+\frac{7\pi^2}{10}\)

\(L(\rho^{20})=2L(\rho^{10})+15L(\rho^4)-10L(\rho^2)+\frac{\pi^2}{5}\)

\(L(\rho^{24})=6L(\rho^{8})+8L(\rho^6)-6L(\rho^4)+\frac{\pi^2}{30}\)

@@ 49번째 줄: / 49번째 줄: @@
 <math>L(\rho^{12})=2L(\rho^6)+3L(\rho^4)+4L(\rho^3)-6L(\rho^2)+\frac{7\pi^2}{10}</math>
-<math>L(\rho^{12})=2L(\rho^6)+3L(\rho^4)+4L(\rho^3)-6L(\rho^2)+\frac{7\pi^2}{10}</math>
+<math>L(\rho^{20})=2L(\rho^{10})+15L(\rho^4)-10L(\rho^2)+\frac{\pi^2}{5}</math>
@@ 55번째 줄: / 55번째 줄: @@
+<h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">Lewin</h5>
+<math>L(\rho^{24})=6L(\rho^{8})+8L(\rho^6)-6L(\rho^4)+\frac{\pi^2}{30}</math>
@@ 113번째 줄: / 115번째 줄: @@
 * http://ko.wikipedia.org/wiki/
 * http://en.wikipedia.org/wiki/
+* http://en.wikipedia.org/wiki/Polylogarithm
+* http://mathworld.wolfram.com/Dilogarithm.html
 * http://www.wolframalpha.com/input/?i=
 * [http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]