查看“︁斯盆司函数”︁的源代码

[[File:Dilog 2d.gif|thumb|'''斯盆司函数'''的二维图像]]

[[File:Dilog 3d.gif|thumb|'''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：

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

== 恒等式 ==
:<math>\operatorname{Li}_2(z)+\operatorname{Li}_2(-z)=\frac{1}{2}\operatorname{Li}_2(z^2)</math><ref name="Zagier">Zagier</ref>
:<math>\operatorname{Li}_2(1-z)+\operatorname{Li}_2\left(1-\frac{1}{z}\right)=-\frac{\ln^2z}{2}</math><ref name="MathWorld">{{cite mathworld|title=Dilogarithm|urlname=Dilogarithm}}</ref>
:<math>\operatorname{Li}_2(z)+\operatorname{Li}_2(1-z)=\frac{{\pi}^2}{6}-\ln z \cdot\ln(1-z) </math><ref name="Zagier"/>
:<math>\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)</math><ref name="MathWorld"/>
:<math>\operatorname{Li}_2(z) +\operatorname{Li}_2(\frac{1}{z}) = - \frac{\pi^2}{6} - \frac{1}{2}\ln^2(-z)</math><ref name="Zagier"/>

== 特殊值恒等式 ==
:<math>\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}</math><ref name="MathWorld"/>
:<math>\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}  </math><ref name="MathWorld"/>
:<math>\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</math> <ref name="MathWorld"/>
:<math>\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</math> <ref name="MathWorld"/>
:<math>\operatorname{Li}_2\left(-\frac{1}{8}\right)+\operatorname{Li}_2\left(\frac{1}{9}\right)=-\frac{1}{2}\ln^2{\frac{9}{8}}</math><ref name="MathWorld"/>
:<math>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</math>

== 特殊值 ==
:<math>\operatorname{Li}_2(-1)=-\frac{{\pi}^2}{12}</math>
:<math>\operatorname{Li}_2(0)=0</math>
:<math>\operatorname{Li}_2\left(\frac{1}{2}\right)=\frac{{\pi}^2}{12}-\frac{\ln^2 2}{2} </math>
:<math>\operatorname{Li}_2(1)=\frac{{\pi}^2}{6}</math>
:<math>\operatorname{Li}_2(2)=\frac{{\pi}^2}{4}-i\pi\ln2</math>
:<math>\operatorname{Li}_2\left(-\frac{\sqrt5-1}{2}\right)=-\frac{{\pi}^2}{15}+\frac{1}{2}\ln^2 \frac{\sqrt5-1}{2} </math> 
:::::::<math>=-\frac{{\pi}^2}{15}+\frac{1}{2}\operatorname{arcsch}^2 2</math>
:<math>\operatorname{Li}_2\left(-\frac{\sqrt5+1}{2}\right)=-\frac{{\pi}^2}{10}-\ln^2 \frac{\sqrt5+1}{2}</math>
:::::::<math>=-\frac{{\pi}^2}{10}-\operatorname{arcsch}^2 2</math>
:<math>\operatorname{Li}_2\left(\frac{3+\sqrt5}{2}\right)=\frac{{\pi}^2}{15}-\frac{1}{2}\ln^2 \frac{\sqrt5-1}{2}</math>
:::::::<math>=\frac{{\pi}^2}{15}-\frac{1}{2}\operatorname{arcsch}^2 2</math>
:<math>\operatorname{Li}_2\left(\frac{\sqrt5+1}{2}\right)=\frac{{\pi}^2}{10}-\ln^2 \frac{\sqrt5-1}{2}</math>
:::::::<math>=\frac{{\pi}^2}{10}-\operatorname{arcsch}^2 2</math>

== 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。
:<math>
\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}
</math>

== 参考文献 ==
<references/>
*{{Cite book | last1=Lewin | first1=L. | title=Dilogarithms and associated functions | publisher=Macdonald | location=London | others=Foreword by J. C. P. Miller | mr=0105524 | year=1958}}
* {{cite journal|first1=Robert
|last1=Morris
|journal=Math. Comp.
|year=1979
|title=The dilogarithm function of a real argument
|url=https://archive.org/details/sim_mathematics-of-computation_1979-04_33_146/page/778
|pages=778–787
|volume=33
|number=146
|doi=10.1090/S0025-5718-1979-0521291-X 
|mr=521291
}}
* {{cite journal
|first=J. H.
|last1=Loxton
|title=Special values of the dilogarithm
|journal=Acta Arith.
|year=1984
|volume=18
|number=2
|pages=155–166
|url=http://pldml.icm.edu.pl/mathbwn/element/bwmeta1.element.bwnjournal-article-aav43i2p155bwm?q=bwmeta1.element.bwnjournal-number-aa-1983-1984-43-2&qt=CHILDREN-STATELESS
|mr=0736728
|access-date=2015-02-02
|archive-date=2015-02-02
|archive-url=https://web.archive.org/web/20150202103557/http://pldml.icm.edu.pl/mathbwn/element/bwmeta1.element.bwnjournal-article-aav43i2p155bwm?q=bwmeta1.element.bwnjournal-number-aa-1983-1984-43-2&qt=CHILDREN-STATELESS
|dead-url=no
}}
* {{cite arxiv
|first=Anatol N.
|last=Kirillov
|title=Dilogarithm identities
|eprint=hep-th/9408113
|year=1994
}}
* {{cite journal
|first1=Carlos
|last1=Osacar
|first2=Jesus
|last2=Palacian
|first3=Manuel
|last3=Palacios
|title=Numerical evaluation of the dilogarithm of complex argument
|url=https://archive.org/details/sim_celestial-mechanics-and-dynamical-astronomy_1995-05_62_1/page/93
|year=1995
|volume=62
|number=1
|pages=93–98
|journal=Celestial Mech. Dynam. Astron.
|doi=10.1007/BF00692071
}}
* {{cite journal
|journal=Front. Number Theory, Physics, Geom. II
|title=The Dilogarithm Function
|first=Don
|last=Zagier
|year=2007
|doi=10.1007/978-3-540-30308-4_1
|pages=3–65
|url=http://maths.dur.ac.uk/~dma0hg/dilog.pdf
|access-date=2015-02-02
|archive-date=2015-03-26
|archive-url=https://web.archive.org/web/20150326183138/http://maths.dur.ac.uk/~dma0hg/dilog.pdf
|dead-url=no
}}

[[Category:特殊函数]]