斯盆司函数 Spence's function

（重定向自Dilogarithm）

Spence Function of Dilog function complex

斯盆司函数最早由欧拉提出，别称Dilogarithm,定义如下：

\operatorname{Li}_2(z) = -\int_0^z{\ln(1-u) \over u}\, \mathrm{d}u \text{, }z \in\mathbb{C} \setminus [1,\infty)

\operatorname{Li}_2(z)+\operatorname{Li}_2(-z)=\frac{1}{2}\operatorname{Li}_2(z^2)

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

\operatorname{Li}_2(z)+\operatorname{Li}_2(1-z)=\frac{{\pi}^2}{6\ln z \cdot\ln(1-z)

\operatorname{Li}_2(-z)-\operatorname{Li}_2(1-z)+\frac{1}{2}\operatorname{Li}_2(1-z^2)=-\frac {{\pi}^2}{12\ln z \cdot \ln(z+1)

\operatorname{Li}_2(z) +\operatorname{Li}_2(\frac{1}{z}) = - \frac{\pi^2}{6} - \frac{1}{2}\ln^2(-z)

\operatorname{Li}_2\left(\frac{1}{3}\right)-\frac{1}{6}\operatorname{Li}_2\left(\frac{1}{9}\right)=\frac{{\pi}^2}{18\frac{\ln^23}{6}

\operatorname{Li}_2\left(-\frac{1}{2}\right)+\frac{1}{6}\operatorname{Li}_2\left(\frac{1}{9}\right)=-\frac{{\pi}^2}{18}+\ln2\cdot \ln3-\frac{\ln^22}{2\frac{\ln^23}{3}

\operatorname{Li}_2\left(\frac{1}{4}\right)+\frac{1}{3}\operatorname{Li}_2\left(\frac{1}{9}\right)=\frac{{\pi}^2}{18}+2\ln2\ln3-2\ln^22-\frac{2}{3}\ln^23

\operatorname{Li}_2\left(-\frac{1}{3}\right)-\frac{1}{3}\operatorname{Li}_2\left(\frac{1}{9}\right)=-\frac{{\pi}^2}{18}+\frac{1}{6}\ln^23

\operatorname{Li}_2\left(-\frac{1}{8}\right)+\operatorname{Li}_2\left(\frac{1}{9}\right)=-\frac{1}{2}\ln^2{\frac{9}{8}}

36\operatorname{Li}_2\left(\frac{1}{2}\right)-36\operatorname{Li}_2\left(\frac{1}{4}\right)-12\operatorname{Li}_2\left(\frac{1}{8}\right)+6\operatorname{Li}_2\left(\frac{1}{64}\right)={\pi}^2

单词	Dilogarithm
释义	Dilogarithm 中文百科斯盆司函数 Spence's function （重定向自Dilogarithm）斯盆司函数最早由欧拉提出，别称Dilogarithm,定义如下： $\operatorname{Li}_2(z) = -\int_0^z{\ln(1-u) \over u}\, \mathrm{d}u \text{, }z \in\mathbb{C} \setminus [1,\infty)$ $\operatorname{Li}_2(z)+\operatorname{Li}_2(-z)=\frac{1}{2}\operatorname{Li}_2(z^2)$ $\operatorname{Li}_2(1-z)+\operatorname{Li}_2\left(1-\frac{1}{z}\right)=-\frac{\ln^2z}{2}$ $\operatorname{Li}_2(z)+\operatorname{Li}_2(1-z)=\frac{{\pi}^2}{6\ln z \cdot\ln(1-z)$ $\operatorname{Li}_2(-z)-\operatorname{Li}_2(1-z)+\frac{1}{2}\operatorname{Li}_2(1-z^2)=-\frac {{\pi}^2}{12\ln z \cdot \ln(z+1)$ $\operatorname{Li}_2(z) +\operatorname{Li}_2(\frac{1}{z}) = - \frac{\pi^2}{6} - \frac{1}{2}\ln^2(-z)$ $\operatorname{Li}_2\left(\frac{1}{3}\right)-\frac{1}{6}\operatorname{Li}_2\left(\frac{1}{9}\right)=\frac{{\pi}^2}{18\frac{\ln^23}{6}$ $\operatorname{Li}_2\left(-\frac{1}{2}\right)+\frac{1}{6}\operatorname{Li}_2\left(\frac{1}{9}\right)=-\frac{{\pi}^2}{18}+\ln2\cdot \ln3-\frac{\ln^22}{2\frac{\ln^23}{3}$ $\operatorname{Li}_2\left(\frac{1}{4}\right)+\frac{1}{3}\operatorname{Li}_2\left(\frac{1}{9}\right)=\frac{{\pi}^2}{18}+2\ln2\ln3-2\ln^22-\frac{2}{3}\ln^23$ $\operatorname{Li}_2\left(-\frac{1}{3}\right)-\frac{1}{3}\operatorname{Li}_2\left(\frac{1}{9}\right)=-\frac{{\pi}^2}{18}+\frac{1}{6}\ln^23$ $\operatorname{Li}_2\left(-\frac{1}{8}\right)+\operatorname{Li}_2\left(\frac{1}{9}\right)=-\frac{1}{2}\ln^2{\frac{9}{8}}$ $36\operatorname{Li}_2\left(\frac{1}{2}\right)-36\operatorname{Li}_2\left(\frac{1}{4}\right)-12\operatorname{Li}_2\left(\frac{1}{8}\right)+6\operatorname{Li}_2\left(\frac{1}{64}\right)={\pi}^2$ 英语百科 Spence's function 斯盆司函数（重定向自Dilogarithm） In mathematics, Spence's function, or dilogarithm, denoted as Li₂(z), is a particular case of the polylogarithm. Two related special functions are referred to as Spence's function, the dilogarithm itself: and its reflection. For $\|z\|<1$ an infinite series also applies (the integral definition constitutes its analytical extension to the complex plane):
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Dilogarithm

斯盆司函数 Spence's function

Spence's function 斯盆司函数