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:

and its reflection. For |z| ≤ 1, 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 z has hyperbolic volume

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.

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 ${\displaystyle \operatorname {Li} _{2}(1)=\pi ^{2}/6}$.

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: