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%I #8 Jan 26 2015 08:10:46
%S 1,6,6,1,9,0,7,8,7,4,7,3,8,1,2,3,3,7,7,4,0,6,5,8,1,6,8,6,1,6,3,0,5,9,
%T 4,9,7,3,4,8,8,6,8,6,7,3,2,5,1,2,5,8,9,1,8,3,4,1,5,0,8,1,9,4,3,4,2,3,
%U 5,4,9,3,1,0,9,3,0,4,5,2,0,6,6,9,3,8,4,8,3,8,0,5,6,8,7,2,3,4,5,1,0,3,8
%N Decimal expansion of Lamb's integral K_1.
%H D. H. Bailey, J. M. Borwein, R. E. Crandall, <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.154.1794">Advances in the theory of box integrals (2010)</a> p. 18.
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/ClausensIntegral.html">Clausen's Integral</a>
%F K_1 = integral_[3..4] arcsec(x)/sqrt(x^2 - 4*x + 3) dx.
%F K_1 = Cl_2(th) - Cl_2(th + Pi/3) - Cl_2(th - Pi/2) + Cl_2(th - Pi/6) - Cl_2(3*th + Pi/3) + Cl_2(3*th + 2*(Pi/3)) - Cl_2(3*th - 5*(Pi/6)) + Cl_2(3*th + 5*(Pi/6)) + (6*th - 5*(Pi/2))*log(2 - sqrt(3)), where Cl_2 is the Clausen function and th = (arctan((16 - 3*sqrt(15))/11) + Pi)/3.
%e 1.6619078747381233774065816861630594973488686732512589...
%p evalf(int(arcsec(x)/sqrt(x^2 - 4*x + 3), x=3..4), 120); # _Vaclav Kotesovec_, Jan 26 2015
%t Cl2[x_] := (I/2)*(PolyLog[2, Exp[-I*x]] - PolyLog[2, Exp[I*x]]); th = (ArcTan[(16 - 3*Sqrt[15])/11] + Pi)/3; K1 = Cl2[th] - Cl2[th + Pi/3] - Cl2[th - Pi/2] + Cl2[th - Pi/6] - Cl2[3*th + Pi/3] + Cl2[3*th + 2*(Pi/3)] - Cl2[3*th - 5*(Pi/6)] + Cl2[3*th + 5*(Pi/6)] + (6*th - 5*(Pi/2))*Log[2 - Sqrt[3]] // Re; RealDigits[K1, 10, 103] // First
%Y Cf. A244920, A244921, A244922, A254133, A254135.
%K nonn,cons,easy
%O 1,2
%A _Jean-François Alcover_, Jan 26 2015
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