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The dilogarithm along the real axis

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.[1] 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.[2] He was at school with John Galt,[3] who later wrote a biographical essay on Spence.

Analytic structure[edit]

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 .



Particular value identities[edit]


Special values[edit]

where is the Riemann zeta function.

In particle physics[edit]

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:

See also[edit]


  1. ^ Zagier p. 10
  2. ^ "William Spence - Biography".
  3. ^ "Biography – GALT, JOHN – Volume VII (1836-1850) – Dictionary of Canadian Biography".
  4. ^ a b c Zagier
  5. ^ a b c d e f g Weisstein, Eric W. "Dilogarithm". MathWorld.


Further reading[edit]

External links[edit]