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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 as
In 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.
Particular value identities
- 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 521291.
- Loxton, J. H. (1984). "Special values of the dilogarithm". Acta Arith. 18 (2): 155–166. MR 0736728.
- Kirillov, Anatol N. (1994). "Dilogarithm identities". arXiv:hep-th/9408113.
- Osacar, Carlos; Palacian, Jesus; Palacios, Manuel (1995). "Numerical evaluation of the dilogarithm of complex argument". Celestial Mech. Dynam. Astron. 62 (1): 93–98. doi:10.1007/BF00692071.
- Zagier, Don (2007). "The Dilogarithm Function". Front. Number Theory, Physics, Geom. II: 3–65. doi:10.1007/978-3-540-30308-4_1.
- Bloch, Spencer J. (2000). Higher regulators, algebraic K-theory, and zeta functions of elliptic curves. CRM Monograph Series 11. Providence, RI: American Mathematical Society. ISBN 0-8218-2114-8. Zbl 0958.19001.