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].
Plots of in the complex plane are illustrated above.
Functional equations for the trilogarithm include
(1)
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Analytic values for include
(2)
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where is Apéry's constant and is the golden ratio.
Bailey et al. showed that
(3)
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