Dilogarithm

In mathematics, the dilogarithm (or Spence's function), denoted as $Li 2 (z)$ , is a particular case of the polylogarithm. Two related special functions are referred to as Spence's function, the dilogarithm itself:

{Li}_{2} (z) = - \int_{0}^{z} \frac{\ln (1 - u)}{u} d u, z \in ℂ

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

{Li}_{2} (z) = \sum_{k = 1}^{\infty} \frac{z^{k}}{k^{2}} .

Alternatively, the dilogarithm function is sometimes defined as

\int_{1}^{v} \frac{\ln t}{1 - t} d t = {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) = Im {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 $(1, \infty)$ . However, the function is continuous at the branch point and takes on the value ${Li}_{2} (1) = π^{2} / 6$ .

Identities

{Li}_{2} (z) + {Li}_{2} (- z) = \frac{1}{2} {Li}_{2} (z^{2}) .

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

{Li}_{2} (z) + {Li}_{2} (1 - z) = \frac{π^{2}}{6} - \ln z \cdot \ln (1 - z) .

The reflection formula.

{Li}_{2} (- z) - {Li}_{2} (1 - z) + \frac{1}{2} {Li}_{2} (1 - z^{2}) = - \frac{π^{2}}{12} - \ln z \cdot \ln (z + 1) .

{Li}_{2} (z) + {Li}_{2} (\frac{1}{z}) = - \frac{π^{2}}{6} - \frac{(\ln (- z))^{2}}{2} .

L (x) + L (y) = L (x y) + L (\frac{x (1 - y)}{1 - x y}) + L (\frac{y (1 - x)}{1 - x y})

. Abel's functional equation or five-term relation where

L (z) = \frac{π^{2}}{6} [{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

{Li}_{2} (\frac{1}{3}) - \frac{1}{6} {Li}_{2} (\frac{1}{9}) = \frac{π^{2}}{18} - \frac{(\ln 3)^{2}}{6} .

{Li}_{2} (- \frac{1}{3}) - \frac{1}{3} {Li}_{2} (\frac{1}{9}) = - \frac{π^{2}}{18} + \frac{(\ln 3)^{2}}{6} .

{Li}_{2} (- \frac{1}{2}) + \frac{1}{6} {Li}_{2} (\frac{1}{9}) = - \frac{π^{2}}{18} + \ln 2 \cdot \ln 3 - \frac{(\ln 2)^{2}}{2} - \frac{(\ln 3)^{2}}{3} .

{Li}_{2} (\frac{1}{4}) + \frac{1}{3} {Li}_{2} (\frac{1}{9}) = \frac{π^{2}}{18} + 2 \ln 2 \cdot \ln 3 - 2 (\ln 2)^{2} - \frac{2}{3} (\ln 3)^{2} .

{Li}_{2} (- \frac{1}{8}) + {Li}_{2} (\frac{1}{9}) = - \frac{1}{2} {(\ln \frac{9}{8})}^{2} .

36 {Li}_{2} (\frac{1}{2}) - 36 {Li}_{2} (\frac{1}{4}) - 12 {Li}_{2} (\frac{1}{8}) + 6 {Li}_{2} (\frac{1}{64}) = π^{2} .

Special values

{Li}_{2} (- 1) = - \frac{π^{2}}{12} .

{Li}_{2} (0) = 0.

Its slope = 1.

{Li}_{2} (\frac{1}{2}) = \frac{π^{2}}{12} - \frac{(\ln 2)^{2}}{2} .

{Li}_{2} (1) = ζ (2) = \frac{π^{2}}{6},

where

ζ (s)

is the Riemann zeta function.

{Li}_{2} (2) = \frac{π^{2}}{4} - i π \ln 2.

\begin{matrix} {Li}_{2} (- \frac{\sqrt{5} - 1}{2}) & = - \frac{π^{2}}{15} + \frac{1}{2} {(\ln \frac{\sqrt{5} + 1}{2})}^{2} \\ = - \frac{π^{2}}{15} + \frac{1}{2} {arcsch}^{2} 2. \end{matrix}

\begin{matrix} {Li}_{2} (- \frac{\sqrt{5} + 1}{2}) & = - \frac{π^{2}}{10} - \ln^{2} \frac{\sqrt{5} + 1}{2} \\ = - \frac{π^{2}}{10} - {arcsch}^{2} 2. \end{matrix}

\begin{matrix} {Li}_{2} (\frac{3 - \sqrt{5}}{2}) & = \frac{π^{2}}{15} - \ln^{2} \frac{\sqrt{5} + 1}{2} \\ = \frac{π^{2}}{15} - {arcsch}^{2} 2. \end{matrix}

\begin{matrix} {Li}_{2} (\frac{\sqrt{5} - 1}{2}) & = \frac{π^{2}}{10} - \ln^{2} \frac{\sqrt{5} + 1}{2} \\ = \frac{π^{2}}{10} - {arcsch}^{2} 2. \end{matrix}

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:

Φ (x) = - \int_{0}^{x} \frac{\ln | 1 - u |}{u} d u = {\begin{matrix} {Li}_{2} (x), & x \leq 1; \\ \frac{π^{2}}{3} - \frac{1}{2} (\ln x)^{2} - {Li}_{2} (\frac{1}{x}), & x > 1. \end{matrix}