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%I #15 Feb 25 2023 09:37:12
%S 4,9,0,7,7,2,7,2,8,9,5,8,3,4,5,1,5,9,1,6,2,7,1,7,2,5,3,2,0,3,3,8,2,6,
%T 4,0,3,8,1,9,2,3,3,4,7,7,5,8,5,8,4,6,5,6,2,4,2,6,2,6,0,1,2,0,7,8,1,3,
%U 6,3,4,1,5,5,4,8,7,8,6,9,9,9,7,1,2,5,7,2,1,8,0,3,5,7,8,9,5,5,2,3,3,4,2
%N Decimal expansion of Lamb's integral K_0.
%H D. H. Bailey, J. M. Borwein, and 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</a> (2010) p. 18.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/InverseTangentIntegral.html">Inverse Tangent Integral</a>.
%F K_0 = integral_[0..1] arctanh(1/sqrt(3 + x^2))/(1 + x^2) dx.
%F K_0 = 3/2*Ti_2(3 - 2*sqrt(2)) + Pi/4*log(1 + sqrt(2)) - Catalan/2, where Ti_2 is Lewin's arctan integral, Ti_2(x) = (i/2)*(Li_2(-i*x) - Li_2(i*x)).
%e 0.490772728958345159162717253203382640381923347758584656...
%p evalf(int(arctanh(1/sqrt(3 + x^2))/(1 + x^2), x=0..1), 120); # _Vaclav Kotesovec_, Jan 26 2015
%t Ti2[x_] := (I/2)*(PolyLog[2, -I*x] - PolyLog[2, I*x]); K0 = (3/2)*Ti2[3 - 2 Sqrt[2]] + Pi/4*Log[1 + Sqrt[2]] - Catalan/2 // Re; RealDigits[K0, 10, 103] // First
%Y Cf. A244920, A244921, A244922, A254134, A254135.
%K nonn,cons,easy
%O 0,1
%A _Jean-François Alcover_, Jan 26 2015
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