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:

${\displaystyle \operatorname {Li} _{2}(z)=-\int _{0}^{z}{\ln(1-u) \over u}\,du{\text{, ))z\in \mathbb {C} }$

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

${\displaystyle \operatorname {Li} _{2}(z)=\sum _{k=1}^{\infty }{z^{k} \over k^{2)).}$

Alternatively, the dilogarithm function is sometimes defined as

${\displaystyle \int _{1}^{v}{\frac {\ln t}{1-t))dt=\operatorname {Li} _{2}(1-v).}$

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

${\displaystyle D(z)=\operatorname {Im} \operatorname {Li} _{2}(z)+\arg(1-z)\log |z|.}$

The function D(z) is sometimes called the Bloch-Wigner function.^[1] 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.^[2] He was at school with John Galt,^[3] 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 ${\displaystyle z=1}$, where it has a logarithmic branch point. The standard choice of branch cut is along the positive real axis ${\displaystyle (1,\infty )}$. However, the function is continuous at the branch point and takes on the value ${\displaystyle \operatorname {Li} _{2}(1)=\pi ^{2}/6}$.

Identities

${\displaystyle \operatorname {Li} _{2}(z)+\operatorname {Li} _{2}(-z)={\frac {1}{2))\operatorname {Li} _{2}(z^{2}).}$^[4]

${\displaystyle \operatorname {Li} _{2}(1-z)+\operatorname {Li} _{2}\left(1-{\frac {1}{z))\right)=-{\frac {(\ln z)^{2)){2)).}$^[5]

${\displaystyle \operatorname {Li} _{2}(z)+\operatorname {Li} _{2}(1-z)={\frac ((\pi }^{2)){6))-\ln z\cdot \ln(1-z).}$^[4]

${\displaystyle \operatorname {Li} _{2}(-z)-\operatorname {Li} _{2}(1-z)+{\frac {1}{2))\operatorname {Li} _{2}(1-z^{2})=-{\frac ((\pi }^{2)){12))-\ln z\cdot \ln(z+1).}$^[5]

${\displaystyle \operatorname {Li} _{2}(z)+\operatorname {Li} _{2}\left({\frac {1}{z))\right)=-{\frac {\pi ^{2)){6))-{\frac {(\ln(-z))^{2)){2)).}$^[4]

Particular value identities

${\displaystyle \operatorname {Li} _{2}\left({\frac {1}{3))\right)-{\frac {1}{6))\operatorname {Li} _{2}\left({\frac {1}{9))\right)={\frac ((\pi }^{2)){18))-{\frac {(\ln 3)^{2)){6)).}$^[5]

${\displaystyle \operatorname {Li} _{2}\left(-{\frac {1}{3))\right)-{\frac {1}{3))\operatorname {Li} _{2}\left({\frac {1}{9))\right)=-{\frac ((\pi }^{2)){18))+{\frac {(\ln 3)^{2)){6)).}$^[5]

${\displaystyle \operatorname {Li} _{2}\left(-{\frac {1}{2))\right)+{\frac {1}{6))\operatorname {Li} _{2}\left({\frac {1}{9))\right)=-{\frac ((\pi }^{2)){18))+\ln 2\cdot \ln 3-{\frac {(\ln 2)^{2)){2))-{\frac {(\ln 3)^{2)){3)).}$^[5]

${\displaystyle \operatorname {Li} _{2}\left({\frac {1}{4))\right)+{\frac {1}{3))\operatorname {Li} _{2}\left({\frac {1}{9))\right)={\frac ((\pi }^{2)){18))+2\ln 2\cdot \ln 3-2(\ln 2)^{2}-{\frac {2}{3))(\ln 3)^{2}.}$ ^[5]

${\displaystyle \operatorname {Li} _{2}\left(-{\frac {1}{8))\right)+\operatorname {Li} _{2}\left({\frac {1}{9))\right)=-{\frac {1}{2))\left(\ln {\frac {9}{8))\right)^{2}.}$^[5]

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

Special values

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

${\displaystyle \operatorname {Li} _{2}(0)=0.}$

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

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

${\displaystyle \operatorname {Li} _{2}(2)={\frac ((\pi }^{2)){4))-i\pi \ln 2.}$

${\displaystyle {\begin{aligned}\operatorname {Li} _{2}\left(-{\frac ((\sqrt {5))-1}{2))\right)&=-{\frac ((\pi }^{2)){15))+{\frac {1}{2))\left(\ln {\frac ((\sqrt {5))+1}{2))\right)^{2}\\&=-{\frac ((\pi }^{2)){15))+{\frac {1}{2))\operatorname {arcsch} ^{2}2.\end{aligned))}$

${\displaystyle {\begin{aligned}\operatorname {Li} _{2}\left(-{\frac ((\sqrt {5))+1}{2))\right)&=-{\frac ((\pi }^{2)){10))-\ln ^{2}{\frac ((\sqrt {5))+1}{2))\\&=-{\frac ((\pi }^{2)){10))-\operatorname {arcsch} ^{2}2.\end{aligned))}$

${\displaystyle {\begin{aligned}\operatorname {Li} _{2}\left({\frac {3-{\sqrt {5))}{2))\right)&={\frac ((\pi }^{2)){15))-\ln ^{2}{\frac ((\sqrt {5))+1}{2))\\&={\frac ((\pi }^{2)){15))-\operatorname {arcsch} ^{2}2.\end{aligned))}$

${\displaystyle {\begin{aligned}\operatorname {Li} _{2}\left({\frac ((\sqrt {5))-1}{2))\right)&={\frac ((\pi }^{2)){10))-\ln ^{2}{\frac ((\sqrt {5))+1}{2))\\&={\frac ((\pi }^{2)){10))-\operatorname {arcsch} ^{2}2.\end{aligned))}$

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:

${\displaystyle \operatorname {\Phi } (x)=-\int _{0}^{x}{\frac {\ln |1-u|}{u))\,du={\begin{cases}\operatorname {Li} _{2}(x),&x\leq 1;\\{\frac {\pi ^{2)){3))-{\frac {1}{2))(\ln x)^{2}-\operatorname {Li} _{2}({\frac {1}{x))),&x>1.\end{cases))}$