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.