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 .
Identities
[edit]- [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
[edit]Special values
[edit]- Its slope = 1.
- 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]Notes
[edit]- ^ Zagier p. 10
- ^ "William Spence - Biography".
- ^ "Biography – GALT, JOHN – Volume VII (1836-1850) – Dictionary of Canadian Biography".
- ^ a b c Zagier
- ^ a b c d e f g Weisstein, Eric W. "Dilogarithm". MathWorld.
- ^ Weisstein, Eric W. "Rogers L-Function". mathworld.wolfram.com. Retrieved 2024-08-01.
- ^ 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
[edit]- Lewin, L. (1958). Dilogarithms and associated functions. Foreword by J. C. P. Miller. London: Macdonald. MR 0105524.
- Morris, Robert (1979). "The dilogarithm function of a real argument". Math. Comp. 33 (146): 778–787. doi:10.1090/S0025-5718-1979-0521291-X. MR 0521291.
- Loxton, J. H. (1984). "Special values of the dilogarithm". Acta Arith. 18 (2): 155–166. doi:10.4064/aa-43-2-155-166. MR 0736728.
- Kirillov, Anatol N. (1995). "Dilogarithm identities". Progress of Theoretical Physics Supplement. 118: 61–142. arXiv:hep-th/9408113. Bibcode:1995PThPS.118...61K. doi:10.1143/PTPS.118.61. S2CID 119177149.
- Osacar, Carlos; Palacian, Jesus; Palacios, Manuel (1995). "Numerical evaluation of the dilogarithm of complex argument". Celest. Mech. Dyn. Astron. 62 (1): 93–98. Bibcode:1995CeMDA..62...93O. doi:10.1007/BF00692071. S2CID 121304484.
- Zagier, Don (2007). "The Dilogarithm Function". In Pierre Cartier; Pierre Moussa; Bernard Julia; Pierre Vanhove (eds.). Frontiers in Number Theory, Physics, and Geometry II (PDF). pp. 3–65. doi:10.1007/978-3-540-30308-4_1. ISBN 978-3-540-30308-4.
Further reading
[edit]- Bloch, Spencer J. (2000). Higher regulators, algebraic K-theory, and zeta functions of elliptic curves. CRM Monograph Series. Vol. 11. Providence, RI: American Mathematical Society. ISBN 0-8218-2114-8. Zbl 0958.19001.