"다이로그 함수(dilogarithm)"의 두 판 사이의 차이

2009년 5월 8일 (금) 03:21 판

\(\operatorname{Li}_2(z) = -\int_0^z{{\ln (1-t)}\over t} dt = \sum_{k=1}^\infty {z^k \over k^2}\)

\(\mbox{Li}_2(x)+\mbox{Li}_2 \left( \frac{1}{x} \right) = -\frac{\pi^2}{6}-\frac{1}{2}(\ln(-x))^2 \qquad\)

\(\mbox{Li}_2 \left(x \right)+\mbox{Li}_2 \left(1-x \right)= \frac{\pi^2}{6}-\ln(x)\ln(1-x)\)

\(\mbox{Li}_2(-x)=-\mbox{Li}_2 \left( \frac{x}{1+x} \right)-\frac{1}{2}(\ln(x+1))^2 \)

다음 값들은 알려져 있음.

\(\mbox{Li}_{2}(0)=0\)

\(\mbox{Li}_{2}(1)=\frac{\pi^2}{6}\)

\(\mbox{Li}_{2}(-1)=-\frac{\pi^2}{12}\)

\(\mbox{Li}_{2}(\frac{1}{2})=\frac{\pi^2}{12}-\frac{1}{2}\log^2(2)\)

\(\mbox{Li}_{2}(\frac{3-\sqrt{5}}{2})=\frac{\pi^2}{15}-\log^2(\frac{1+\sqrt{5}}{2})\)

\(\mbox{Li}_{2}(\frac{-1+\sqrt{5}}{2})=\frac{\pi^2}{10}-\log^2(\frac{1+\sqrt{5}}{2})\)

\(\mbox{Li}_{2}(\frac{1-\sqrt{5}}{2})=-\frac{\pi^2}{15}+\frac{1}{2}\log^2(\frac{1+\sqrt{5}}{2})\)

\(\mbox{Li}_{2}(\frac{-1-\sqrt{5}}{2})=-\frac{\pi^2}{10}+\frac{1}{2}\log^2(\frac{1+\sqrt{5}}{2})\)

The dilogarithm is the only mathematical function with a sense of humor

@@ 21번째 줄: / 21번째 줄: @@
 <h5>Special values</h5>
-<math>0,\pm 1, {1 \over2}, \pm 1 \pm{{\sqrt5} \over {2}} </math> 에서의 값은 알려져 있음.
+다음 값들은 알려져 있음.
 <math>\mbox{Li}_{2}(0)=0</math>
@@ 31번째 줄: / 31번째 줄: @@
 <math>\mbox{Li}_{2}(\frac{1}{2})=\frac{\pi^2}{12}-\frac{1}{2}\log^2(2)</math>
-<math>\mbox{Li}_{2}(\frac{1}{2})=\frac{\pi^2}{12}-\frac{1}{2}\log^2(2)</math>
+<math>\mbox{Li}_{2}(\frac{3-\sqrt{5}}{2})=\frac{\pi^2}{15}-\log^2(\frac{1+\sqrt{5}}{2})</math>
-<math>\mbox{Li}_{2}(0)=0</math>
-<math>\mbox{Li}_{2}(0)=0</math>
+<math>\mbox{Li}_{2}(\frac{-1+\sqrt{5}}{2})=\frac{\pi^2}{10}-\log^2(\frac{1+\sqrt{5}}{2})</math>
-<math>\mbox{Li}_{2}(0)=0</math>
+<math>\mbox{Li}_{2}(\frac{1-\sqrt{5}}{2})=-\frac{\pi^2}{15}+\frac{1}{2}\log^2(\frac{1+\sqrt{5}}{2})</math>
-<math>\mbox{Li}_{2}(0)=0</math>
+<math>\mbox{Li}_{2}(\frac{-1-\sqrt{5}}{2})=-\frac{\pi^2}{10}+\frac{1}{2}\log^2(\frac{1+\sqrt{5}}{2})</math>
+* [[황금비]]
@@ 68번째 줄: / 64번째 줄: @@
 <h5>재미있는 사실</h5>
-* Don Zagier remarked that "The dilogarithm is the only mathematical function with a sense of humor".
+* Don Zagier
+<blockquote>
+The dilogarithm is the only mathematical function with a sense of humor
+</blockquote>