Dilogarithm

Short description: Special case of the polylogarithm

The dilogarithm along the real axis

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

${\mathrm{Li}}_2(z)=-\underset{0}{\overset{z}{\int }}\frac{\ln (1-u)}{u}du\text{, }z\in \mathbb{ℂ}$

and its reflection. For |z| < 1, an infinite series also applies (the integral definition constitutes its analytical extension to the complex plane):

${\mathrm{Li}}_2(z)=\sum _{k=1}^{\mathrm{\infty }}\frac{z^k}{k^2}.$

Alternatively, the dilogarithm function is sometimes defined as

$\underset{1}{\overset{v}{\int }}\frac{\ln t}{1-t}dt={\mathrm{Li}}_2(1-v).$

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

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

The function D(z) is sometimes called the Bloch-Wigner function. 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. He was at school with John Galt, 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 $(1,\mathrm{\infty })$. However, the function is continuous at the branch point and takes on the value ${\mathrm{Li}}_2(1)={\pi }^2/6$.

Identities

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

${\mathrm{Li}}_2(1-z)+{\mathrm{Li}}_2(1-\frac{1}{z})=-\frac{(\ln z{)}^2}{2}.$

${\mathrm{Li}}_2(z)+{\mathrm{Li}}_2(1-z)=\frac{{\pi }^2}{6}-\ln z\cdot \ln (1-z).$

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

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

Particular value identities

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

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

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

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

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

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

Special values

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

${\mathrm{Li}}_2(0)=0.$

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

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

${\mathrm{Li}}_2(2)=\frac{{\pi }^2}{4}-i\pi \ln 2.$

$\begin{array}{rl}{\mathrm{Li}}_2(-\frac{\sqrt{5}-1}{2})& =-\frac{{\pi }^2}{15}+\frac{1}{2}{(\ln \frac{\sqrt{5}+1}{2})}^2\\ & =-\frac{{\pi }^2}{15}+\frac{1}{2}{\mathrm{arcsch}}^{2}2.\end{array}$

$\begin{array}{rl}{\mathrm{Li}}_2(-\frac{\sqrt{5}+1}{2})& =-\frac{{\pi }^2}{10}-{\ln }^2\frac{\sqrt{5}+1}{2}\\ & =-\frac{{\pi }^2}{10}-{\mathrm{arcsch}}^{2}2.\end{array}$

$\begin{array}{rl}{\mathrm{Li}}_2\left(\frac{3-\sqrt{5}}{2}\right)& =\frac{{\pi }^2}{15}-{\ln }^2\frac{\sqrt{5}+1}{2}\\ & =\frac{{\pi }^2}{15}-{\mathrm{arcsch}}^{2}2.\end{array}$

$\begin{array}{rl}{\mathrm{Li}}_2\left(\frac{\sqrt{5}-1}{2}\right)& =\frac{{\pi }^2}{10}-{\ln }^2\frac{\sqrt{5}+1}{2}\\ & =\frac{{\pi }^2}{10}-{\mathrm{arcsch}}^{2}2.\end{array}$

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:

$\mathrm{\Phi }(x)=-\underset{0}{\overset{x}{\int }}\frac{\ln |1-u|}{u}du=\{\begin{array}{ll}{\mathrm{Li}}_2(x),& x\le 1;\\ \frac{{\pi }^2}{3}-\frac{1}{2}(\ln x{)}^2-{\mathrm{Li}}_2(\frac{1}{x}),& x>1.\end{array}$

See also

• Markstein number

References

• Lewin, L. (1958). Dilogarithms and associated functions. Foreword by J. C. P. Miller. London: Macdonald. 
• Morris, Robert (1979). "The dilogarithm function of a real argument". Math. Comp. 33 (146): 778–787. doi:10.1090/S0025-5718-1979-0521291-X. 
• Loxton, J. H. (1984). "Special values of the dilogarithm". Acta Arith. 18 (2): 155–166. doi:10.4064/aa-43-2-155-166. 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. 
• Kirillov, Anatol N. (1995). "Dilogarithm identities". Progress of Theoretical Physics Supplement 118: 61–142. doi:10.1143/PTPS.118.61. Bibcode: 1995PThPS.118...61K. 
• Osacar, Carlos; Palacian, Jesus; Palacios, Manuel (1995). "Numerical evaluation of the dilogarithm of complex argument". Celest. Mech. Dyn. Astron. 62 (1): 93–98. doi:10.1007/BF00692071. Bibcode: 1995CeMDA..62...93O. 
• Zagier, Don (2007). "The Dilogarithm Function". Frontiers in Number Theory, Physics, and Geometry II. pp. 3–65. doi:10.1007/978-3-540-30308-4_1. ISBN 978-3-540-30308-4. http://maths.dur.ac.uk/~dma0hg/dilog.pdf. 

Further reading

• Bloch, Spencer J. (2000). Higher regulators, algebraic K-theory, and zeta functions of elliptic curves. CRM Monograph Series. 11. Providence, RI: American Mathematical Society. ISBN 0-8218-2114-8. 

External links

• NIST Digital Library of Mathematical Functions: Dilogarithm
• Weisstein, Eric W.. "Dilogarithm". http://mathworld.wolfram.com/Dilogarithm.html.