Wikipedia / Dilogarithm
Wikipedia / Dilogarithm
In mathematics, the dilogarithm (or Spence's function), denoted as , is a particular case of the polylogarithm. Two related special functions are referred to as Spence's function, the dilogarithm itself:
and its reflection. For , an infinite series also applies (the integral definition constitutes its analytical extension to the complex plane):
Alternatively, the dilogarithm function is sometimes defined as
In hyperbolic geometry the dilogarithm can be used to compute the volume of an ideal simplex. Specifically, a simplex whose vertices have cross ratio has hyperbolic volume
The function 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
.
Identities
The reflection formula.
. is the Rogers L-function (an analogous relation is satisfied also by the quantum dilogarithm)
Particular value identities
Special values
Its slope = 1.
is the Riemann zeta function.
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: