Trilogarithm
المؤلف:
Bailey, D. H.; Borwein, P. B.; and Plouffe, S
المصدر:
"On the Rapid Computation of Various Polylogarithmic Constants." Math. Comput. 66
الجزء والصفحة:
...
13-8-2019
1841
Trilogarithm
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)
|
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](http://mathworld.wolfram.com/images/equations/Trilogarithm/NumberedEquation2.gif) |
(2)
|
where 
is Apéry's constant and 
is the golden ratio.
Bailey et al. showed that
 |
(3)
|
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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