Dilogarithm
The dilogarithm 
is a special case of the polylogarithm 
for 
. Note that the
notation 
is unfortunately similar to that
for the logarithmic integral 
. There are
also two different commonly encountered normalizations for the 
function,
both denoted 
, and one of which is known as the
Rogers L-function.
The dilogarithm is implemented in the Wolfram Language as PolyLog[2, z].
The dilogarithm can be defined by the sum
 |
(1)
|
or the integral
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(2)
|
Plots of 
in the complex
plane are illustrated above.
The major functional equations for the dilogarithm are given by
 |
(3)
|
 |
(4)
|
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(5)
|
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(6)
|
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(7)
|
A complete list of 
which can be evaluated in closed
form is given by
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(8)
|
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(9)
|
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(10)
|
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(11)
|
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(12)
|
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(13)
|
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(14)
|
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(15)
|
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(16)
|
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(17)
|
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(18)
|
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(19)
|
where 
is the golden
ratio (Lewin 1981, Bailey et al. 1997; Borwein et al. 2001).
There are several remarkable identities involving the dilogarithm function. Ramanujan gave the identities
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(20)
|
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(21)
|
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(22)
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(23)
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(24)
|
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(25)
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(26)
|
(Berndt 1994, Gordon and McIntosh 1997) in addition to the identity for 
,
and Bailey et al. (1997) showed that
 |
(27)
|
Lewin (1991) gives 67 dilogarithm identities (known as "ladders"), and Bailey and Broadhurst (1999, 2001) found the amazing additional dilogarithm identity
 |
(28)
|
where 
is the largest positive root of the polynomial in Lehmer's
Mahler measure problem and 
is the Riemann zeta function.
Abramowitz, M. and Stegun, I. A. (Eds.). "Dilogarithm." §27.7 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 1004-1005, 1972.
Andrews, G. E.; Askey, R.; and Roy, R. Special Functions. Cambridge, England: Cambridge University Press, 1999.
Bailey, D. H.; Borwein, P. B.; and Plouffe, S. "On the Rapid Computation of Various Polylogarithmic Constants." Math. Comput. 66, 903-913, 1997.
Bailey, D. H. and Broadhurst, D. J. "A Seventeenth-Order Polylogarithm Ladder." 20 Jun 1999. http://arxiv.org/abs/math.CA/9906134.
Bailey, D. H. and Broadhurst, D. J. "Parallel Integer Relation Detection: Techniques and Applications." Math. Comput. 70, 1719-1736, 2001.
Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 323-326, 1994.
Borwein, J. M.; Bradley, D. M.; Broadhurst, D. J.; and Lisonek, P. "Special Values of Multidimensional Polylogarithms." Trans. Amer. Math. Soc. 353, 907-941, 2001.
Bytsko, A. G. "Fermionic Representations for Characters of 
, 
, 
and 
Minimal Models and Related Dilogarithm and
Rogers-Ramanujan-Type Identities." J. Phys. A: Math. Gen. 32,
8045-8058, 1999.
Erdélyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. "Euler's Dilogarithm." §1.11.1 in Higher Transcendental Functions, Vol. 1. New York: Krieger, pp. 31-32, 1981.
Gordon, B. and McIntosh, R. J. "Algebraic Dilogarithm Identities." Ramanujan J. 1, 431-448, 1997.
Kirillov, A. N. "Dilogarithm Identities." Progr. Theor. Phys. Suppl. 118, 61-142, 1995.
Lewin, L. Dilogarithms and Associated Functions. London: Macdonald, 1958.
Lewin, L. Polylogarithms and Associated Functions. New York: North-Holland, 1981.
Lewin, L. "The Dilogarithm in Algebraic Fields." J. Austral. Math. Soc. Ser. A 33, 302-330, 1982.
Lewin, L. (Ed.). Structural Properties of Polylogarithms. Providence, RI: Amer. Math. Soc., 1991.
Nielsen, N. "Der Eulersche Dilogarithmus und seine Verallgemeinerungen." Nova Acta Leopoldina, Abh. der Kaiserlich Leopoldinisch-Carolinischen Deutschen Akad. der Naturforsch. 90, 121-212, 1909.
Watson, G. N. "A Note on Spence's Logarithmic Transcendent." Quart. J. Math. Oxford Ser. 8, 39-42, 1937.
Weisstein, Eric W. "Dilogarithm." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Dilogarithm.html
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