In mathematics, Spence's function, or dilogarithm, 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 an infinite series also applies (the integral definition constitutes its analytical extension to the complex plane):Alternatively, the dilogarithm function is sometimes defined asIn hyperbolic geometry the dilogarithm occurs as the hyperbolic volume of an ideal simplex whose ideal vertices have cross ratio . 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.