Thursday, October 9, 2014

Dilogarithm Identities

• $\displaystyle\operatorname{Li}_2\!\left(\tfrac12\right)=\frac{\pi^2}{12}-\frac{\ln^22}2$
• $\displaystyle\operatorname{Li}_2\!\left(\tfrac23\right)=\frac{\pi^2}6+\ln2\cdot\ln3-\ln^23-\operatorname{Li}_2\!\left(\tfrac13\right)$
• $\displaystyle\operatorname{Li}_2\!\left(\tfrac14\right)=\frac{\pi^2}6-2\ln^22+2\ln2\cdot\ln3-\ln^23-2\operatorname{Li}_2\!\left(\tfrac13\right)$
• $\displaystyle\operatorname{Li}_2\!\left(\tfrac34\right)=\ln^22-2\ln^22+2\operatorname{Li}_2\!\left(\tfrac13\right)$
• $\displaystyle\operatorname{Li}_2\!\left(\tfrac35\right)=\frac{\pi^2}6-\ln2\cdot\ln3+\ln2\cdot\ln5+\ln3\cdot\ln5-\ln^25-2\operatorname{Li}_2\!\left(\tfrac25\right)$
• $\displaystyle\operatorname{Li}_2\!\left(\tfrac45\right)=\frac{\pi^2}6+2\ln2\cdot\ln5-\ln^25-\operatorname{Li}_2\!\left(\tfrac15\right)$
• $\displaystyle\operatorname{Li}_2\!\left(\tfrac16\right)=\frac{\pi^2}{12}-\frac{\ln^22}2-2\ln2\cdot\ln3+2\ln2\cdot\ln5+\ln3\cdot\ln5-\ln^25+\operatorname{Li}_2\!\left(\tfrac13\right)-\operatorname{Li}_2\!\left(\tfrac15\right)-\operatorname{Li}_2\!\left(\tfrac25\right)$
• $\displaystyle\operatorname{Li}_2\!\left(\tfrac56\right)=\frac{\pi^2}{12}-\frac{\ln^22}2-\ln^23-\ln2\cdot\ln5+\ln^25-\operatorname{Li}_2\!\left(\tfrac13\right)+\operatorname{Li}_2\!\left(\tfrac15\right)+\operatorname{Li}_2\!\left(\tfrac25\right)$
• More to follow...