dilogarithm

Image:Mplwp dilogarithm.svg

In mathematics, the dilogarithm (or Spence's function), denoted as {{math|Li₂(z)}}, is a particular case of the polylogarithm. Two related special functions are referred to as Spence's function, the dilogarithm itself:

: $\operatorname{Li}_2(z) = -\int_0^z{\ln(1-u) \over u}\, du \text{, }z \in \Complex$

and its reflection.

For {{math|{{abs|z}} ≤ 1}}, an infinite series also applies (the integral definition constitutes its analytical extension to the complex plane):

: $\operatorname{Li}_2(z) = \sum_{k=1}^\infty {z^k \over k^2}.$

Alternatively, the dilogarithm function is sometimes defined as

: $\int_{1}^{v} \frac{ \ln t }{ 1 -t } dt = \operatorname{Li}_2(1-v).$

In hyperbolic geometry the dilogarithm can be used to compute the volume of an ideal simplex. Specifically, a simplex whose vertices have cross ratio {{mvar|z}} has hyperbolic volume

: $D(z) = \operatorname{Im} \operatorname{Li}_2(z) + \arg(1-z) \log|z|.$

The function {{math|D(z)}} is sometimes called the Bloch-Wigner function.Zagier p. 10 Lobachevsky's function and Clausen's function are closely related functions.

William Spence, after whom the function was named by early writers in the field, was a Scottish mathematician working in the early nineteenth century.{{Cite web| url=https://mathshistory.st-andrews.ac.uk/Biographies/Spence/|title = William Spence - Biography}} He was at school with John Galt,{{Cite web|url=http://www.biographi.ca/009004-119.01-e.php?BioId=37522|title = Biography – GALT, JOHN – Volume VII (1836-1850) – Dictionary of Canadian Biography}} who later wrote a biographical essay on Spence.

Analytic structure

Using the former definition above, the dilogarithm function is analytic everywhere on the complex plane except at $z = 1$ , where it has a logarithmic branch point. The standard choice of branch cut is along the positive real axis $(1, \infty)$ . However, the function is continuous at the branch point and takes on the value $\operatorname{Li}_2(1) = \pi^2/6$ .

Identities

: $\operatorname{Li}_2(z)+\operatorname{Li}_2(-z)=\frac{1}{2}\operatorname{Li}_2(z^2).$ Zagier

: $\operatorname{Li}_2(1-z)+\operatorname{Li}_2\left(1-\frac{1}{z}\right)=-\frac{(\ln z)^2}{2}.$ {{MathWorld|title=Dilogarithm|urlname=Dilogarithm}}

: $\operatorname{Li}_2(z)+\operatorname{Li}_2(1-z)=\frac{{\pi}^2}{6}-\ln z \cdot\ln(1-z).$ The reflection formula.

: $\operatorname{Li}_2(-z)-\operatorname{Li}_2(1-z)+\frac{1}{2}\operatorname{Li}_2(1-z^2)=-\frac{{\pi}^2}{12}-\ln z \cdot \ln(z+1).$

: $\operatorname{Li}_2(z) +\operatorname{Li}_2\left(\frac{1}{z}\right) = - \frac{\pi^2}{6} - \frac{(\ln(-z))^2}{2}.$

: $\operatorname{L}(x)+\operatorname{L}(y)=\operatorname{L}(xy)+\operatorname{L}(\frac{x(1-y)}{1-xy})+\operatorname{L}(\frac{y(1-x)}{1-xy})$ .{{Cite web |last=Weisstein |first=Eric W. |title=Rogers L-Function |url=https://mathworld.wolfram.com/ |access-date=2024-08-01 |website=mathworld.wolfram.com |language=en}}{{Cite journal |last=Rogers |first=L. J. |date=1907 |title=On the Representation of Certain Asymptotic Series as Convergent Continued Fractions |url=http://doi.wiley.com/10.1112/plms/s2-4.1.72 |journal=Proceedings of the London Mathematical Society |language=en |volume=s2-4 |issue=1 |pages=72–89 |doi=10.1112/plms/s2-4.1.72}} Abel's functional equation or five-term relation where $\operatorname{L}(z)=\frac{\pi}{6}[\operatorname{Li}_2(z)+\frac12\ln(z)\ln(1-z)]$ is the Rogers L-function (an analogous relation is satisfied also by the quantum dilogarithm)

Particular value identities

: $\operatorname{Li}_2\left(\frac{1}{3}\right)-\frac{1}{6}\operatorname{Li}_2\left(\frac{1}{9}\right)=\frac{{\pi}^2}{18}-\frac{(\ln 3)^2}{6}.$

: $\operatorname{Li}_2\left(-\frac{1}{3}\right)-\frac{1}{3}\operatorname{Li}_2\left(\frac{1}{9}\right)=-\frac{{\pi}^2}{18}+\frac{(\ln 3)^2}{6}.$

: $\operatorname{Li}_2\left(-\frac{1}{2}\right)+\frac{1}{6}\operatorname{Li}_2\left(\frac{1}{9}\right)=-\frac{{\pi}^2}{18}+\ln2\cdot \ln3-\frac{(\ln 2)^2}{2}-\frac{(\ln 3)^2}{3}.$

: $\operatorname{Li}_2\left(\frac{1}{4}\right)+\frac{1}{3}\operatorname{Li}_2\left(\frac{1}{9}\right)=\frac{{\pi}^2}{18}+2\ln2\cdot\ln3-2(\ln 2)^2-\frac{2}{3}(\ln 3)^2.$

: $\operatorname{Li}_2\left(-\frac{1}{8}\right)+\operatorname{Li}_2\left(\frac{1}{9}\right)=-\frac{1}{2}\left(\ln{\frac{9}{8}}\right)^2.$

: $36\operatorname{Li}_2\left(\frac{1}{2}\right)-36\operatorname{Li}_2\left(\frac{1}{4}\right)-12\operatorname{Li}_2\left(\frac{1}{8}\right)+6\operatorname{Li}_2\left(\frac{1}{64}\right)={\pi}^2.$

Special values

: $\operatorname{Li}_2(-1)=-\frac{{\pi}^2}{12}.$

: $\operatorname{Li}_2(0)=0.$ Its slope = 1.

: $\operatorname{Li}_2\left(\frac{1}{2}\right)=\frac{{\pi}^2}{12}-\frac{(\ln 2)^2}{2}.$

: $\operatorname{Li}_2(1) = \zeta(2) = \frac{{\pi}^2}{6},$ where $\zeta(s)$ is the Riemann zeta function.

: $\operatorname{Li}_2(2)=\frac{{\pi}^2}{4}-i\pi\ln2.$

: $\begin{align}
\operatorname{Li}_2\left(-\frac{\sqrt5-1}{2}\right)
&=-\frac{{\pi}^2}{15}+\frac{1}{2}\left(\ln\frac{\sqrt5+1}{2}\right)^2 \\
&=-\frac{{\pi}^2}{15}+\frac{1}{2}\operatorname{arcsch}^2 2.
\end{align}$

: $\begin{align}
\operatorname{Li}_2\left(-\frac{\sqrt5+1}{2}\right)
&=-\frac{{\pi}^2}{10}-\ln^2 \frac{\sqrt5+1}{2} \\
&=-\frac{{\pi}^2}{10}-\operatorname{arcsch}^2 2.
\end{align}$

: $\begin{align}
\operatorname{Li}_2\left(\frac{3-\sqrt5}{2}\right)
&=\frac{{\pi}^2}{15}-\ln^2 \frac{\sqrt5+1}{2} \\
&=\frac{{\pi}^2}{15}-\operatorname{arcsch}^2 2.
\end{align}$

: $\begin{align}
\operatorname{Li}_2\left(\frac{\sqrt5-1}{2}\right)
&=\frac{{\pi}^2}{10}-\ln^2 \frac{\sqrt5+1}{2} \\
&=\frac{{\pi}^2}{10}-\operatorname{arcsch}^2 2.
\end{align}$

In particle physics

Spence's Function is commonly encountered in particle physics while calculating radiative corrections. In this context, the function is often defined with an absolute value inside the logarithm:

: $\operatorname{\Phi}(x) = -\int_0^x \frac{\ln|1-u|}{u} \, du =
\begin{cases}
\operatorname{Li}_2(x), & x \leq 1; \\ \frac{\pi^2}{3} - \frac{1}{2}(\ln x)^2 - \operatorname{Li}_2(\frac{1}{x}), & x > 1.
\end{cases}$

Notes

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

|pages=778–787

|volume=33

|number=146

|doi=10.1090/S0025-5718-1979-0521291-X

|mr=521291

|doi-access=free

}}

{{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

|doi=10.4064/aa-43-2-155-166

|doi-access=free

}}

{{cite journal

|first=Anatol N.

|last=Kirillov

|title=Dilogarithm identities

|arxiv=hep-th/9408113

|year=1995

|doi=10.1143/PTPS.118.61

|volume=118

|journal=Progress of Theoretical Physics Supplement

|pages=61–142

|bibcode=1995PThPS.118...61K

|s2cid=119177149

}}

{{cite journal

|first1=Carlos

|last1=Osacar

|first2=Jesus

|last2=Palacian

|first3=Manuel

|last3=Palacios

|title=Numerical evaluation of the dilogarithm of complex argument

|year=1995

|volume=62

|number=1

|pages=93–98

|journal=Celest. Mech. Dyn. Astron.

|doi=10.1007/BF00692071

|bibcode=1995CeMDA..62...93O

|s2cid=121304484

}}

{{ cite book

|first=Don |last=Zagier

|year=2007

|chapter=The Dilogarithm Function

|title=Frontiers in Number Theory, Physics, and Geometry II

|editor1=Pierre Cartier |editor2=Pierre Moussa |editor3=Bernard Julia |editor4=Pierre Vanhove

|pages=3–65

|url=http://maths.dur.ac.uk/~dma0hg/dilog.pdf

|doi=10.1007/978-3-540-30308-4_1

|isbn=978-3-540-30308-4

}}

External links

[http://dlmf.nist.gov/25.12 NIST Digital Library of Mathematical Functions: Dilogarithm]
{{MathWorld|title=Dilogarithm|urlname=Dilogarithm}}

Category:Special functions