Dilogarithm

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 ${\displaystyle |z|<1}$ 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 ${\displaystyle \operatorname {Li} _{2}(z)}$ occurs as the hyperbolic volume of an ideal simplex whose ideal vertices have cross ratio ${\displaystyle z}$. Lobachevsky's function and Clausen's function are closely related functions.

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