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Source: Dilogarithm

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:

Li

2

⁡
(
z
)
=
−

∫

0

z

ln
⁡
(
1
−
u
)

u

d
u

,

z
∈

C

{\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):

Li

2

⁡
(
z
)
=

∑

k
=
1

∞

z

k

k

2

.

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

Alternatively, the dilogarithm function is sometimes defined as

∫

1

v

ln
⁡
t

1
−
t

d
t
=

Li

2

⁡
(
1
−
v
)
.

{\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

D
(
z
)
=
Im
⁡

Li

2

⁡
(
z
)
+
arg
⁡
(
1
−
z
)
log
⁡

|

z

|

.

{\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. 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.