login
Decimal expansion of integral_{0..1} Li_3(x) dx, where Li_3 is the trilogarithm function.
4

%I #7 Nov 03 2014 09:26:43

%S 5,5,7,1,2,2,8,3,6,3,1,1,3,6,7,8,4,8,9,2,7,3,2,2,9,9,4,8,6,5,4,2,4,8,

%T 0,1,5,4,6,0,3,6,3,9,1,1,3,3,7,0,0,4,4,4,0,5,6,7,1,3,3,2,5,9,7,1,8,3,

%U 0,7,3,5,3,8,3,1,1,2,2,1,6,3,5,2,8,2,6,9,7,2,9,8,9,5,7,6,5,5,2,8,4,2

%N Decimal expansion of integral_{0..1} Li_3(x) dx, where Li_3 is the trilogarithm function.

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/Trilogarithm.html">Trilogarithm</a>

%F int_{0..1} Li_3(x) dx = 1 - zeta(2) + zeta(3) = 1 - Pi^2/6 + zeta(3).

%F Compare with the same integral of the dilogarithm:

%F int_{0..1} Li_2(x) dx = zeta(2) - 1 = Pi^2/6 - 1 = 0.644934...

%e 0.5571228363113678489273229948654248015460363911337...

%t RealDigits[1 - Zeta[2] + Zeta[3], 10, 102] // First

%Y Cf. A002117, A013661.

%K nonn,cons,easy

%O 0,1

%A _Jean-François Alcover_, Nov 03 2014