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| {{redirect|Li2|the molecule with formula Li<sub>2</sub>|dilithium}}
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| {{See also|polylogarithm#Dilogarithm}}
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| {{technical|date=November 2012}}
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| In [[mathematics]], '''Spence's function''', or '''dilogarithm''', denoted as Li<sub>2</sub>(''z''), is a particular case of the [[polylogarithm]]. Two related [[special functions]] are referred to as Spence's function, the dilogarithm itself:
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| ::<math>
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| \operatorname{Li}_2(z) = -\int_0^z{\ln(1-u) \over u}\, \mathrm{d}u \text{, }z \in\mathbb{C} \setminus [1,\infty)
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| </math>
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| and its reflection.
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| For <math>|z|<1</math> an infinite series also applies (the integral definition constitutes its analytical extension to the complex plane):
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| ::<math>
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| \operatorname{Li}_2(z) = \sum_{k=1}^\infty {z^k \over k^2}.
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| </math>
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| Alternatively, the dilogarithm function is sometimes defined as
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| ::<math>
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| \int_{1}^{v} \frac{ \ln t }{ 1 -t } \mathrm{d}t = \operatorname{Li}_2(1-v).
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| </math>
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| In [[hyperbolic geometry]] the dilogarithm <math>\operatorname{Li}_2(z)
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| </math> occurs as the [[hyperbolic volume]] of an [[ideal simplex]] whose ideal vertices have [[cross ratio]] <math>z</math>. '''[[Lobachevsky's function]]''' and '''[[Clausen's function]]''' are closely related functions.
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| William Spence, after whom the function was named by early writers in the field, was a Scottish mathematician working in the early nineteenth century.<ref>http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Spence.html</ref> He was at school with [[John Galt (novelist)|John Galt]],<ref>http://www.biographi.ca/009004-119.01-e.php?BioId=37522</ref> who later wrote a biographical essay on Spence.
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| ==Identities==
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| :<math>\operatorname{Li}_2(z)+\operatorname{Li}_2(-z)=\frac{1}{2}\operatorname{Li}_2(z^2)</math><ref name="Zagier">Zagier</ref>
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| :<math>\operatorname{Li}_2(1-z)+\operatorname{Li}_2\left(1-\frac{1}{z}\right)=-\frac{\ln^2z}{2}</math><ref name="MathWorld">{{MathWorld|title=Dilogarithm|urlname=Dilogarithm}}</ref>
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| :<math>\operatorname{Li}_2(z)+\operatorname{Li}_2(1-z)=\frac{{\pi}^2}{6}-\ln z \cdot\ln(1-z) </math><ref name="Zagier"/>
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| :<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"/>
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| :<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"/>
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| ==Particular value identities==
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| :<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"/>
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| :<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"/>
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| :<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"/>
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| :<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"/>
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| :<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"/>
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| :<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>
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| ==Special values==
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| :<math>\operatorname{Li}_2(-1)=-\frac{{\pi}^2}{12}</math>
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| :<math>\operatorname{Li}_2(0)=0</math>
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| :<math>\operatorname{Li}_2\left(\frac{1}{2}\right)=\frac{{\pi}^2}{12}-\frac{\ln^2 2}{2} </math>
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| :<math>\operatorname{Li}_2(1)=\frac{{\pi}^2}{6}</math>
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| :<math>\operatorname{Li}_2(2)=\frac{{\pi}^2}{4}-i\pi\ln2</math>
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| :<math>\operatorname{Li}_2\left(-\frac{\sqrt5-1}{2}\right)=-\frac{{\pi}^2}{15}+\frac{1}{2}\ln^2 \frac{\sqrt5-1}{2} </math>
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| :::::::<math>=-\frac{{\pi}^2}{15}+\frac{1}{2}\operatorname{arcsch}^2 2</math>
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| :<math>\operatorname{Li}_2\left(-\frac{\sqrt5+1}{2}\right)=-\frac{{\pi}^2}{10}-\ln^2 \frac{\sqrt5+1}{2}</math>
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| :::::::<math>=-\frac{{\pi}^2}{10}-\operatorname{arcsch}^2 2</math>
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| :<math>\operatorname{Li}_2\left(\frac{3+\sqrt5}{2}\right)=\frac{{\pi}^2}{15}-\frac{1}{2}\ln^2 \frac{\sqrt5-1}{2}</math>
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| :::::::<math>=\frac{{\pi}^2}{15}-\frac{1}{2}\operatorname{arcsch}^2 2</math>
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| :<math>\operatorname{Li}_2\left(\frac{\sqrt5+1}{2}\right)=\frac{{\pi}^2}{10}-\ln^2 \frac{\sqrt5-1}{2}</math>
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| :::::::<math>=\frac{{\pi}^2}{10}-\operatorname{arcsch}^2 2</math>
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| *{{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}}
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| * {{cite journal|first1=Robert
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| |last1=Morris
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| |journal=Math. Comp.
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| |year=1979
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| |title=The dilogarithm function of a real argument
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| |pages=778–787
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| |volume=33
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| |number=146
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| |doi=10.1090/S0025-5718-1979-0521291-X
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| |mr=521291
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| }}
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| * {{cite journal
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| |first=J. H.
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| |last1=Loxton
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| |title=Special values of the dilogarithm
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| |journal=Acta Arith.
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| |year=1984
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| |volume=18
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| |number=2
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| |pages=155–166
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| |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
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| |mr=0736728
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| }}
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| * {{cite arxiv
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| |first=Anatol N.
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| |last=Kirillov
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| |title=Dilogarithm identities
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| |eprint=hep-th/9408113
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| |year=1994
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| }}
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| * {{cite journal
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| |first1=Carlos
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| |last1=Osacar
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| |first2=Jesus
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| |last2=Palacian
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| |first3=Manuel
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| |last3=Palacios
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| |title=Numerical evaluation of the dilogarithm of complex argument
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| |year=1995
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| |volume=62
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| |number=1
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| |pages=93–98
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| |journal=Celestial Mech. Dynam. Astron.
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| |doi=10.1007/BF00692071
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| }}
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| * {{ cite journal
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| |journal=Front. Number Theory, Physics, Geom. II
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| |title=The Dilogarithm Function
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| |first=Don
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| |last=Zagier
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| |year=2007
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| |doi=10.1007/978-3-540-30308-4_1
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| |pages=3–65
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| |url=http://maths.dur.ac.uk/~dma0hg/dilog.pdf
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| }}
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| ==Further reading==
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| * {{cite book | last=Bloch | first=Spencer J. | authorlink=Spencer Bloch | title=Higher regulators, algebraic ''K''-theory, and zeta functions of elliptic curves | series=CRM Monograph Series | volume=11 | location=Providence, RI | publisher=[[American Mathematical Society]] | year=2000 | isbn=0-8218-2114-8 | zbl=0958.19001 }}
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| == External links ==
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| *[http://dlmf.nist.gov/25.12 NIST Digital Library of Mathematical Functions: Dilogarithm]
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| * {{MathWorld|title=Dilogarithm|urlname=Dilogarithm}}
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| [[Category:Special functions]]
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