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A261024 Decimal expansion of Cl_2(2*Pi/3), where Cl_2 is the Clausen function of order 2.

%I A261024 #29 Feb 16 2025 08:33:26
%S A261024 6,7,6,6,2,7,7,3,7,6,0,6,4,3,5,7,5,0,0,1,4,1,3,5,0,3,6,1,8,3,0,1,3,5,
%T A261024 2,3,9,6,1,1,2,6,2,0,5,0,2,0,1,9,9,8,6,1,3,4,4,9,9,2,7,3,7,8,5,1,0,6,
%U A261024 4,9,8,4,1,7,2,1,6,2,6,8,1,4,2,4,3,1,3,5,6,9,4,8,5,5,0,4,4,6,3,2,9,7,2,4,1
%N A261024 Decimal expansion of Cl_2(2*Pi/3), where Cl_2 is the Clausen function of order 2.
%H A261024 G. C. Greubel, <a href="/A261024/b261024.txt">Table of n, a(n) for n = 0..10000</a>
%H A261024 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/ClausenFunction.html">Clausen Function</a>.
%H A261024 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/ClausensIntegral.html">Clausen's Integral</a>.
%H A261024 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/BarnesG-Function.html">Barnes G-Function</a>.
%H A261024 Wikipedia, <a href="https://en.wikipedia.org/wiki/Clausen_function">Clausen function</a>.
%H A261024 Wikipedia, <a href="https://en.wikipedia.org/wiki/Barnes_G-function">Barnes G-function</a>.
%F A261024 Equals 2*Pi*log(G(2/3)/G(1/3)) - 2*Pi*LogGamma(1/3) + (2*Pi/3)*log(2*Pi/sqrt(3)), where G is the Barnes G function.
%e A261024 0.676627737606435750014135036183013523961126205020199861344992737851...
%t A261024 Cl2[x_] := (I/2)*(PolyLog[2, Exp[-I*x]] - PolyLog[2, Exp[I*x]]); RealDigits[Cl2[2*Pi/3] // Re, 10, 105] // First
%o A261024 (PARI)
%o A261024 clausen(n, x) = my(z = polylog(n, exp(I*x))); if (n%2, real(z), imag(z));
%o A261024 clausen(2, 2*Pi/3) \\ _Gheorghe Coserea_, Sep 30 2018
%Y A261024 Cf. A006752 (Cl_2(Pi/2) = Catalan's constant), A143298 (Cl_2(Pi/3) = Gieseking's constant), A261025 (Cl_2(Pi/4)), A261026 (Cl_2(3*Pi/4)), A261027 (Cl_2(Pi/6)), A261028 (Cl_2(5*Pi/6)).
%Y A261024 Cf. A252798, A252799, A263416, A263417.
%K A261024 nonn,cons,easy
%O A261024 0,1
%A A261024 _Jean-François Alcover_, Aug 07 2015