**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：

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

恒等式
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{\displaystyle \operatorname {Li} \_{2}(z)+\operatorname {Li} \_{2}(-z)={\frac {1}{2}}\operatorname {Li} \_{2}(z^{2})}
{\displaystyle \operatorname {Li} \_{2}(1-z)+\operatorname {Li} \_{2}\left(1-{\frac {1}{z}}\right)=-{\frac {\ln ^{2}z}{2}}}
{\displaystyle \operatorname {Li} \_{2}(z)+\operatorname {Li} \_{2}(1-z)={\frac {{\pi }^{2}}{6}}-\ln z\cdot \ln(1-z)}
{\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)}
{\displaystyle \operatorname {Li} \_{2}(z)+\operatorname {Li} \_{2}({\frac {1}{z}})=-{\frac {\pi ^{2}}{6}}-{\frac {1}{2}}\ln ^{2}(-z)}

特殊值恒等式
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{\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}}}
{\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}}}
{\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} 
{\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} 
{\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}}}
{\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
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{\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
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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}}}

参考文献
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1. 1 2 3  Zagier
2. 1 2 3 4 5 6 7  Weisstein, Eric W. (编).. at *MathWorld*--A Wolfram Web Resource. Wolfram Research, Inc. （英语）.

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