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

and its reflection. For $|z| < 1$, 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 $z$ has hyperbolic volume
 * $$D(z) = \operatorname{Im} \operatorname{Li}_2(z) + \arg(1-z) \log|z|.$$

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 $$\operatorname{Li}_2(1) = \pi^2/6$$.

Identities

 * $$\operatorname{Li}_2(z)+\operatorname{Li}_2(-z)=\frac{1}{2}\operatorname{Li}_2(z^2).$$
 * $$\operatorname{Li}_2(1-z)+\operatorname{Li}_2\left(1-\frac{1}{z}\right)=-\frac{(\ln z)^2}{2}.$$
 * $$\operatorname{Li}_2(z)+\operatorname{Li}_2(1-z)=\frac{{\pi}^2}{6}-\ln z \cdot\ln(1-z).$$
 * $$\operatorname{Li}_2(-z)-\operatorname{Li}_2(1-z)+\frac{1}{2}\operatorname{Li}_2(1-z^2)=-\frac{{\pi}^2}{12}-\ln z \cdot \ln(z+1).$$
 * $$\operatorname{Li}_2(z) +\operatorname{Li}_2\left(\frac{1}{z}\right) = - \frac{\pi^2}{6} - \frac{(\ln(-z))^2}{2}.$$

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} $$