A254135 - OEIS

%I #13 Aug 06 2024 09:28:05

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

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

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

%N Decimal expansion of Lamb's integral K_2.

%H D. H. Bailey, J. M. Borwein, and R. E. Crandall, <a href="https://citeseerx.ist.psu.edu/pdf/7f8e05b56fe9a987cf8f50f5f71414954bd94c1e">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_2 = integral_[0..Pi/4] sqrt(1 + sec(x)^2)*arctan(1/sqrt(1 + sec(x)^2)) dx.

%F K_2 = (1/2)*Ti_2(-2 + sqrt(3)) + (Pi/8)*log(2 + sqrt(3)) + Pi^2/32, where Ti_2 is Lewin's arctan integral, Ti_2(x) = (i/2)*(Li_2(-i*x) - Li_2(i*x)).

%e 0.69266081515264750650943118588427245846713483280766884258...

%p evalf(int(sqrt(1 + sec(x)^2)*arctan(1/sqrt(1 + sec(x)^2)), x=0..Pi/4), 120); # _Vaclav Kotesovec_, Jan 26 2015

%t Ti2[x_] := (I/2)* (PolyLog[2, -I *x] - PolyLog[2, I *x]); K2 = (1/2)*Ti2[-2 + Sqrt[3]] + (Pi/8)*Log[2 + Sqrt[3]] + Pi^2/32 // Re; RealDigits[K2, 10, 103] // First

%Y Cf. A244920, A244921, A244922, A254133, A254134.

%K nonn,cons,easy

%O 0,1

%A _Jean-François Alcover_, Jan 26 2015