"다이로그 함수(dilogarithm)"의 두 판 사이의 차이

2009년 5월 7일 (목) 19:48 판

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

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

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

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

0,\pm 1, 1 \over2,

Don Zagier remarked that "The dilogarithm is the only mathematical function with a sense of humor".

@@ 7번째 줄: / 7번째 줄: @@
 <h5>함수방정식</h5>
-<math>\mbox{Li}_2(-x)=-\mbox{Li}_2 \left( \frac{x}{1+x} \right)-\frac{1}{2}(\ln(x+1))^2 </math>
+<math>\mbox{Li}_2(x)+\mbox{Li}_2 \left( \frac{1}{x} \right) = -\frac{\pi^2}{6}-\frac{1}{2}(\ln(-x))^2 \qquad</math>
-<math>\mbox{Li}_2(x)+\mbox{Li}_2 \left( \frac{1}{x} \right) = -\frac{\pi^2}{6}-\frac{1}{2}(\ln(-x))^2 \qquad</math>
+<math>\mbox{Li}_2 \left(x \right)+\mbox{Li}_2 \left(1-x \right)=  \frac{\pi^2}{6}-\ln(x)\ln(1-x)</math>
+<math>\mbox{Li}_2(-x)=-\mbox{Li}_2 \left( \frac{x}{1+x} \right)-\frac{1}{2}(\ln(x+1))^2 </math>
-<math>\mbox{Li}_2 \left(x \right)+\mbox{Li}_2 \left(1-x \right)=  \frac{\pi^2}{6}-\ln(x)\ln(1-x)</math>
-:<math></math> for <math>~x\not\in~]0;1[,<br> </math><br> and for ''x'' ≥ 1 also<br> :<math>\mbox{Li}_2(x)+\mbox{Li}_2 \left( \frac{1}{x} \right) = \frac{\pi^2}{3}-\frac{1}{2}(\ln(x))^2-i \pi \ln(x).</math>
+<h5>Special values</h5>
+,\pm 1, 1 \over2,
@@ 66번째 줄: / 68번째 줄: @@
 <h5>관련된 다른 주제들</h5>
 * [[수학사연표 (역사)|수학사연표]]
+* [[황금비]]
+* [[리만제타함수|리만제타함수와 리만가설]]
@@ 77번째 줄: / 79번째 줄: @@
 *  Frontiers in number theory, physics, and geometry II<br>
 ** Cartier P., Julia B., Moussa P., Vanhove P.
+*  Polylogarithms and associated functions<br>
+** Lewin L
 *  도서내검색<br>
 ** http://books.google.com/books?q=