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The trilogarithm 
,
sometimes also denoted 
,
is special case of the polylogarithm 
for 
. Note that the notation 
for the trilogarithm is unfortunately similar to that
for the logarithmic integral 
.
The trilogarithm is implemented in the Wolfram Language as PolyLog[3, z].
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Plots of 
in the complex plane are illustrated above.
Functional equations for the trilogarithm include
 |
(1)
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Analytic values for 
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](/images/equations/Trilogarithm/NumberedEquation2.svg) |
(2)
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where 
is Apéry's constant and 
is the golden ratio.
Bailey et al. showed that
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(3)
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