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Dilogarithm

 
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 Mathematica 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) which 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).

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
(19)
Li_2(-1/2)+1/6Li_2(1/9)=-1/(18)pi^2+ln2ln3-1/2(ln2)^2-1/3(ln3)^2
(20)
Li_2(1/4)+1/3Li_2(1/9)=1/(18)pi^2+2ln2ln3-2(ln2)^2-2/3(ln3)^2
(21)
Li_2(-1/3)-1/3Li_2(1/9)=-1/(18)pi^2+1/6(ln3)^2
(22)
Li_2(-1/8)+Li_2(1/9)=-1/2(ln9/8)^2
(23)

(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)).
(24)

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),
(25)

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.


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