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| {{redirect|Li2|the molecule with formula Li<sub>2</sub>|dilithium}}
| | I would like to introduce myself to you, I am Andrew and my wife doesn't like it at all. Ohio is exactly where his home is and his family loves it. Distributing manufacturing has been his profession for some time. The preferred pastime for him and his children is style and he'll be starting some thing else along with it.<br><br>Also visit my web site; good psychic ([http://ltreme.com/index.php?do=/profile-127790/info/ ltreme.com]) |
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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. | |
| 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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I would like to introduce myself to you, I am Andrew and my wife doesn't like it at all. Ohio is exactly where his home is and his family loves it. Distributing manufacturing has been his profession for some time. The preferred pastime for him and his children is style and he'll be starting some thing else along with it.
Also visit my web site; good psychic (ltreme.com)