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Trilogarithm

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Trilogarithm

The trilogarithm Li_3(z), sometimes also denoted L_3, is special case of the polylogarithm Li_n(z) for n=3. Note that the notation Li_3(x) for the trilogarithm is unfortunately similar to that for the logarithmic integral Li(x).

The trilogarithm is implemented in the Wolfram Language as PolyLog[3, z].

TriLogReIm
TriLogContours

Plots of Li_3(z) in the complex plane are illustrated above.

Functional equations for the trilogarithm include

Li_3(z)+Li_3(-z)=1/4Li_3(z^2) Li_3(-z)-Li_3(-z^(-1))=-1/6(lnz)^3-1/6pi^2lnz Li_3(z)+Li_3(1-z)+Li_3(1-z^(-1)) =zeta(3)+1/6(lnz)^3+1/6pi^2lnz-1/2(lnz)^2ln(1-z).
(1)

Analytic values for Li_3(x) include

Li_3(-1)=-3/4zeta(3) Li_3(0)=0 Li_3(1/2)=1/(24)[-2pi^2ln2+4(ln2)^3+21zeta(3)] Li_3(1)=zeta(3) Li_3(phi^(-2))=4/5zeta(3)+2/3(lnphi)^3-2/(15)pi^2lnphi
(2)

where zeta(3) is Apéry's constant and phi is the golden ratio.

Bailey et al. showed that

(35)/2zeta(3)-pi^2ln2 =36Li_3(1/2)-18Li_3(1/4)-4Li_3(1/8)+Li_3(1/(64)) 2(ln2)^3-7zeta(3) =-24Li_3(1/2)+18Li_3(1/4)+4Li_3(1/8)-Li_3(1/(64)) 10(ln2)^3-2pi^2ln2 =-48Li_3(1/2)+54Li_3(1/4)+12Li_3(1/8)-3Li_3(1/(64)).
(3)

See also

Dilogarithm, Polylogarithm

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References

Bailey, D. H.; Borwein, P. B.; and Plouffe, S. "On the Rapid Computation of Various Polylogarithmic Constants." Math. Comput. 66, 903-913, 1997.Lewin, L. Polylogarithms and Associated Functions. New York: North-Holland, pp. 154-156, 1981.

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Trilogarithm

Cite this as:

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

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