Dilogarithm
المؤلف:
Abramowitz, M. and Stegun, I. A.
المصدر:
"Dilogarithm." §27.7 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover
الجزء والصفحة:
...
9-8-2019
3126
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
 |
(2)
|
Plots of 
in the complex plane are illustrated above.
The major functional equations for the dilogarithm are given by
A complete list of 
which can be evaluated in closed form is given by
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
 |
(20)
|
 |
(21)
|
 |
(22)
|
 |
(23)
|
 |
(24)
|
 |
(25)
|
 |
(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.
REFERENCES:
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.
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