斯盆司函数

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：

${\displaystyle \operatorname {Li} _{2}(z)=-\int _{0}^{z}{\ln(1-u) \over u}\,\mathrm {d} u{\text{, }}z\in \mathbb {C} \setminus [1,\infty )}$

斯盆司函数的二维图像

斯盆司函数在复数域上的图像

恒等式

${\displaystyle \operatorname {Li} _{2}(z)+\operatorname {Li} _{2}(-z)={\frac {1}{2}}\operatorname {Li} _{2}(z^{2})}$^[1]

${\displaystyle \operatorname {Li} _{2}(1-z)+\operatorname {Li} _{2}\left(1-{\frac {1}{z}}\right)=-{\frac {\ln ^{2}z}{2}}}$^[2]

${\displaystyle \operatorname {Li} _{2}(z)+\operatorname {Li} _{2}(1-z)={\frac {{\pi }^{2}}{6}}-\ln z\cdot \ln(1-z)}$^[1]

${\displaystyle \operatorname {Li} _{2}(-z)-\operatorname {Li} _{2}(1-z)+{\frac {1}{2}}\operatorname {Li} _{2}(1-z^{2})=-{\frac {{\pi }^{2}}{12}}-\ln z\cdot \ln(z+1)}$^[2]

${\displaystyle \operatorname {Li} _{2}(z)+\operatorname {Li} _{2}({\frac {1}{z}})=-{\frac {\pi ^{2}}{6}}-{\frac {1}{2}}\ln ^{2}(-z)}$^[1]

特殊值恒等式

${\displaystyle \operatorname {Li} _{2}\left({\frac {1}{3}}\right)-{\frac {1}{6}}\operatorname {Li} _{2}\left({\frac {1}{9}}\right)={\frac {{\pi }^{2}}{18}}-{\frac {\ln ^{2}3}{6}}}$^[2]

${\displaystyle \operatorname {Li} _{2}\left(-{\frac {1}{2}}\right)+{\frac {1}{6}}\operatorname {Li} _{2}\left({\frac {1}{9}}\right)=-{\frac {{\pi }^{2}}{18}}+\ln 2\cdot \ln 3-{\frac {\ln ^{2}2}{2}}-{\frac {\ln ^{2}3}{3}}}$^[2]

${\displaystyle \operatorname {Li} _{2}\left({\frac {1}{4}}\right)+{\frac {1}{3}}\operatorname {Li} _{2}\left({\frac {1}{9}}\right)={\frac {{\pi }^{2}}{18}}+2\ln 2\ln 3-2\ln ^{2}2-{\frac {2}{3}}\ln ^{2}3}$ ^[2]

${\displaystyle \operatorname {Li} _{2}\left(-{\frac {1}{3}}\right)-{\frac {1}{3}}\operatorname {Li} _{2}\left({\frac {1}{9}}\right)=-{\frac {{\pi }^{2}}{18}}+{\frac {1}{6}}\ln ^{2}3}$ ^[2]

${\displaystyle \operatorname {Li} _{2}\left(-{\frac {1}{8}}\right)+\operatorname {Li} _{2}\left({\frac {1}{9}}\right)=-{\frac {1}{2}}\ln ^{2}{\frac {9}{8}}}$^[2]

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

特殊值

${\displaystyle \operatorname {Li} _{2}(-1)=-{\frac {{\pi }^{2}}{12}}}$

${\displaystyle \operatorname {Li} _{2}(0)=0}$

${\displaystyle \operatorname {Li} _{2}\left({\frac {1}{2}}\right)={\frac {{\pi }^{2}}{12}}-{\frac {\ln ^{2}2}{2}}}$

${\displaystyle \operatorname {Li} _{2}(1)={\frac {{\pi }^{2}}{6}}}$

${\displaystyle \operatorname {Li} _{2}(2)={\frac {{\pi }^{2}}{4}}-i\pi \ln 2}$

${\displaystyle \operatorname {Li} _{2}\left(-{\frac {{\sqrt {5}}-1}{2}}\right)=-{\frac {{\pi }^{2}}{15}}+{\frac {1}{2}}\ln ^{2}{\frac {{\sqrt {5}}-1}{2}}}$

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

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

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

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

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

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

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

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。

${\displaystyle \operatorname {\Phi } (x)=-\int _{0}^{x}{\frac {\ln |1-u|}{u}}\,du={\begin{cases}\operatorname {Li} _{2}(x),&x\leq 1;\\{\frac {\pi ^{2}}{3}}-{\frac {1}{2}}\ln ^{2}(x)-\operatorname {Li} _{2}({\frac {1}{x}}),&x>1.\end{cases}}}$