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:

The dilogarithm along the real axis

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

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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  .

Identities

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 [4]
 [5]
 [4] The reflection formula.
 [5]
 [4]
 .[6][7] Abel's functional equation or five-term relation where   is the Rogers L-function (an analogous relation is satisfied also by the quantum dilogarithm)

Particular value identities

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 [5]
 [5]
 [5]
  [5]
 [5]
 

Special values

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  Its slope = 1.
 
  where   is the Riemann zeta function.
 
 
 
 
 

In particle physics

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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

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Notes

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  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.
  6. ^ Weisstein, Eric W. "Rogers L-Function". mathworld.wolfram.com. Retrieved 2024-08-01.
  7. ^ Rogers, L. J. (1907). "On the Representation of Certain Asymptotic Series as Convergent Continued Fractions". Proceedings of the London Mathematical Society. s2-4 (1): 72–89. doi:10.1112/plms/s2-4.1.72.

References

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Further reading

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