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.
Using the former definition above, the dilogarithm function is analytic everywhere on the complex plane except at , where it has a logarithmic branch point. The standard choice of branch cut is along the positive real axis . However, the function is continuous at the branch point and takes on the value .
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: