斯盆司函数

斯盆司函数的二维图像

斯盆司函数在复数域上的图像

斯盆司函数（英文：Spence's function）也叫二重对数函数（英文：Dilogarithm），最早由欧拉提出，定义如下：

${\mathrm{Li}}_2(z)=-\underset{0}{\overset{z}{\int }}\frac{\ln (1-u)}{u}\mathrm{d}u\text{, }z\in ℂ\setminus [1,\mathrm{\infty })$

恒等式

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

${\mathrm{Li}}_2(1-z)+{\mathrm{Li}}_2(1-\frac{1}{z})=-\frac{{\ln }^{2}z}{2}$

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

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

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

特殊值恒等式

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

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

${\mathrm{Li}}_2\left(\frac{1}{4}\right)+\frac{1}{3}{\mathrm{Li}}_2\left(\frac{1}{9}\right)=\frac{{\pi }^2}{18}+2\ln 2\ln 3-2{\ln }^{2}2-\frac{2}{3}{\ln }^{2}3$

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

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

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

特殊值

${\mathrm{Li}}_2(-1)=-\frac{{\pi }^2}{12}$

${\mathrm{Li}}_2(0)=0$

${\mathrm{Li}}_2\left(\frac{1}{2}\right)=\frac{{\pi }^2}{12}-\frac{{\ln }^{2}2}{2}$

${\mathrm{Li}}_2(1)=\frac{{\pi }^2}{6}$

${\mathrm{Li}}_2(2)=\frac{{\pi }^2}{4}-i\pi \ln 2$

${\mathrm{Li}}_2(-\frac{\sqrt{5}-1}{2})=-\frac{{\pi }^2}{15}+\frac{1}{2}{\ln }^2\frac{\sqrt{5}-1}{2}$

$=-\frac{{\pi }^2}{15}+\frac{1}{2}{\mathrm{arcsch}}^{2}2$

${\mathrm{Li}}_2(-\frac{\sqrt{5}+1}{2})=-\frac{{\pi }^2}{10}-{\ln }^2\frac{\sqrt{5}+1}{2}$

$=-\frac{{\pi }^2}{10}-{\mathrm{arcsch}}^{2}2$

${\mathrm{Li}}_2\left(\frac{3+\sqrt{5}}{2}\right)=\frac{{\pi }^2}{15}-\frac{1}{2}{\ln }^2\frac{\sqrt{5}-1}{2}$

$=\frac{{\pi }^2}{15}-\frac{1}{2}{\mathrm{arcsch}}^{2}2$

${\mathrm{Li}}_2\left(\frac{\sqrt{5}+1}{2}\right)=\frac{{\pi }^2}{10}-{\ln }^2\frac{\sqrt{5}-1}{2}$

$=\frac{{\pi }^2}{10}-{\mathrm{arcsch}}^{2}2$

在粒子物理中

斯盆司函数在粒子物理领域中计算辐射校正时比较常见。此时该函数常用一个含绝对值的对数表达式定义。

$\mathrm{\Phi }(x)=-\underset{0}{\overset{x}{\int }}\frac{\ln |1-u|}{u}du=\{\begin{array}{cc}{\mathrm{Li}}_2(x),& x\le 1;\\ \frac{{\pi }^2}{3}-\frac{1}{2}{\ln }^2(x)-{\mathrm{Li}}_2(\frac{1}{x}),& x>1.\end{array}$

参考文献

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