Dilogarithm

The dilogarithm along the real axis

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

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

and its reflection. For Template:Math, 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 Template:Mvar has hyperbolic volume

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

The function Template:Math 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

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

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

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

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

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

$\mathrm{L}(x)+\mathrm{L}(y)=\mathrm{L}(xy)+\mathrm{L}(\frac{x(1-y)}{1-xy})+\mathrm{L}(\frac{y(1-x)}{1-xy})$. Abel's functional equation or five-term relation where $\mathrm{L}(z)=\frac{\pi }{6}[{\mathrm{Li}}_2(z)+\frac{1}{2}\ln (z)\ln (1-z)]$ is the Rogers L-function (an analogous relation is satisfied also by the quantum dilogarithm)

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

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

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

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

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

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

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

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

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

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

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:

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

See also

• Markstein number

References

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Further reading

• Template:Cite book

External links

• NIST Digital Library of Mathematical Functions: Dilogarithm
• Template:MathWorld