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Dilogarithm

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Dilogarithm

The dilogarithm Li_2(z) is a special case of the polylogarithm Li_n(z) for n=2. Note that the notation Li_2(x) is unfortunately similar to that for the logarithmic integral Li(x). There are also two different commonly encountered normalizations for the Li_2(z) function, both denoted L(z), 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

Li_2(z)=sum_(k=1)^infty(z^k)/(k^2)
(1)

or the integral

Li_2(z)=int_z^0(ln(1-t)dt)/t.
(2)
DiLogReIm
DiLogContours

Plots of Li_2(z) in the complex plane are illustrated above.

The major functional equations for the dilogarithm are given by

Li_2(x)+Li_2(-x)=1/2Li_2(x^2)
(3)
Li_2(1-x)+Li_2(1-x^(-1))=-1/2(lnx)^2
(4)
Li_2(x)+Li_2(1-x)=1/6pi^2-(lnx)ln(1-x)
(5)
Li_2(-x)-Li_2(1-x)+1/2Li_2(1-x^2)=-1/(12)pi^2-(lnx)ln(x+1).
(6)

A complete list of Li_2(x) with real arguments x that can be evaluated in closed form is given by

Li_2(-1)=-1/(12)pi^2
(7)
Li_2(0)=0
(8)
Li_2(1/2)=1/(12)pi^2-1/2(ln2)^2
(9)
Li_2(1)=1/6pi^2
(10)
Li_2(-phi)=-1/(10)pi^2-(lnphi)^2
(11)
=-1/(10)pi^2-(csch^(-1)2)^2
(12)
Li_2(-phi^(-1))=-1/(15)pi^2+1/2(lnphi)^2
(13)
=-1/(15)pi^2+1/2(csch^(-1)2)^2
(14)
Li_2(phi^(-2))=1/(15)pi^2-(lnphi)^2
(15)
=1/(15)pi^2-(csch^(-1)2)^2
(16)
Li_2(phi^(-1))=1/(10)pi^2-(lnphi)^2
(17)
=1/(10)pi^2-(csch^(-1)2)^2,
(18)

where phi is the golden ratio (Lewin 1981, Bailey et al. 1997; Borwein et al. 2001).

Two-term identities involving irrational numbers include

Li_2(sqrt(2)-1)-Li_2(1-sqrt(2))=(pi^2)/8-1/2ln^2(1+sqrt(2))
(19)

(Lima 2012, Campbell 2021) and

Li_2(phi^(-3))-Li_2(-phi^(-3))=(phi^3(pi^2-18ln^2phi))/(3(phi^6-1))
(20)
Li_2(i(2-sqrt(3)))-Li_2(-i(2-sqrt(3)))=i/6[8K-piln(2+sqrt(3))]
(21)
Li_2(3^(-1/2)i)-Li_2(-3^(-1/2)i)=(i[3psi_1(1/6)+15psi_1(1/3)-6sqrt(3)piln3-16pi^2])/(36sqrt(3)),
(22)

where phi is the golden ratio, K is Catalan's constant, and psi_1(z) is the trigamma function (Campbell 2021).

Another identity for a dilogarithm with complex argument is

Im[Li_2(i(sqrt(2)-1))]=(psi_1(1/8)+psi_1(3/8))/(32sqrt(2))+pi/(16)ln(3-2sqrt(2))-(pi^2)/(8sqrt(2)),
(23)

where Im[z] denotes the imaginary part of z and psi_1(x) is the trigamma function (Campbell 2021; W. Gosper, Sep. 19, 2021).

There are several remarkable identities involving the dilogarithm function. Ramanujan gave the identities

Li_2(1/3)-1/6Li_2(1/9)=1/(18)pi^2-1/6(ln3)^2
(24)
Li_2(-1/2)+1/6Li_2(1/9)=-1/(18)pi^2+ln2ln3-1/2(ln2)^2-1/3(ln3)^2
(25)
Li_2(1/4)+1/3Li_2(1/9)=1/(18)pi^2+2ln2ln3-2(ln2)^2-2/3(ln3)^2
(26)
Li_2(-1/3)-1/3Li_2(1/9)=-1/(18)pi^2+1/6(ln3)^2
(27)
Li_2(-1/8)+Li_2(1/9)=-1/2(ln9/8)^2
(28)

(Berndt 1994, Gordon and McIntosh 1997) in addition to the identity for Li_2(phi^(-1)), and Bailey et al. (1997) showed that

pi^2=36Li_2(1/2)-36Li_2(1/4)-12Li_2(1/8)+6Li_2(1/(64)).
(29)

Lewin (1991) gives 67 dilogarithm identities (known as "ladders"), and Bailey and Broadhurst (1999, 2001) found the amazing additional dilogarithm identity

0=Li_2(alpha_1^(-630))-2Li_2(alpha_1^(-315))-3Li_2(alpha_1^(-210))-10Li_2(alpha_1^(-126))-7Li_2(alpha_1^(-90))+18Li_2(alpha_1^(-35))+84Li_2(alpha_1^(-15))+90Li_2(alpha_1^(-14))-4Li_2(alpha_1^(-9))+339Li_2(alpha_1^(-8))+45Li_2(alpha_1^(-7))+265Li_2(alpha_1^(-6))-273Li_2(alpha_1^(-5))-678Li_2(alpha_1^(-4))-1016Li_2(alpha_1^(-3))-744Li_2(alpha_1^(-2))-804Li_2(alpha_1^(-1))-22050(lnalpha_1)^2+2003zeta(2),
(30)

where alpha_1=(x^(10)+x^9-x^7-x^6-x^5-x^4-x^3+x+1)_2 approx 1.17628 is the largest positive root of the polynomial in Lehmer's Mahler measure problem and zeta(z) is the Riemann zeta function.

See also

Abel's Duplication Formula, Abel's Functional Equation, Clausen Function, Inverse Tangent Integral, L-Algebraic Number, Legendre's Chi-Function, Logarithm, Polylogarithm, Rogers L-Function, Spence's Function, Spence's Integral, Trilogarithm, Watson's Identities

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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. https://arxiv.org/abs/math/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 M(3,t), M(4,5), M(5,6) and M(6,7) Minimal Models and Related Dilogarithm and Rogers-Ramanujan-Type Identities." J. Phys. A: Math. Gen. 32, 8045-8058, 1999.Campbell, J. M. "Some Nontrivial Two-Term Dilogarithm Identities." Irish Math. Soc. Bull., No. 88, 31-37, 2021.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, England: 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.Lima, F. M. S. "New Definite Integrals and a Two-Term Dilogarithm Identity." Indag. Math. 23, 1-9, 2012.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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Dilogarithm

Cite this as:

Weisstein, Eric W. "Dilogarithm." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Dilogarithm.html

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